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Resonant triad interactions in symmetric systems
Physica D 54 (1992) 267-310 North-Holland
Resonant triad interactions in symmetric systems John G u c k e n h e i m e r and Alex Mahalov Cornell University, Ithaca, NZ, USA
Received 4 February 1991 Revised manuscript received 9 August 1991 Accepted 27 August 1991 Communicated by A.C. Newell
We analyse resonant triad interactions in symmetricsystems. The case of coupling at quadratic order in amplitudes and the case of coupling at cubic order are considered. We study modulated travellingwaves and heteroclinic cyclesin amplitude equations and we discuss their stability types. We describe possible dynamical regimes and spatial patterns arising from triad interactions of long waves in rapidly rotating Hagen-Poiseuille flow.
1. Introduction Nonlinear wave interactions are an important phenomenon in fluid dynamics [7] as well as in plasma physics [21] and nonlinear optics [4]. We describe the phenomenon in the context of fluids. The Navier-Stokes equations governing fluid motion are nonlinear. If one expands a fluid velocity in a Fourier series, then products of individual modes appear in the equations and these serve to excite other modes and to couple sets of modes. The resulting interactions can be ordered by powers of amplitude. If the initial field has small amplitude that is scaled by a small parameter e, then the solution can be expanded as a formal power series in e. Each interaction occurs with an amplitude proportional to e ~. Here k is called the order of the interaction. The strongest interactions are resonant triads. These are triples of modes whose spatial wave vectors sum to zero and they have order k = 2. For this reason, one suspects that strong nonlinear effects will be observed in systems which are close to marginal stability for a set of modes that form a resonant triad. Fluid examples where this occurs can be found in ref. [7]. ~ The dynamics associated with resonant interactions can be complex and chaotic [23]. There have been several studies of these nonlinear systems with primary emphasis upon the study of conservative systems that include only the effects of linear excitation and quadratic coupling. The Navier-Stokes equations are not conservative, and one of our observations is that third order coupling terms are an essential part of the qualitative features of resonant triad interactions in dissipative systems. The third order expansions of the amplitude equations for resonant triad interactions in shear flows were derived from the Navier-Stokes equations by Usher and Craik [22]. Our approach to understanding the dynamics of resonant triad interactions relies heavily upon considerations of symmetry. In our analysis, we neglect most coupling between the phases of the interacting modes and examine only the effects of the amplitudes of the modes. More precisely, we assume that there is a symmetry on the phase spaces of the systems we study that serves to restrict the phase dependence of the system to a single linear combination of the phases. We also assume that there 0167-2789/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
268
J. Guckenheimer, A. Mahaloc / Resonant triad interactions
is an additional reflection symmetry. In applications, these symmetries can be approximate, resulting from coordinate changes that reduce the original system to a "normal form", or they can be a consequence of spatial symmetries in the underlying system of PDE's. The consequence of symmetry upon the observed dynamical behavior of "generic" systems have been intensively investigated during the past several years [2, 3]. Two effects of symmetry are particularly dramatic: (1) symmetry can force the existence of invariant subspaces in the phase space of a system, (2) within the context of symmetric systems, new features of a flow can become structurally stable. Both of these aspects play an important role in the systems studied here. In particular, the subspaces corresponding to an individual nonzero mode are invariant. Since the quadratic interaction terms of the system vanish on these subspaces, the cubic terms of the resonant triad equations are needed to achieve structurally stable behavior within the invariant subspaces of the pure modes. The amplitude equations for resonant triad interactions studied in this paper are dZl
dt =YlZ2Z3WzI( [d'l÷elllZll2WelElzEl2÷el3]z3l
2)
'
dz2
dt = "Y2ZlZ~ + z2( Iz2 + e21[zl [2 + e221z21 z ÷ e23[z3[Z),
dz 3
dt = Y3z1z~ ÷ z3(]'£3 ÷ e31[Zl[2 ÷ e32[z212 + e33[z3[2) '
(1.1)
w h e r e / z j and ekj are real parameters and yj = _+1. These equations are equivariant with respect to the symmetry operations ( ZI, Z2, Z3) ~ ( Z~, Z~, Z~ ), ( Z 1 , 22 , 2 3 ) --'~ ( e x p ( i x D
z~,exp(ix2) z2,exp(ix3) z3),
provided X~ =X2 +X3.
Symmetries of this type are obtained from partial differential equations with spatial translation and reflection invariance. We note that there is a single relation between phases. It implies that the normal form is invariant under symmetry group of a two-dimensional torus T 2. We note that in the special case when two identical waves participate in the formation of a resonant triad, the amplitude equations have an invariant subspace corresponding to 2 : 1 resonance with 0(2) symmetry studied by Armbruster et al. [2], and Proctor and Jones [20]. This paper contains a partial analysis of the qualitative dynamics of the resonant triad equations. We place particular emphasis upon the important case in which there is a further symmetry that interchanges two of the modes in the resonant triad. For example, this occurs when a plane travelling wave interacts with two oblique travelling waves that are reflections of each other with respect to a plane containing the streamwise direction of the initial wave. For this restricted situation, there is an invariant subspace on which the equations of motion are identical to those studied by Armbruster et al. [2]. Much of our analysis for the general resonant triads uses the methods developed there. However, there is a much wider range of possibilities for the dynamical states of resonant triads than is found in the 0(2) symmetric systems studied in ref. [2]. In particular, multiple modulated travelling wave solutions are possible in some parameter ranges. Numerical evidence strongly suggests the presence of chaotic attractors in other parameter ranges. The scaling that is appropriate for the quadratic and cubic terms of the resonant triad equations depends upon the physical problem being studied. Normally, scaling of the variables at small amplitude
J. Guckenheimer, A. Mahalov / Resonant triad interactions
269
makes the cubic coefficients small relative to the quadratic coefficients. However~ there are problems, such as channel flow between fixed plates, in which the quadratic interaction terms of the equations vanish. If one changes the boundary conditions for channel flow to allow sliding of the boundaries relative to each other in the flow direction, then the quadratic interaction terms become nonzero. Thus, varying the relative velocity of the boundaries and the applied pressure gradient driving the channel flow produces a two parameter family of flows in which there will be regions where the quadratic interaction coefficients are small relative to the cubic coefficients. Thus both the limits of cubic coefficients vanishing and quadratic coefficients vanishing are relevant for fluid flows. The case of vanishing quadratic coupling coefficients reduces the triad equations to the normal form for a triple Hopf bifurcation with nonresonant frequencies [17]. We do not discuss in this paper the case of resonant triad interactions that allow spatial modulation of wave amplitudes. An inverse scattering transform for the conservative three-wave equations in time and one spatial dimension has been developed by Zakharov and Manakov [25, 26] and Kaup [12]. We refer to Kaup et al. [13] for a review. Three wave interactions with spatial modulation in dissipative systems are governed by coupled Ginzburg-Landau equations. Third order terms in these equations are essential to capture the qualitative features of the dynamics of resonant interactions. This paper is organized as follows. In section 2 we set up the resonant triad equations and discuss some of their basic properties. In section 3 we study modulated travelling waves. We use the method of averaging to find periodic orbits that are preserved after linear and cubic terms are included in the equation. Section 4 gives a full discussion of the dynamics on the real subspace and subspaces obtained from this subspace by applying elements of the symmetry group. We consider both the weak and the strong resonance case building on the earlier work of Guckenheimer and Holmes, [10], Melbourne [17] and Melnikov [18]. In sections 5 and 6 we are concerned with the existence of heteroclinic nets for the full system of equations and discuss their stability. In section 7 we analyze resonant patterns that occur in fast rotating Hagen-Poiseuille flow. We use the results obtained in the previous sections to describe different kinds of time-dependent behavior that can occur.
2. Preliminary remarks The normal form truncated at third order for an 0(2) equivalent vector field involving three modes forming a resonant triad can be written as
dz 1 dt
= Sl°Z2z3
-b
Zl(ld, 1 -b $111z112'+ Sl2lZ2l 2 + Sl3lZ312),
dz2 dt = s20z1z ~ d- z2(l& 2 q-- s21 [z 112 d- $221z212 d- s231z312), dz3 dt = S30ZlZ ~ + Z3(/~3 + S31 ]Z1[2 + S32[Z2[2 + S33[Z3[2).
(2.1) ,
Here ski are real coefficients and /z 1, ]/,2 and ~l,3 are unfolding parameters. Assuming that sk0 ~ 0 (k = 1, 2,3), we can rescale to obtain (1.1).
J. Guckenheimer, A. Mahalov / Resonant triad interactions
270
Letting z~. = r~ exp(i0y), the system (1.1) may be rewritten as d/1
dt = y~r2r3 cos 4' + r,(l~, + e~lr2~ + e,2r ~ + e,3r~),
d/2
dt = Y2rlr3c°s 4, + r2(/z2 +e2~r? + e22r~ + e23r32) '
dr3
2
2
dt -- 3,3rlr2 cos 4' + r3(/~ 3 + e3~r21+ e32r 2 + e33r 3 ),
d4'
dt = -
(
r2r 3
r~r 3
YIT+Y2-~-2
r~r 2 ~
+Y3T]
(2.2)
sin4,'
with 4' = 0~ - 02 - 03. Thus, in polar coordinates the system is reduced to four real dimensions. In our analysis we also use the real Cartesian form d/1
dt =3,~(x2x3-Y2Y3) + x~(Ix~ + e,r2~ + e~2r ~ +e~3r3~),
dYl dt =3q(x2Y3 +Y2x3) +Yl(/~ +e~lr21 +e~2r~ +e~3r32)' dx 2
dt =3,2(x~x3 +Y~Y3) +x2(/~2 +e2~r? + e22r22 +e23r~),
dY2 dt
= 3,2(-xIY3 +fix3)
dx 3
dt = 3,3(x~x2 + r
+Y2(/'/'2
+e2~r? + e22r~ + e23r32)'
y2) + x3( 3 + e3~r? + e32r~ + e33r32),
dy3 dt = 3'3(-x~y2 + y~x2) + Y3(/z3 + e31r2~+ e32r~ + e33r3~),
(2.3)
where r~2 = x~ + y~2. Reflectional symmetry implies that the purely real system (Yl = 0, Y2 = 0, Y3 = 0) is invariant. Furthermore, any 4,-invariant rotation of it is invariant. For example, the rotation Z l - ' exP(½iar) Zl, z 2 -~ ~ , z 3 --->exP(½i~r) z 3 transforms the real subspace to the Yl, x2, Y3 subspace (x~ = 0, Y2 = 0, x 3 -- 0). The scaling rj = esi, gj = e2vy with appropriate scaling of time transforms the polar system to ds 1
dr = ?~s2s~ cos ~ + es~( Ix~ + e.s2~ + e12s~ + e~s~ ),
ds 2 :
cos
+
+
+
ds~
+
d~" = 3,3s1s2 cos ~ + E83(]3,3 + e315 ~ + e32s ~ + e338 ~ ),
d4,
[
s2s3 + 3,~23 +
d~" = - [3,~-~-~
sts2 ~
3,3-~-3 )sin4,"
J. Guckenheimer, A. Mahalov / Resonant triad interactions
271
The limit e = 0 has three integrals known as Manley-Rowe relations (see, for instance, ref. [24]). These are E l ---- ~ l S l2 -- ~2S2, 2
E 2 ~ Y2s~ - yaS~,
L = SLS2S3 sin
~b.
Steady solutions or fixed points of (2.3) correspond to fixed points of (2.2) with ~b = 0 or xr. In addition to the trivial solution r~ -- r~ = r 3 0, we have pure modes (solutions for which two amplitudes are zero and one amplitude is nonzero) and mixed modes (solutions for which at least two amplitudes are nonzero). Fixed points of (2.2) with ~b different from zero or ~r correspond to travelling waves, in which the phase difference remains constant. Periodic orbits of (2.2) on the subspace ~b = 0, rr correspond to singly periodic standing waves while periodic orbits with ~b different from 0 and ~r correspond to doubly periodic, modulated travelling waves. A quasiperiodic solution of (2.2) corresponds to a triply periodic modulated travelling wave of (2.3). The origin is always a steady solution with eigenvalues/~1, /z~ and /~3, each of multiplicity two. We have invariant subspaces {z~--0, z 3 --0}, {z~ = 0, z3 = 0} and {z~--0, zz--0}. On these subspaces the system reduces to the equation (j -- 1, 2, 3) =
dzj
dt
--
zy(~y + e~ylz~.l~).
When/zy/e~. < 0, there are equilibria (pure modes) along the circles Iz~l = X/-/~-/e~y. The eigenvalues of these equilibria are 0 and -2/~. within the corresponding invariant subspace. The following proposition is proved as in ref. [2].
Proposition 2.1. If yt, Y2 and 'Y3 are positive (negative) then all solutions of (2.2), (2.3) are asymptotic to the invariant subspace {4, = O} ({~b = ~-}) or start on the invariant subspace {~b = "rr} ({4' = 0}). Proof Let us assume without loss of generality that y~ = Y2 ----')/3 ) 0. We consider the phase equation d~b/dt = - ( 3 q r 2 r 3 / r ~ + y2rtr3/r2 + y3rlr2/r3)sin ~b. Since the quantity in parenthesis is positive off the coordinate planes, we have that dd~/dt < 0 for ~b in the interval (0,at) and ddp/dt > 0 for ~b in the interval (rr, 2~r). Thus, d,(t) ~ 0 = 2~ unless solutions start with ~b(0) = "rr. The proposition is proved.
3. Modulated travelling waves
Let us consider the system ds 1
d~" = -s2s~ cos dp + es~(P~l + e~s~ + e~2s~ + e13532),
ds 2
= sis3 c o s ,
+
s2( 2 +
+
+
ds 3 dq" -~ - s 1 s 2 c o s ~ -+- Es3(/.g3 "+ e31s~ -j- e32522 "~ e33s32),
dck
( s2s3 s s3 slS2)sinck. =
-
-
+
s2
s3
(3.1)
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272
The system (3.1) appears in the rotating pipe flow problem. Integrals of this system in the case e = 0 are E~ = s~ + s~,
E~ = s~ + s~,
E3 = s2~ - s~ = E 1 - E2,
L = s~s2s3 sin 4~.
The phase space is the product of a circle S l (varying ~b) with the positive quadrant of the (s 1, s2, s 3) space. We note that the boundary of the phase space is not invariant under the flow and that the points (0, s~, s3, ~b) and (0, s2, s3, ~b + ~r), (Sl, 0, s 3, ~b) and (sl, 0, s 3, ~b + ~), (sl, sz, 0, ~b) and (Sl, s~, 0, ~ + ~) should be identified so that a trajecto~ exiting the b o u n d a ~ at (0, s2,s3, ~) reenters at the point (0, s~, s3, ~ + ~). The vector field is singular at these boundaries. The intersections of level surfaces of E~, E~ and L are transverse except at the b o u n d a ~ of the phase space given by s~s~s 3 = 0 and the surface {~ = 7~,1 s~s~~ - s~s~ + s~s ~ 3~ -- 0}. The latter surface corresponds to steady states of (3.1). On this surface we have L ~ = g(E1, E~) or L 2 = h(E~, E3) , where g( E , , E 2 )
= ~[-2E~+ 3E~E2+ 3E~E~- 2E~ + 2(E~-E,E 2+ E})3/z],
h( E , , E 3 ) = ~ [ 2 E ~ -
3E~E 3 - 3 E I E ~ + 2E~ + 2 ( E ~ - E , E 3 + E~)3/z].
We note that L ~ ~ g(E~, Ez) throughout the phase space. The case E~ = 0 (E~ = E~) corresponds to 2 : 1 resonance considered in ref. [2]. In this case we have L z ~ g(E~, E~)= ~E~. ~ t p = s~. Then for e = 0 we have do ]2 ~] =4[o(EI-p)(o-E,
+E2) -LZ].
(3.2)
Evolution equations for E~, E z and L for the perturbed solutions then may be written in terms of p, dE~d, = 2 e ( a l + ~10 + ~102),
~dE2 = 2 e ( a 2 + O z P + y 2 p 2 ) ,
~d L = e L ( ~ + Ap),
where O/1 ----"]d, 2 g 1 ÷ ( e 2 2 - e } ~ ) E ~
+ e23E1E2,
f l l = 1"/~1-- /"/'2 ÷ ( e l 2 ÷ e21 +
2e23-
el3 -
2ez2)E1 +
(el3 -
ez3)E2,
"Y1 ----e l l ÷ el3 ÷ e22 - e l 2 - e21 - e23,
a 2 = (/z2 - / x 3 ) E 1 +/x3E ~ + (ez~ + e33 - e~3 - e32)E~l + (ez3 + e32 - 2e33)E1E2 ÷ e33E~, ~ 2 = / ~ 3 - - / ~ z + ( e ~ l + 2 e z 3 + 2 e 3 e --
2e~z -
e31 - 2 e 3 3 ) E ~ + (e31 + 2e33 - ez3 - e 3 ~ ) E ~ ,
T2 = e22 + e31 W e33 - e21 - e23 - e32, ~ = ~1 + ~2 + ~3 + (el2 +
e~
+ e32 - el3 -
e~3 -
e33)E 1 +(el3 .
A=
e l l + e21 + e31 + el3 + e23 + e33 -- e l 2 -- e22 -- e32.
We define P ( p ) = -O 3 + ( 2 E 1 - E 2 ) p 2 + E ~ ( E 2 - e l ) p - L 2.
+ e23 + e 3 3 ) E 2 ,
(3.3)
J. Guckenheimer, A. Mahalov / Resonant triad interactions
273
We use the averaging method in our analysis [2, 9, 11]. For small e > 0 periodic solutions will be found near an unperturbed closed orbit when the following function with three components:
AE~ =
for dE~ dr
'
[ r d___Ez AE2 = J0 d r '
/rdL AL = J0 ~-~
has a nondegenerate zero. Here T is the period along the unperturbed orbit. The three components of this function can be written in terms of elliptic integrals
AEI = fCCtI + /31P + VlP 2 dp,
AE~= f ca~+/3zp+y2p~ dp,
zXL= I
dp,
where b = b(E~, E2, L) and c = c(E1, E2, L) are the middle and the largest roots of the cubic polynomial P(p). Using the identity f~ d ( ~ / f f - ~ ) = 0, we get c
p2
c
~
£ ¢N3S
fbc dp
V P)
Let us define I i = f~pi/lff~(P) do" Then we obtain
AE~
~---
1
2
[a~ + ~yxE~( E2-E~)]Io + [/31+ ~ y l ( E E ~ - E 2 ) ] I ~ ,
A E 2 = [a 2 + ½y2E~(E 2 - E~)]I 0 + [/32 + ~yE(EE~- E2)] Ia,
A L = ~ L I 0 +ALIa.
(3.4)
For nonzero L the condition AE 1 = 0, AE 2 = 0, AL = 0 becomes f~(E1, E2) = I [ % + ½y~EI(E 2 - El) ] - ~[/3~ + ~3q(EE 1 - E2) ] = 0,
(3.5a)
f z ( E , , Ez) = A[a z + ½7zEa(E2 - E,)] - 6[/3z + ]Yz(2E~ - E2)] = 0,
(3.5b)
I~ 6 f 3 ( E " E 2 ' L ) = ~0 + ~- = 0 .
(3.5c)
Eq. (3.5c) gives us the ratio of elliptic integrals I1/I o = - ~ / A . We note that the functions f l and f2 are independent of L. We can solve the system of equations (3.5) in two steps. First, we find positive values of E1 and E 2 satisfying (3.5a) and (3.5b). Second, we substitute this solution in (3.5c) and find L such that I i / I o = - ~ / A . The r a t i o - ~/A does not depend directly on L. Proposition 3.1 shows that I1/I o is an increasing function of L. These two facts allow us to write necessary and sufficient conditions for the solvability of the equation I~/I o = -15/A by checking the endpoints in the variation of the function I1/I o with respect to L.
Proposition 3.1. ~L ~o > 0 .
(3.6)
274
J. Guckenheimer, A. Mahalou / Resonant triad interactions
Proof. We have (O/OL)(I~/I o) = [(OI~/~L)I o -l~OIo/OL]/Io z. If we define Ji = (1/L)(OIi/OL), then we obtain
0 (I~1 Jllo-I~Jo a--~ ~o = L Ig
(3.7)
We need the following identities:
-LZJi + E,(E 2 - El)J/+ 1q- (2E, - E2) Ji+ 2 - Ji+3 = li, 3
- ½E,(E 2 - E,)Ji+ l - (2E, - E2)Ji+ 2 + 7Ji+3
~-
(3.8)
- (i + l ) I i,
(3.9a)
3
- ½ E , ( E 2 - E , ) J o - ( 2 E l - Ez)J, + ~J2 = 0.
(3.9b)
The validi~ of these relations is proved by direct evaluation of the left hand sides using the definition of
Jk, P(P, L) and the fact that b(E1, E2, L), c(E,, Ez, L) are roots of the polynomial P(p, L). From (3.8) and (3.9a) with i = 0, I together with (3.9b) we obtain five linear equations for Jo, J,, Jz, J3 and J4 treating I 0 and I, as parameters. These allow us to express J0 and J, as functions of I 0 and I,. ~ e determinant of this linear system is E z '~ - 2 E ~ E ~ + E ~ E ~ - 4 E ~ L Z + 6 E ~ E 2 L ~ + 6 E , E ~ L 2 - 4 E ~ L 2 - 2 7 L 4 ) .
D=-~(E,
In the n e ~ lemma we show that the d e t e ~ i n a n t D is always negative except for the steady states.
Lemma 2.1. D < 0 except for the surface of steady states {~ = ~~ , ~z z - ~
+ s~
= 0t, on w~i~n
D=0.
Pr~f
Since L = p sin ~ ( p = s~szs3), we have
D = - ~' t~~' ~2~
~e~e~ + e?e~)
~(-4E~p~sin~6 + 6E~E~p ~sinZ4 + 6E,E~p ~sin26 - 4E~pZsin~6 -
-H~
27p 4 sin46)
- H z sin~,
where
4~ H , = ~~tt~4~2 = , = 2 _ 2E~E~ + EZE , 21,
H2 = ~(-4E~pZ +6E?Ezp2 +6E,E~pZ-4E~p z - 27p').
If H 2 > 0 then D < - H , = - ( E ~ E 2 - E , E ~ ) z < 0. If H 2 ~ 0 then - H 2 sin2~ ~ - H 2 and, therefore, ~ ~
1 ~2~2 -n, ~= -~(~.,~.~_ ~ _
+
+
~] [(,,+~)~+~]
and we have equali~ only in the case ~ = ~w. 1 From our analysis it follows that D = 0 when ~ = ~, 2 2 2 2 2 2 sxsz -s~s 3 + s2s3 = 0, and D < 0 in all other cases. ~ m m a 3.1 is proved.
and
J. Guckenheimer, A. Mahalov / Resonant triad interactions
275
Now we can conclude our proof of proposition 3.1. If we are not at the steady state, the determinant D is nonzero and we can express Jo and J~ in terms of I o and I v We find that 1 Jo = --~( PlI1 --P210),
1 J~ = - ~ ( P211 +P3Io) '
pl = 2E2x - 2E~E 2 + 2E~,
P2 = 2E~ - 3E~E 2 + E~E~ + 9L 2,
(3.10)
P3 = 4E3~Ez - 2E4~ - 2E~E22- 12E, L2 + 6E~ L2"
From (3.7) and (3.10) we obtain a Riccatti equation for I1/Io:
~-~ ~oo = - ~ -
Pl ~o
- 2p2 ~o
"
With an easy calculation one can show that the discriminant of the quadratic expression for I ~ / I o is 3D. Thus, from lemma 3.1 it follows that the discriminant is always negative except for the steady states. Now it follows that (a/OL)(I1/I o) > 0 since Pl > 0. Proposition 3.1 is proved. Now we find bounds on the ratio I ~ / I o. Fix El, E 2 and let L 2 vary. The variable L 2 can be varied in the interval 0 < L 2 < g ( E 1, E2), where g( E 1 , E 2 ) -- 2~(-2E~3 + 3E2~E~ ÷ 3E~E22 - 2E~ ÷ 2(E2~- E1E 2 ÷ E22)3/2).
We note that D = 0 at L 2 = g(E~, E2). We find that
~-~ = P-~ -- ~(2E1- E~ + ¢ ~ - el~2 + ~ ) Pl
Io
at the right endpoint L z = g(E 1, E2). At the left endpoint (L ~= O) the integrals I 0 and I~ can also be evaluated. We have if E~ <_E2,
/I=2V/~-q-~E
E2
÷
~-~
Io = --~
'K[~/~-~
K
i f E 2 < E 1.
if E~ _
(3.11)
Here K is the complete elliptic integral of the first kind and E is the complete elliptic integral of the
276
J. Guckenheimer, A. Mahalov / Resonant triad interactions
second kind. From (3.11) we obtain the value for I 1 / I 0 a t L ~. = 0:
I1
--'°,, ~
(
I-~ = E2 -- E1 + E 1
if E 1_
Since l , / I o is an increasing function of L and - 8 / h is independent of L, the equation l l / I o = - 8 / h has a solution for L ~ between 0 and g(E1, E 2) if and only if
E2 -El --
I+E
/K E, ) < (~E~)(~E~
~ < ~ I [2 - ~ +E z X~:l
E , I ' - ~ E, V/[I,~-~~]
+I
]
(3.12)
in the case E 2 _
El
E-~ - 1 + E
(E~E~) / K (E~E~) <
(~
I[
A ? , < X [2E~ - 1 +
~//(E1 )2_ NE1 + 1 ] ~22
(3.13)
in the case E l _
F(x)=x- l +
e(v;)
G(x)= ½(2-x +
/x2-x +1),
H(x)= ½(2x- l +
/x2-x + l ),
that appear in (3.12) and (3.13). Graphs of these functions are shown in figs. 1 and 2 respectively. We see that F(x) < G(x) and F(x) < H(x) for x in the interval [0, 1]. The periodic orbit with parameters (El, E 2) and with appropriate L is preserved under cubic perturbations if and only if f~(E1, E 2) = 0, f2(E1, E 2) = 0 and one of the inequalities (3.12) or (3.13) is satisfied. The functions f~(E 1, E 2) and f 2 ( g l , g 2) are quadratic in E 1 and E 2. This implies that no more than four periodic orbits can survive under perturbations. Thus, we have proved the following theorem.
Theorem 3.1. Suppose that the second order curves given by the equations fl(El,
E2) = ~[o/1 -]- ½ T , E , ( E 2 - E,)]
- ~[~I -]- ] T I ( 2 E I - E2)] = 0,
f 2 ( E l , E2) = ~[0/2 -~" }~2EI(E2 - El)] - ~[~2 ~- ~T2(2El -
E2)] =
0
(3.14)
are in generic position (i.e. have nondegenerate intersection). Suppose that ( E 1, E 2) is a solution of (3.14) satisfying (3.12) in the case 0 _
J. Guckenheimer, A. Mahalov / Resonant triad interactions
277
11
0.8 ¸
0.6
0.4-
0.2-
Fig. 1. Graphs of the f u n c t i o n s F ( x ) and G(x). Note that F(x) < G(x) for x between 0 and 1 (0 < x < 1).
0.50.4-
0.30.20".i~
,
,
0.4
0.6
0.8
1
Fig. 2. Graphs of the functions F(x) and H(x). Note that F(x) < H(x) for x between 0 and 1 (0 < x < 1).
Let us consider the special case studied in ref. [2]. The system of equations (3.1) has an invariant subspace defined by s 1 = s 3 if/~i =/z3, et~ + el3 = e3r + e33 , e~2 = e32. Let us denote e~.l + e~3 = e31 + e33 by e~l and e2~ + e23 by e2v Then the system of equations restricted to this invariant subspace becomes dsl
d z = -s~s2 cos ~b + es~( l~x + ellS~ + et2s~ ),
ds 2
dz = s12 cos d' + es2(/z2 + e21s21+ e22s22),
d~b
d---~ = -
1
sa / - 2 s 2 + s 2]/ sin ~b.
(3.15)
278
J. Guckenheiraer, A. Mahalov / Resonant triad interactions
In this case we have fl(E1, E 2) = fE(El, E2) and we define Solutions of the equation f(E) = 0 are E =
- 3 ( 2 / ~ 1 d- /22) 4ell + 2e12 + 2e21 + e22 '
f(E) = f l ( E l , E 2) = f2(El, E 2) at E = E l = E 2.
g ~- /22 - / 2 1 el2 - e22 •
From theorem 3.1 it follows that the periodic orbit on the energy surface E survive perturbations if and only if ~
2
0 < - ~-~ < 7.
(3.16)
In the case E = -3(2/21 + / 2 2 ) / ( 4 e l l + 2e12 + 2e21 + e22) we find that - ~ / A E -- 3. Thus, in this case the inequality (3.16) cannot be satisfied. In the case E = (/22 - / 2 1 ) / ( e 1 2 - e22) the inequality (3.16) becomes 2 0 < (/21e22 - ~2e12)/,~(/2 2 - / 2 1 ) < ~. W e also note t h a t E = (/22 - / 2 1 ) / ( e 1 2 - e22) s h o u l d b e positive. Thus, the perturbed system of equations (3.15) possesses one periodic orbit if and only if 0 < /21e22- - /22e12 < ~-, 'h'(/22 --/21)
//,2--/1,1
> 0.
(3.17)
e12 -- e22
The inequalities (3.17) can be easily analyzed and we obtain the following
Theorem 3.2. Suppose that one of the conditions listed in the left column of table 1 is satisfied. Then the perturbed equation (3.15) possesses a periodic orbit in the region of the parameter space defined by the inequalities given in the right column of table 1.
4. The system of equations (2.1) on the real subspace The real subspace of (2.1) and subspaces obtained from the real subspace by applying elements of the symmetry group play a very important role in the overall dynamics. In this paragraph we discuss the dynamics of (2.1) restricted to the real subspace. In some cases symmetries in the problem result in the fact that second order coupling coefficients in the amplitude equations are identically zero. For example, symmetry with respect to the midplane requires quadratic coupling coefficients to vanish identically in the equations describing resonant triad interactions of the Tollmien-Schlichting waves in channel Poiseuille flow [7, 15]. In other cases, quadratic coupling coefficients are small compared with cubic coupling coefficients and, therefore, quadratic terms can be treated as a small perturbation. In the case of channel flow the symm.etry can be broken in a natural way by sliding one boundary of the channel with respect to the other with a small constant velocity. Then the base flow is called plane Couette-Poiseuille flow. In the case of small symmetry-breaking perturbations or in the case when quadratic coupling coefficients are small compared with the cubic coupling coefficients, the quadratic terms can be treated as a perturbation of a system with
279
J. Guckenheimer, A. Mahalov / Resonant triad interactions Table 1 Regions of existence of periodic orbits in the parameter space.
A > O, e12 < O, {3. + el2 > O, el2-- e22 > 0
e22 ~.~ > O, P-2 > e~--~Zl
A > O, el2 < O, {A + e~2 < O, et2 - e22 > 0
//'1 > O, e22 < ~2 < e22 + {A e~---~/-~ e~2 + ~ 1
A <0,
e12<0,
e12-e22>O
~1 > O,
A
e12>O ,
{A+eI2O
~1 >0,
A>O,
e12
{A+el2>O,
e12-e22
~x>O, ~2< e ~
A >0,
e12<0,
{A+el2
e12--e22
~1 > O, e22 + {A e22 e~x+{A ~t <~2 < ~e12 ~
A <0,
e12
e12-e22
A <0,
e12>O,
{A +e12
A>O,
e12>O,
e~2-e22>O
A>O,
e12
{A+e~2>O,
A <0,
e12>O,
{A+e12
A >0,
e~2>O,
e~2--e22
A > 0,
e12 < 0,
{A + e12 > 0,
e12- e22 < 0
~1 <0,
A <0,
e12>0,
{A+e12>0,
e12-e22<0
~1 < O,
e22 + ~2~1 < ~2 < e2~ el2 + el 2 ~1
~<0,
e2~ ~2< et2~ ~
e22 + ~ ~ e~2 + {A ~ <~2 < ~ e22 + ~A ~2 > el 2 + { ~ 1 e22
~>0,
e2~ <~2 < ~e22 + {A e~2~l e~2+ {A ~
~1 >0,
e22 + {A ~2 < et 2 + ~ ~
~<0,
e22 + ~A e22 e~2+~A~l <~2 < ~e~2 ~
e~2-e22>O
~<0,
~2 > ~ ~12 + {A ~1
e12-e22>O
~ < O, ~2 > e2~ e12 ~1
e12-e22
~22 ~1 < O, ~ 1
A < O, e~2 > O, {A + e~2 < O, e~2- e22 < 0
~ e22 + ~A ~ ~2 < el 2 + { ~ 1
e22 + {A ~2 < el 2 + { ~ 1
J. Guckenheiraer,A. Mahalov / Resonant triad interactions
280
cubic terms only. In the case when quadratic coupling coefficients are not small we use the rescaling of amplitudes zi -~ ezy and unfolding parameters /xy ~ evy to get a system of equations in which the odd degree terms can be considered as a small perturbation. In both cases the unperturbed case can be analyzed and used to understand the dynamics of the full system. The general case lies between these two systems in which the quadratic terms are small compared with the odd degree terms and vice versa. We find modulated travelling waves and heteroclinic cycles in perturbations of the system with only quadratic terms, and we find heteroclinic cycles in perturbations of the system with only cubic terms. We conjecture that in the general case both modulated travelling waves and heteroclinic cycles are possible.
4.1. Weak resonance We assume that symmetry in a problem that forces quadratic coupling coefficients to be identically zero is broken. Then the system of amplitude equations describing resonant triad interactions has the following form: dz 1
dt
---//'lZ1 q- es~°z2z3 + SllZ1 [ZI [2 + s12z11z212 + s~3z~ [z312,
dz 2
dt = Iz:z2 + es2°z~z~ + s21z2lz112 + s22z2[z212 q- S23Z21Z312'
dz3 dt ---/d,3z3 q- Es30z1z ~ --ks31z 3 Iz I 12 + $32z3 Iz212 + s33z31z312. Here e is a small parameter in the problem. In the case of channel Couette-Poiseuille flow e is proportional to the size of a symmetry breaking perturbation (sliding). This problem is studied in ref. [15]. We note that the coupling coefficients in this problem are complex. Nevertheless, for e = 0 the phases are decoupled from the amplitudes and we obtain a three-dimensional system of ordinary differential equations with real coefficients for the amplitudes r~. = Iz~l, The results obtained in this paragraph can be applied to this system of equations. For e = 0 the system of equations with 0(2) symmetry restricted to the real subspace becomes dx 1
dt
= Izlx~ + SllX31 "b Sl2X1X ~ + s13xlx~,
dX2dt =b~2x2 +s21x2x~ +s22x~ +s23x2x~' dx 3
dt
=/z3x3 + s31x3x~ -[-s32x3x~ -[-$33x33"
(4.1)
It turns out that this system of equations has the same form as the system of equations studied in refs. [10, 17]. The system of equations has two-dimensional invariant subspaces {x 3 = 0}, {x 2 = 0} and {x~ = 0}, and is equivariant with respect to the transformation x~--> -x~. Solutions remain in an octant, and it suffices to consider this system of equations in the domain {x~ > 0, x 2 > 0, x 3 > 0}. The axes x~, x 2, x 3 are invariant under the flow. Provided that ~1/s11 < 0, /z2/s22 < 0, /~,3/$33 < 0, we have three steady
J. Guckenheimer, A. Mahalor' / Resonant triad interactions
281
Fig. 3. Heteroclinic cycle between the steady states A, B and C (schematic).
states
.=(0, lying on the invariant lines x 1, x 2 and x 3. We give necessary and sufficient conditions for the existence of a heteroclinic cycle between the steady states .4, B and C with the property t h a t each heteroclinic connection lies in the corresponding two-dimensional invariant subspace as shown in fig. 3. We also investigate stability of heteroclinic cycles.
Lemrna 4.1. A heteroclinic cycle between the steady states A, B and C can exist only if all the coefficients sij (i, j = 1, 2, 3) have the same sign.
Proof. Guckenheimer and Holmes [9] gives a full description for the dynamics of the system of equations (4.1) restricted to the two-dimensional invariant subspaces {x 3 = 0}, {x 2 = 0} and {x I = 0}. From their results it follows that heteroclinic orbits connecting steady states in the two-dimensional invariant subspaces exist only in the three cases la, lb and 5a in their classification. We will call a heteroclinic connection corresponding to their case la a type 1 connection; a heteroclinic connection corresponding to their case lb a type 2 connection; and a heteroclinic connection corresponding to their case 5a a type 3 connection. In type 1 and type 2 connection, all the coefficients have the same sign (either positive or negative). First, we show that if a heteroclinic cycle connects A, B and C, then in the two-dimensional invariant subspaces {x I = 0}, {x 2 = 0} and {x 3 = 0} heteroclinic connections of type 1 or type 2 but not type 3 occur. The proof is by contradiction. Suppose that a type 3 connection occurs in at least one two-dimensional invariant subspace. Then the following two cases are possible: (a) Type 3 connection occurs in at least two subspaces. (b) Type 3 connection occurs in exactly one subspace. Suppose that a type 3 connection occurs in at least two subspaces. Then there are subspaces with the type 3 connection that are adjacent to each other. By inspection of the type 3 connection we find that the heteroclinic orbit always goes in the direction from the steady state that is unstable on the invariant axis on which it sits to the steady state that is stable on the invariant axis on which it sits. Let us now consider the steady state lying on the invariant line separating two adjacent two-dimensional subspaces where the heteroclinic connection is of type 3. Since we have a heteroclinic orbit in one of the subspaces, the heteroclinic orbit should enter this steady state. Therefore, this steady state must be stable on the corresponding invariant line. On the other hand, the heteroclinic orbit in the other subspace is leaving this steady state. Therefore, since the heteroclinic connection in this subspace is of type 3, the steady
282
J.
Guckenheimer,A. Mahalov / Resonant triad interactions
state must be unstable on the invariant line. It cannot be stable and unstable at the same time, so a contradiction is obtained. Therefore, case (a) is impossible. Now we suppose that case (b) is realized. Without loss of generality we assume that a heterodinic connection of type 3 is in the two-dimensional invariant subspace {x3 = 0} and heteroclinic connections either of type 1 or of type 2 occur in the subspaces {xt = 0} and {x 2 --- 0}. Inspecting the type 3 connection, we find that the unfolding parameters /£1 and /£2 always have opposite signs in the case of a heteroclinic connection. On the other hand, inspecting the type 1 and the type 2 connections we find that the unfolding parameters /£1 and /£3, /£2 and /£3 always have the same sign implying that /£1 and /z 2 are of the same sign. Thus, we have a contradiction in the case (b) as well. We conclude that each two-dimensional invariant subspace has either a type 1 or a type 2 connection. It remains to show that the signs of all the coefficients are the same. So far we have shown that signs of the coefficients in each two-dimensional invariant subspaces are the same. Without loss of generality let us assume that all the coefficients in the invariant subspace {x 3 = 0} are positive. Then we have a 1 > 0, b 2 > 0. Since the signs of the coefficients in the subspace {x 2 = 0} are the same, then c 3 has the same sign as a 1 and, therefore, c 3 > 0. Thus, a 1 > 0, b 2 > 0, c 3 > 0. Now it follows immediately that all the coefficients a~., b~ and c~. are positive. The lemma is proved. Thus, in our derivation of conditions for the existence of a heteroclinic cycle we can restrict ourselves to the case when all the coefficients sii have the same sign. Assume that all the coefficients are negative. Necessary and sufficient conditions for the case when the coefficients are positive are obtained from this case by reversing time t ~ - t and the unfolding parameters/£1 " ~ ' --/£1, /£'/'2" ~ --/£2, /£3 - - ~ --/£3" Let a I ----- - - S l l , b I =
-s12
, c 1 =
-s13
a 3 =
-$32
~c3 =
-$33.
-$31
, b 3 =
~
a 2 =
-$21
, b 2 =
-$22
~c2 =
-$23
,
Then we have the following system of equations: dXl d"~- = / £ 1 X l
_ a2x
~ -
b ~ x l x ~ - ClXlX ~,
dx2 dt =/£2x2 _ a2x2x21 - b2x32 - c2x2x~, dx3 dt
=/£3x3
_ a3x3x ~ - b3x3x ~ - c3x ~.
Theorem 4.1. Suppose that a j, b~. and c~ are positive numbers. There exists a heteroclinic cycle between the steady states A--(/£~1
,0,0),
B=(0,
/£~2-~2,0), a n d C = ( 0 , 0 , / £ ~ 3 3 )
if one of the sets of conditions listed in the middle column of table 2 is satisfied. A heteroclinic cycle exists in the region of the parameter space defined by the inequalities given in the right column of table 2.
Proof. The eigenvalues corresponding to the steady state A = ( ~ l V / - ~ - ~ , 0 , 0 ) are -2/£1,/£ 2 (a2/al)/£1,/£ 3 - ( a 3 / a l ) / £ 1 , the eigenvalues corresponding to the steady state B = (0, V/--~z/b2,0) are
J. Guckenheimer,A. Mahalov / Resonant triad interactions
283
Table 2 Regions of existence of heteroclinic cycles in the parameter space. la
a2
b-~a-~>l,
cl a3 ~-3~-i > 1 ,
c2 b3 ~-~2>1,
a2 ct b3 al c3b2 < 1
g,Z>~-lp,,,
~ > b~z~z, ~ > ~
lb
b~ a 2
b~ a~ > l, cS a~ > l,
c~ a~
c2 b3 ~ > 1 ,
a 3 b~ c 2
a~ b~ c~ < l
a3 ~3>~1,
b~ cz ~ > b-~u,z, u,2 > ~3~3
2a
bl a2
c~ a~
c 2 b~ ~ < 1 ,
a2 c3 < 1
~2 > ~~2,
2b
3a
3b
bt a2
b2a~
>1,
b~ a 2
bS~>x,
c3a~ < 1 , cI a 3
c2 b 3
~aC<~,
E~
b~ a 2 b~a~
c~ ~ a3 a, > 1 ,
EE
bl a2 b~a~
c~a~>l,
Cl a3
c2 b3 ~<1,
c2 b3
4a
b~ a 2 cI a3 bS.S<~, cSaS
4b
b~ a 2 b2 al < 1,
5a
5b
6a
6b
cI a3 ~c~ ~, < 1,
b~ a 2 cI a3 b~a~
b2atb~a~< 1 , b~ a 2
bS~S
cI a3
b~ c 3
b~
~,>E~,
Ec~<~ b 2 c~
<1
EE
b2 a3 ~ < 1
c 2 b3
b2 g2>~g3,
a~ c 2
c3
E(<~
~>l, c2 b3
b2
~> ~ '
~>~,,
a~ b 3 > 1,
E ~ c2 b3 ~ < 1 ,
at <
E E
~
a t b 2 c3
a~b~c~
c 2 b3 ~ < 1 ,
c2 b3
b 2 at c3
~bta3c~ <1 b3 cI
~>1,
~>~,
E~<~
~, > ~ ' al ~1>~2,
~2>~,
b3
~>~,
C3
~ > ~'~ c3
bz /.I,2 > ~3~3, C1
~ > ~,
hi'3 > ~1~1 C3
~ > ~2t~
a3
~ > ~(~,
a1
~ > ~2~
a! C2 ~ > ~,~, ~ > ~,~
b3
~ > ~2~,
b~ "
t~ > ~ - ~
b~
c~
u-~ > b~3~,~, ~ > ~ al /.t 1 > ~3/./,3, b:,
c3 /.1,3 ). ~2~/,2 c1
~2> ~~1,
/~1 > ~3~3 ~ c2 /.,I,2)" ~3#,3
cS aS > L E ~ > ~, ~ ~ < ~
Cl a3
c2 b3
a3 c2
~ > ~,,
al ~./'1 ~> ~2#'2,
7a
bt a 2 bEal >1,
cI a3 -c3 - ~al <1,
c2 b3 ~>1,
b~ c 2 ~c ~1 < 1
b~ ~ > ~g2,
~2 > ~3~3, ,3 > ( ~
7b
b~ a 2 b2a~ > 1 ,
cI a3 ~ < 1 ,
c2 b3 ~>1,
a 2 b3 ~ < 1
a2 ~2>~1,
~>~3,
~3>~
8a
bl a2
b~a~>l,
~c~ ~al> 1 ,
~
Cl //~1 ~> ~33~3,
C3 ~3 > ~2/'t2
bl a 2 bz a 1 > 1,
cI a3 ~ ~ > 1,
c~ a~
c2 b3
c2 b 3
~ E
a2 cl
<1,
~E
< 1,
~ E
a 3 bl
a3
al
m > ~3,
c1
b~ ~
~
bt a~ < 1,
c~a~C~a3 <1,
<1,
a~c~
b3
a2
<1
~ > a~"l,
< 1
~ > ~,
-
-
a3
172
at
bl
t~t > ~ ' 2 ,
C3
b3
b2
~2 > ~ 3
284
J. Guckenheimer, A. Mahalov /
Resonant triad interactions
I.Xl-(bl/b2)Iz2,
- 2 / z 2, Iz3-(b3/b2)tz2, the eigenvalues corresponding to the steady state C = (0,0, #3VC#-----3~)are ~i -(c~/c3)ix3, ~ 2 - (c2/c3)P.3, --2/z3- It is clear that the steady states A, B and C are stable on the corresponding invariant lines since the eigenvalues in these directions are -2/.~1, -2/x 2 and - - 2 g , 3. The remaining two eigenvalues are responsible for the stability in the two dimensional invariant subspaces. We have shown that if a heteroclinic cycle exists then heteroclinic connections in two-dimensional invariant subspaces are either of type 1 or of type 2. There are the following possibilities: (1) The heteroclinic connection in each invariant subspace is of the type 1. (2) The heteroclinic connection in one invariant subspace is of type 2 and heteroclinic connections in two other invariant subspaces are of type 1. (3) The heteroclinic connections in all three two-dimensional invariant subspaces are of type 1. (4) The heteroclinic connections in two invariant subspaces are of the type 2 and the heteroclinic connection in one invariant subspace is of type 1. We consider case (1): all heteroclinic connections are of type 2. Then the coefficients satisfy [9]
b_~_la__22 b2 al > 1 ,
¢1 a3 c--~a-~-> 1 '
c2 b3 c3 b2 > 1 .
Analyzing the type 2 connection we find that in case (1) a necessary and sufficient condition for the existence of a heteroclinic cycle is that each steady state A, B and C has at least one unstable eigenvalue in each two-dimensional invariant subspace. This is equivalent tO the condition that the following logical expression is true: AND
, We find that this logical expression is true if and only if one of the following logical expressions is true:
a3
bl
[(#3 > ~ll g . , ) A N D (/.Ll > "b-~2/.d.2)AND (~2
> C2 ~3)]"
The first of these expressions is valid if and only if (a2/alXCl/c3Xb3/b 2) < 1. The second expression is valid if and only if (a3/alXbl/,b2Xc2/c 3) < 1. Thus, we proved the statements la and lb of the theorem. Now we consider case (2): in one two-dimensional invariant subspace the heteroclinic connection is of type 2 and in two other two-dimensional invariant subspaces heteroclinic connections are of type 1. Without loss of generality we assume that the heteroclinic connection in the subspace {x 3 = 0} is of type 2 and the heteroclinic connections in the subspaces {x 2 = 0} and {x 1 = 0} are of type 1. Then we have (bl/b2)(a2/a l) > 1, (Cl/C3Xa3/al) < 1, (c2/c3)(b3/b2) < 1. Necessary and sufficient conditions for the existence of a heteroclinic cycle are that there be a heteroclinic connection in each subspace and that each steady state A, B, C has at least one unstable eigenvalue. The condition that we are in the region of existence of a heteroclinic orbit in the first subspace is [/z 2 > (a2/al)izl] OR [/x 2 < (b2/bl)~z~]. The
J. Guckenheiraer,A. Mahalov/Resonant triadinteractions
285
condition that we are in the region of existence of a heteroclinic orbit in the second~ subspace is [/x3 >(c3/c~)lx~] OR [/x 3 <(a3/a~)laq]. The condition that we are in the region of existence of a heteroclinic orbit in the third subspace is [/~3 > (c3/c2)l-~2] OR [/~3 < (b3/b2)tx2]. Now we can easily find conditions guaranteeing that at least one eigenvalue is unstable for each of the steady states A, B, C. The condition that the steady state A has at least one unstable eigenvalue is [~2>(a2/al)l~] OR [/z3> (a3/a~)pq]. The condition that the steady state B has at least one unstable eigenvalue is [l~ > (bl/b2)~2] OR [~3 > (b3/b2)lx2]. The condition that the steady state C has at least one unstable eigenvalue is [/x~ > (¢1/c3)]A,3] OR []'/'2 > (C2/¢3)]'/'3 ]" Then a necessary and sufficient condition for the existence of a heteroclinic cycle in the case (2) is that the following logical expression is true:
~,,)] ~o ~,)o~(~,> ~,~,)]~o
~--~/Zl) O R ( / x 3 < a 3 > C3
b3
> bl
b3
a3
<
C2
~.~to~(.~ ~.~tl.
It turns out that this logical expression is true if the following logical expression is true:
ANo(~3>~2)]o. ~ / x l ) AND a~ I(.~~°~ (", > ~"~t c3
b~ b2 > ¢3 [(/~1> ~2/z2) AND (/z2> ~3~3 ) A N D (~/,3 t The fiLst alternative holds if and only if (a2/a3Xc3/c2) < l. The second logical expression holds if and only if (bl/b3Xc3/c 1) < 1. This proves statement 2 of the theorem. Statements 3 and 4 are proved with similar arguments. Now we consider case (3): all three heteroclinic connections are of type 1. This implies that b___~a__~2 b2 al <1,
cl a 3 c-~a~-< 1 '
¢2 b3 c3 b2 <1.
We have a hctcroclinic cycle if each steady state A, B, C has at least one unstable eigcnvaluc and we arc in regions of the parameter space (/z l,/z 2,/~a) where heteroclinic connections in each two-dimensional invariant subspaccs occur. This is equivalent to the following logical expression: a2
~2/z2) OR bl
b2
> ca b3
AND [(~,~ ~,)°' O. (~, > ~,)]c, ~D
~O[(~. >a2~,)O. (~.> ~,)]~O a, ~o[(~,>c~~)o~ (~>~)]. ~
2%
J. Guckenheimer,
A. Mahalov /Resonant
triad interactions
This logical expression is equivalent to
From the first expression we get the condition (a,/a,Xb,/b,Xc,/c,) < 1. From the second expression we get the condition (b,/b,Xa,/a,Xc,/c,) < 1. Thus, we have proved the statements 5a and 5b of the theorem. Now we consider case (4): two invariant subspaces have heteroclinic connections of type 2 and one subspace has a heteroclinic connection of type 1. Without loss of generality we assume that the subspace {x3 = 0) has a heteroclinic connection of type 1. The coefficients satisfy the following relations: b,
a2
iqiy
c’fL>l, Cl,
c3
??>l.
a,
There is a heteroclinic
3
2
cycle if
(lL3>%1”2)AND(1”2>~P~)AND(~~‘~~3)oR P3 > $,
AND )
PI>
$2
AND )
(
(
From the first expression we find that the coefficients must satisfy (b,/b,Xc,/c,) < 1. From the second expression we find that the coefficients must satisfy (a,/a,)(c,/c,) < 1. Thus, we proved the statements 6a and 6b of the theorem. Statements 7 and 8 can be proved in a similar fashion. Theorem 4.1 is proved. The paper of Guckenheimer and Holmes [lo] contains the first example of structurally stable heteroclinic cycles. In our notation, their sufficient conditions for the existence of a heteroclinic cycle are a,=b2=c3=a,b,=c2=a3=b,c,=a2=b,=c,~,=~2=~~=1,c>a>b.Wefindthatifbc/a2>1 then condition lb of the theorem 4.1 is satisfied, if bc/a2 < 1 condition 5a of theorem 4.1 for the existence of a heteroclinic cycle is satisfied. But one of these conditions is always satisfied. Thus, we have a heteroclinic cycle. Melbourne [17] also gives sufficient conditions for the existence of a heteroclinic cycle. In our notation, his sufficient conditions are a, = 1, b, = 1, c3 = 1, b, < 1, c2 < 1, a3 < 1, a2 + b, > 2, c2 + b3 > 2, c, + a3 > 2. From these inequalities it follows that a2 > 1, b, > 1, c, > 1. Then we obtain a,b,c, < 1, bl/b3
a3/a2
<
b3,
c2/cI
<
a2,
(l/a2Xl/b3Xl/cl)
< 1, a3/a2 < l/c,,
b, < cJcZ, ~3 < b,/b,. Thus,
the fourth condition in lb, 2b 3b, 4a, 5a, 6b, 7a and 8b of theorem 4.1 is satisfied. Therefore, conditions of theorem 4.1 is necessarily satisfied and we have a heteroclinic cycle. Now we discuss stability of heteroclinic cycles. We have
one of the
Theorem 4.2. Suppose that one of the conditions of theorem 4.1 for the existence of heteroclinic cycles is satisfied. Then the regions of the parameter space where these heteroclinic cycles are attracting (repelling) are given in the right column of table 3.
Z Guckenheimer, A. M a h a l o v / R e s o n a n t
triad interactions
287
Table 3 Regions of stability of heteroclinic cycles in the p a r a m e t e r space. la
Attracting (fig. 4, la)
lb
Attracting (fig. 4, lb)
2a
Attracting: inside the cone S (fig. 4, 2a). Repelling: inside the cone U (fig. 4, 2a).
•: (~,- ~)(~-~)(~-~) - (~- ~,/(~- ~)(~,- ~/~0 2b
t? I
bl
c2
Attracting: inside the cone S (fig. 4, 2b). Repelling: inside the cone U (fig. 4, 2b). (~12.ttt
•: 3a
b3
''b 3
'[c,
-,~#tr~,~-l)t
_p.,)
~
a3
_
('-~,~,t(~, ~ ) ( ~ - ~/~°
Attracting: inside the cone S (fig. 4, 3a). Repelling: inside the cone U (fig. 4, 3a).
•: (~,- ~)(~-~,)(~-~1 - (~- :+,/(~- ~)(~,- ~t ~0 3b
Attracting: inside the cone S (fig. 4, 3b). Repelling: inside the cone U (fig. 4, 3b). a2 b3
(c~-~,t
4a
a3
b3
c1
b~
c2
Attracting: inside the cone S (fig. 4, 4a). Repelling: inside the cone U (fig. 4, 4a). a2 b3
a3 _ b~ c2 = 0 (~-~,/ -(~-~,~,/(", ~"~)(-~-~t
4b
Attracting: inside the cone S (fig. 4, 4b). Repelling: inside the cone U (fig. 4, 4b). a3 c2
,(~,-~t(~-~,)(~-~1-(~-~,1(~ 5a
_ b.3
cl
~t(~,-~1 ~°
Repelling (~g. 4, 5a).
5b
Re~elling (fig. 4, 5b).
6a
Attracting: inside the cone S (fig. 4, 6a). Re~elling: inside ~he cone U (fig. 4, 6a).
,: (~,_~(~_~,~(~_~_(~_ ~,~(1 ~(~,_~0 a3
6b
c2
-- b 3
Attracting: inside the cone S (fig. 4, 6a). Re~elling: inside ~he ~ n e U (fig. 4, 6b). a2 b3
.: (~,-~(~.~-,)(~-~,~-(~-~.,~(~, 7a
7b
Attracting: inside the cone S (fig. 4, 7a). Repelling, inside the cone U (fig. 4, 7a). [ a2 ~[ b3
.:t~,-~t~-~)(~ -~,)- (~- ~,)(~,-
b~ ~)(~_ ~c~, ~o
_~/_(~_:+~/(,_~~ ) t ~,,, - ~ ) - 0~,,
Attracting: inside the ~ n e S (fig. 4, 8a). Repelling: inside the ~ n e U (fig. 4, 8a). [a 3 ,[b, )(c 2
.: t ~ , - ~ ) t ~ - ~ , 8b
c~ s~)(.~_~0
_ b~
Attracting: inside the cone S (fig. 4, 7b). Repelling: inside the cone U (fig. 4, 7b). [ a3 ~[ bI "~[ c 2
•:tz,.,- :)t~.2-.,jt~ 8a
c1
~
)~
~)~
- ~)t~,-~)~o
Attracting: inside the cone S (fig. 4, 8b). Repelling: inside the cone U (fig. 4, 8b).
~, ~0 .: (~.,-.~(~~-~)(~ _.,/_(~_~.,~(.,_~,~ ) t ~. ~ - ~)
288
J. Guckenheimer, A. Mahalov / Resonant triad interactions
lb
la a2 P'2=~1 P'I
~2
/-,~~1/
P'2
--',
I."/'
b2 P2= ~11pI
,
gl
gl
~
I
Cl
al b2
b 1 c2
a~
c3
a2 b 3
b2 c 3
a3
2a
2b a2 " 2 = ~ 1 P'I
~2
o~~ "
" gl a1 _
b2 P'2=~1 "1
"2
| / ~- ~ 1
,
~
I
~
a 1 c2
_
a3
a2 c 3
bl
Cl
-
-
-
b3
3a
gl
-
c3
3b b2 P-2=~1"1P'I
"2
a2 P'2=~1"1pI
"2 -
•
°
i
~
c1 -
/ ~°~ ~"! :,,,~,
b 1 c2
al b 2
a.~_l
b2 c 3
a2 b 3
a3
-
c3
Fig. 4. Regions of the parameter space where the heteroclinic cycle is attracting (repelling).
J. Guckenheimer, .4. Mahalov /Resonant triad interactions
289
4b
4a a2 P-2=~1 P-1
I~2
b2 p.2 = ~--~1.u.1
P-2 "
S
~
/
~"
PI a 1 c2
c1
~
•
I
a1
bl
a3
b3
_ _
- -
--r-
a2 c 3
c3
5b
5a a2 P2=~1 P'I
P2
b2 P'2
P2=~11 PI
i .
.
~
/ I
g~
g~ a 1 b2
c1
a1
a 2 b3
c3
a3
b 1 c2 b2 c 3
- -
6b
6a b2 P'2~11 P'I
~2
a2 P.2=a~- P-1
I~2
I ~'1
PI c1
b1
a1 c2
c3
b3
a2 c 3
- -
- -
al
- -
a3
F i g . 4. ( C o n t i n u e d . ) 4
290
J. Guckenheimer, A. Mahalov / Resonant triad interactions 7a
7b b2 .2=~11 "1
~2
I ', us,,
_ _~~ ~
a2 ~2=~1 ~1
~2
.
~ bl c2
~ ¢1
b2 c 3
c3
gl
8a
al
gl al b2
a3
a2 b 3
8b a2 P-2=~1 P-1
P'2
b2 P2 =~1-1P1
P2
,,
i"i "
gl
~'
~'1
c1
a 1 c2
b1
al
~
~
c3
a2 c3
b3
a3
Fig. 4. (Continued.)
Proof. Guckenheimer and Holmes [10], and Melbourne [17] have shown that a sufficient condition for the stability of a heteroclinic cycle is that the product of unstable eigenvalues is less than the absolute value of the product of stable eigenvalues. In case la the unstable eigenvalues are /x2 -(a2/al)/~l, /~3- (b3/b2)/'~2, ~ 1 - (Cl/C3)~t~3 and the stable eigenvalues are /-~3- (a3/al)tzl, Ixl- (b~/b2)lz2, 1~2- (c2/c3)lz3. Thus, the heteroclinic cycle is stable if f(/xl,/z2,/~3) > 0 where c2
a2
b3
Cl
~,,~,~,~(~,-~t( ~' -~1t(~-~/-(~ ~,/(~ ~4(~, ~4 ~
- -
We have
~
a2 ct b3 a 1 a~ c 3 b2 (~2/z2-/a'l)( b2
(-~/.~
--/'~2) c3
_
_
.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
291
From the inequalities b~ a 2 b-~a-~- > 1 ,
cI a3
c2 b3
a2 c 1 b3
c-~a-~ > 1 ,
c3b2 > 1 ,
a~cab 2
<1
(theorem 4.1, la) it follows that f(]/'l,/'/'2, ~'/'3) > 0 and, therefore, the heteroclinic cycle is attracting. Case lb can be considered using a similar argument. Now we turn our attention to case 2a. In this case the unstable eigenvalues are /z 2 - ( a 2 / a ~ ) l x ~, l z 3 - (b3/b2)l.~2, I - ~ - (q/c3)!~3 and the stable eigenvalues are / ~ 3 - (a3/a~)Iz~, l z ~ - ( b l / b 2 ) ~ 2 , ]2,2 (c2/c3)1x 3. Thus, the heteroclinic cycle is stable if f ( ~ l , Ix2,/z3) > 0 where -
-
a3 bl ¢2 a2 b3 c~ f(/zl,/Z2,/z3) = (~11/zl- ]-~3)(~2]-/,2- ]-/q)(~33]-~3- ]/,2 ) - ( ] ~ 2 - ~ 1 ~/'1)(]-~3 - ~2]d'2)(~t~l - ~33]~3) • The function f(//"l, ]d'2, ~U'3) is homogeneous in the variables /-h, ~2,/z3. Therefore, solutions of the inequality f(/zl,/~2,/x3) > 0 form a cone. Without loss of generality we can assume that /~3 = 1. The function f(/x~,/~2,/z 3) is defined in the interior of the triangle bounded by the lines ~u,1 = a l / a 3, ix 2 = c 2 / c 3 and ~2 = a2/a~lx~ (see fig. 4, 2a). It takes positive values on the line ~ = (a2/al)iz~ and negative values on the line ~ = a~/a 3 a n d / z 2 = c2/c 3. The function is zero at the points (a~/a 3, a2/a 3) and ( ( c 2 / c 3 X a l / a 2 ) , c 2 / c 3 ) . The curve f(pq,/z2,/x 3) = 0 passes through these points and divides the interior of the triangle into two regions. In the region S the function is positive and the heteroclinic cycle is stable. In the region U the function is negative and the heteroclinic cycle is unstable (see fig. 4, 2a). The remaining statements of theorem 4.2 are proved using similar arguments. •
4.2. Strong resonance Now we consider the system restricted to the real subspace without making the assumption that the quadratic terms are small. Then we have
dx 1
d)" = Y ' x z x ~ + x , ( l ~ , + e,,x~ + e,2x ~ + e13x~),
dx 2 dt = ~2x~x3 + x2(/~2 + e2~x~ + e22x~ + e23x~)'
dx3 dt = "~3XlX2 + X3(IA'3 -b e3~x2~ + e32x 22 + e33x 3). 2
(4.2)
Here we consider the system with "~1 < 0, ~/2 > 0, ~/3 < 0 and the system with 3q > 0, Y2 > 0, Y3 > 0. These systems appear in the rotating pipe flow problem. The scaling x I = ex, x 2 = ey, x 3 - ez, l,~j = e2vj with appropriate scaling of time transforms the system (4.2) to
dx ~ = TlYZ + dy
~
dz
= ~2 XZ +
ex(I.~l +
ellx 2 + el2Y 2 + el3z2),
eY(/~'2 + e21 X2 + e22Y 2 + e2322),
d--'~ = y3xy + e z ( Ix3 + e31x2 + e32y2 + e33z2)"
(4.3)
292
J. Guckenheimer, A. Mahalov / Resonant triad interactions
We treat this system as a perturbation of the system with quadratic terms only. The limit e = 0 has the form dx d--~ = T l Y Z '
dy ~
=
"~2XZ,
dz ~
=
(4.4)
~/3XY.
L e m m a 4.2. If two of the coefficients y~. in (4.4) have different signs then the system of equation~ (4.14) is
equivalent to the system of equations describing the dynamics of a rigid body. Proof. First we show that via x ~ ax, y -~ by, z ~ cz the system of equations (4.4) can be transformed to
the system of equations dx d--~=pyz,
dy
-d-~z = q x z ,
dz
-d-~z = rxy
(4.5)
w i t h p + q + r = O.
We have p = y l b c / a , q = y z a c / b , r = y 3 a b / c . The condition p +.q + r = 0 b e c o m e s 'Yl(bc) 2 + "y2(gC)2 + y3(ab) 2 = 0. Let us assume without loss of generality that Yl < 0, Y2 > 0, ~3 < 0. If we define a = c = 1 and b = ~/Y2/([Yl[ + [Y3[) then the condition p + q + r = 0 is satisfied. The equations describing the dynamics of a rigid body are dml d'c
12- I3
dm 2
= I-~-~3 mEm3,
~
I3 - I 1 = I--~l
dm3 m3m~'
dr
IlI 2 ~m~m2"
(4.6)
In order to show that (4.5) can be written in the form (4.6) it remains to find positive I1, 12 and 13 SO that 1 ia
1 12 = p ,
1 11
1 13 = q ,
1 12
--
_
1 11
--
.~_
r.
•
(4.7)
Let us again assume without loss of generality that p < 0, q > 0, r < 0. If 11 = 2 / 3 q , 12 = 2/(21pl + q), then (4.7) is satisfied. Thus, the system (4.4) is equivalent to the system (4.6) describing the dynamics of a rigid body. Lemma 4.2 is proved.
I a = 2/q,
Now we can describe the dynamics of (4.4). The phase space of (4.4) is three-dimensional. Consider a level surface of an integral of the motion. (a) The case "Y1 < 0, "Y2> 0, 3/3 < 0. The phase portrait has the form shown in fig. 5~ We have heteroclinic orbits connecting two opposite poles of the ellipsoid belonging to t h e y axis. Four heteroclinic orbits divide the ellipsoid into four regions. Each region is filled with a faiiiily'of periodic , orbits. : ~' ; :
Fig. 5. Phase Pot.trait of (4.4). correspon.ding to the case "Yl dO, ~/2 > O, "Y3<~0.
Z Guckenheimer,A. Mahalov/Resonant triad interactions
293
(b) The case Yl > 0, Y2 > 0, "Y3 < 0. This case is similar to the case (a) but now we have a heteroclinic connection between points lying on the z axis. In the m~. variables the system of equations (4.3) becomes
dml 12 -- 13 dr = ~ mEm3 + e :ml(I£1+ c11m~ + c~2m~ + c l 3 m ~ ) ' din2 = ~
dr
din3
dr =
~
m3ml + em2(/~2 +c21m~ + c22m~ + c23m~)'
mlm 2 + em3(/x 3 +c31m~2 +c32m~ +c33m~) ,
(4.8)
t~
where 11 = 2b/3y2, I 2 = 2 b / ( y 2 + 213qlb2), I 3 = 2b/y2, b = Cy2/(13ql + 13'31) • We also have cij= eij( j --/:2), Ci2 ~- bEei2 . Without loss of generality we assume that 11 > I 2 > I 3. We are interested in the heteroclinic orbit of the unperturbed equations connecting the point A = (0, ¢ - - ~ 2 / e 2 2 , 0 ) and the point B = (0, -- ¢ - / . t E / e E 2 , 0) lying on the m 2 axis. This heteroclinic orbit is given by
qb~(t)=l
-Pq s e c h ( - v / ~ l t ) ,
q~(t)=ltanh(-f~-~lt),
q~(t)=l
- rq s e c h ( - ¢ - ~ l t ) , (4.9)
w h e r e 1 = ¢ - g,2/e22, p = (I2 - I3)/I213, q = (I3 -
11)/1311, r
= (I1 - I2)/Ili2"
We write (4.8)jn the form dm
dt' = f ( m ) + eg(m), ,,
(4.10)
,
where d m / d t = f ( m ) is the equation describing the dynamics of a three dimensional rigid body. Thus,
[ Pm2m3 I~ f(m)=
[qm3ml k ITl'llgl'12
p=
i2_i3 13 --11 11 -12 1213 , q = 1311 , r = 1112
A and B are steady states of (4.10). They lie on the sphere of radius ¢ - ~ 2 / e 2 2
(fig. 6). The
A
B
Fig. 6. Heteroclinic orbit between the steady states A = ' ( 0 , - ¢ - ~ / e ~ , 0 ) and B = ( 0 , - ~ , 0 ) Of the per~rbed system. A and 8 lie on the sphere of radius ¥/"~/~2/e22.
z Guckenheimer,A. Mahalov / Resonant triad interactions
294
i
/ ' / Fig. 7. Splitting of the heteroclinic orbit. The splitting distance is measured in the plane P. This plane is spanned by the vectors VHt(q°(t )) and VHz(q°(t )).
unperturbed equation (e -- 0) has two integrals of motion:
H,( m ) = m~ + m2~ + m~,
ml2 m2~ m~ H2(m) = -~l + ~ + I-~-"
We define L(m) = m l m 2 m 3 and we find that f ( m ) = I VL where I = diag{p, q, r} is a diagonal matrix. We study the splitting of the heteroclinic orbit q°(t) given by (4.9) under the perturbation g(tn) using the technique originally proposed by Melnikov for planar systems [9, 18]. For the perturbed system we have
a:(t) = a°(t) + ~a~(t) + : ( ~ )
a : ( t ) = a ° ( t ) + ~a~(t) + : ( ~ )
t e [o, +~o),
t~(-~,o], (4.11)
where q~(t) and q~(t) satisfy dq~ dt = D f ( q ° ( t ) ) q ~ ( t ) + g ( q ° ( t ) ) t > - O '
da~ dt = D f ( q ° ( t ) ) q ~ ( t ) + g ( q ° ( t ) ) '
t<_O.
At each point q°(t) of the heteroclinic orbit we define a plane spanned by the vectors VHt(q°(t)) and VH2(q°(t)). The heteroclinic orbit q°(t) splits into two parts: q~(t) and q~(t). We measure the splitting in the plane (see fig. 7) span[ V H , ( q ° ( t ) ), VH2(q°(t) )] using the following functions: At = (q~ _ q : , V H 1 ) ,
A2 = (qS~ _ q : , V H 2 ) .
Using (4.11), we obtain
At = (qS~ _qU~,VHt) = e ( q ~ - q ~ , V g 1) + ~'(e2),
d2
=
(qS~-qu~,vg2) = e ( q ~ - q ~ , V H 2) + ~'(e2).
We define d t = (q~ - q[, VHt), d 2 = (q~ - q~, VH2). These functions measure splitting of the heteroclinlc orbit. We have d t = d~ - d~, d 2 = d~ - d~ where d~ -- (q[,VHt), d~ = (q~,VH1), d~ = (q~,VH2), d~= (q~,VH2). Now we find differential equations for d~,d~,d~,d~. Without loss of generality we
J. Guckenheimer, A. Mahalov / Resonant triad interactions
295
consider d~ and d[. We have d
=
d
=(vn,(q°(t)),g(q°(t,))) +(vn,(q°(t)),Df(q°(t))q~(t))
Now we show that
(VHl(q°(t)), Df(q°(t)) q~(t, to)) +(d2H,(q°(t)) f(q°(t)), q~(t)) = O. We have f = I VL, D r = I dZL (d2L is the Hessian matrix). We note that I and d~L are symmetric matrices. Therefore, we have
(VH~(q°(t)), Df(q°(t)) q~(t))
= ( d 2 L I VH~, q~),
f(qO( t)),q~(t))
=(
Lemma 4.3. d2L I VH 1 ÷ d2H1 1 VL =
0.
Proof.
mlm2),
( d2H~(q°(t))
We have VL = (mzm 3, mlm3, dEHl = 2E (E is the identity matrix),
d2L
--
(0 m 3
0
m 2
m 1
1
d2H~I VL,q~).
VH~ = 2(m~, m 2, m3), I = diag{p, q, r}. Then we obtain
•
Substituting these expressions for the matrices, we get
d2LIVH~+d2HIlVL=2
m3 ,m 2
0 mi m~ 0
/qm /+2
qmlm31=O ,
~ rm3 ]
rmlm2 ]
since p + q + r = 0. Lemma 4.3 is proved. Using lemma 4.3, we obtain get (d/dt)(d~)= (VH~(q°(t)),
d
d'-}"(d~) = ( V H 2 ( q ° ( t ) ) ,
(d/dt)(d~)= (VHl(q°(t)), g(q°(t))). We can apply a similar g(q°(t))). The function d~ satisfies the following equation: g( q°( t ) )) +(VH2(q°(t)), Df ( q°( t ) ) q~( t ))
+( d2H2( q°( t ) ) f ( q°( t ) ), q~( t )). Lemma 4.4. d2LI
VH 2 ÷ d2H2 I VL = 0.
argument to
296
J. Guckenheimer, A. Mahalov / Resonant triad interactions
Proof.
We use the expressions for VL, d2L, I obtained in the proof of lemma 4.3. Here
VH 2 = 2
(1
, i2 , i3
, d2H2--2diag i 1 , i 2 , i 3
and
pm2m3 I d2LIVH 2+d2H2IVL=2
~-~ + ~ + ~3 t~'~ 1m 2 ]
since
,
~ = (I~ -I~)/I~Ia, q = (I~ -I1)/I311, r = (I~ -I~)/I~I~.
~ m m a 4.4 is proved.
Using lemma 4.4, we obtain
~(d~)=(VH2(q°(t)),g(o°(t))),
d
~(d~)=(VH2(q°(t),g(q°(t))).
Thus, the functions d[, d[, d~ and d~ satis~ the following differential equations: -ff-7(d[) = ( V H , ( q ° ( t ) ) ,
g(q°(t))),
d ~'~(d~) = ( VHI(q°(t)),
g(q°(t))),
ff-Tt(d-~) = ( VH2(q°(t) ),
g(q°(t) )),
d ~7 (d~) = ( 7H2(q°(t),
g(qO(t))).
Integrating the first equation from 0 to + 0% we get +~
d~( +~)-d~(O) = fo (VHl(q°(t))'g(q°(t)))dt Integrating the second equation from -oo to O, we get
d~(O)-d~(-oo) = fo ( Hl(qO(t))g(qO(t)))dt" -oo
We note that d~( + ~) = d~( - ~) = O. Thus, dl(0)
= d~(0)
- d~(0)
~ f_+~(VHl(q°(t))
g(
q°(t ), t )1 dt.
Similarly, d2(0 ) = d ~ ( 0 ) - d ~ ( O ) - -
f_~VH~(q°(t)),g(q°(t))~
dr.
J. Guckenheimer,A. Mahalov / Resonant triad interactions
297
Therefore, the following two functions measure the splitting of a heteroclinic orbit: df(/z~,~2,/z3) = L+~(VH~(q°(t)) g(q°(t))) dt, ,
d2(/z~2,~3) =
(4.12)
VH2(q°(t)) g(q°(t))) dt. _
:
We have
VH~(qO(t)) = 2(q~O(t), q 2o( t ) , q 3o( t ) ) ,
0 1 o 1 o ) VH2(q°(t)) = 2 ( 1-~ql(t),~2q2(t),7~3q3(t) •
(4.13)
We substitute (4.9) and (4.13) in (4.12). Using the identities (a > 0)
f + ~ s e c h 2 ( - a t ) dt = -~, f7 s e c h 2 ( - a t ) t a n h 2 ( - a t ) dt = -~-d' ~ - -
o~
- -
we evaluate the integrals (4.12). We find that the condition following: ~
f +? sech4( - a t )
dt = 43a
--
d1(~1,/~2,~3)=0 is
equivalent to the
3pqe22/x 1 + [~eHp 2 - q2e22 + 2e33r2 - p q ( e l 2 + e21) + 2pr(e~3 + e31) - qr(e23 + e32)]tz2
"
+3rqez21z3 = O. (,
The condition d2(/Xl, tz2, ~3) --- 0 can be written in the form
3pqe22
:
r 2en/92
[
r,
q2e22 2e33 r2 - - ;2 + 13
[ e~2 e21] ,, [ e13 e31] _ q r [ e23 e321] PO~ i1 + i2 } q- zPr~ 7-~ + i3 ) ~ i2 + ~
1]~1"2
3rqe22 + - - - ~ 3 tx3=O. Thus, we proved the following theorem.
Theorem 4.3. For e > 0 sufficiently small bifurcations with a heteroclinic orbit are given approximately by the equations dl(/xl,/x2, ~3) = 0, d 2 ( ~ , ~2,/~3)--0. These two conditions are satisfied on a straight line passing through the origin in the parameter space (/x~, ~2, ~3). This line is defined as the intersection of two planes '. II1:
3pqe2zN 1 + [2e~lp 2 --q2e2z + 2e33 r2 - p q ( e12 + e2, ) + 2pr( e13 + %,) -qr(e23 + e32)]/x2 + 3rqe221~3 -= O, 112: ,,
3pqe22 [ 2enp2 i1 ~1 + i1 3qe22 + ---1-~3/x3 = 0.
q 2e22 2e33r2 _ p q ( e12 e2, ( e13 e3,1 ~I1 + T 2 ) + 2 p r ~ I 1 + ia]-qr( I2 + I3
e23
e32
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J. Guckenheimer, A. Mahalov / Resonant triad interactions
5. Existence of heteroclinic cycles Reflection symmetry of (2.3) implies that the purely real subspace is invariant as is any ~b-invariant rotation of it. Thus, the following are invariant subspaces of (2.3): (Xl, x2, x3), (Yl, x2, Y3), (Xl, Y2, Y3), (Y~, Y2, x3). In order to show the existence of a heteroclinic cycle it is enough to show the existence of connecting orbits in the real subspace. Then the heteroclinic cycles in the (Yl, x2, Y3) subspace are obtained by applying the element of the symmetry group. The system of equations (2.3) restricted to the real subspace is dx 1 dt = - x 2 x 3 -]-Xl(/~l + e1'x21 +e12x~2 +e13x32)' dx 2 dt = XlX3 + x2(/x2 + e21x~ + e22x22 + e23x~) ' dx 3 dt = - g i g 2 +xa(/z3 + e31x~ + e32x~ + e33x~)"
(5.1)
We want to establish the existence of heteroclinic connections between the steady states (0, ~ , 0 ) and (0, - ~ ,0) lying on the x 2 axis. If ~z~ =/~3, ell + e~3 = e3~ + e33, e~2 = e32 then the system of equations (5.1) has two-dimensional invariant subspaces x a = x I and x 3 = - x r Let us assume that the conditions of proposition 5.1 of ref. [2] are satisfied. Then the heteroclinic cycles in the two-dimensional invariant subspaces exist and they are locally asymptotically stable in the real subspace. Thus, we have proved the following. Theorem 5.1. Suppose that the conditions of proposition 5.1 of ref. [2] are satisfied and ~t~1 1~3' ell + e l 3 =e31 +e33 , el2 =e32. Then there is a heteroclinic net for the system of equations (2.3) connecting two pair of points on the torus of pure modes. This net consist of four heteroclinic cycles in the real subspace and four heteroclinic cycles in the (y~, x 2, Y3) subspace which is conjugate to the real subspace under symmetry transformation. =
6. Stability of heteroclinic nets In this section we discuss stability of heteroclinic nets (cycles). Heteroclinic cycles for the unperturbed system exist in the case ~b = 0, xr with E 1 = E 2. Eq. (3.2) with L = 0 (~b = 0, xr) and E 1 = E 2 becomes
~dp -g!
- 4p2(E_p) •
(6.1)
The solution of this equation corresponding to heteroclinic cycles is p(~-) = E sechZ(v~- ~').
(6.2)
We want to establish conditions that guarantee local asymptotic stability of the heteroclinic net lying on
J. Guckenheimer, A. Mahalov / Resonant triad interactions
299
the level set El = _ /~2 + Fl '
E2 = _/~_~_2+ F2"
e22
After substitution of
U = eL
/d,1 "~- /'g3
(6.3)
e22
p(r) --
from (6.2), eq. (3.3) for L then becomes
/z_~2( /~2(e e22 el2 + e32) -- --e22, 11 + eEl + e3~ + e13 + e23 + e3a -- e12 -- e22 -- e32)
X s e c h 2 ( ¢ - ~2/e22 ~)) + ~ ( L F I , L F 2 ) .
(6.4)
Since s e c h 2 ( ~ - ~ 2 / e 2 2 7 ) exponentially converges to zero at infiniff, the behavior of L(r) at infini~ is ~ n t r o l l e d by the sign of the term ~l + ~3 -- (~2/e22~e12 + e32)- If this t e ~ is negative then L(¢) ~ 0 e ~ o n e n t i a l l y when • ~ + ~. From (3.3) we also obtain equations for F~ and F 2. These equations are
~ [' F~=ze[[~2 ~e 2 3 - ~ 2 ) F I - ~ 2 ~ F 2 ]
+
H~,
~2 F '~ = 2 e [ ( - ~ z - ~ 3 + e2~ ~2 (e23 +ea2))Fl+(~3_e22
(e23 +e32 ) ) F2 ] + H 2 "
~ e t e ~ s H~ and H 2 have s e c h Z ( ~ - ~ 2 / e 2 2 ~) as a factor and, therefore, they converge to zero at infinity at an e ~ o n e n t i a l rate. Then the behavior of F~ and F 2 at infini~ is controlled by the linear t e ~ s . The m a t r ~ corresponding to the linear terms is
-~
- ~ + (~/e~z)(e~3 + e~)
~ - (~/e~)(e~
+ e3~ ) "
The eigenvalues of the matrN M are - ~ and ~3 - ~ e 3 a / e ~ • F~(r),F~(~) ~ 0 at an exponential rate when r ~ + ~ if and only if both eigenvalues are negative. Combining these results with our results on the behavior of L(~), we obtain the following
Theorem 6.1. If a heteroclinic net formed by heteroclinic cycles exists for the scaled equations, e ~: 0 is sufficiently small and /'~'1 @ //'3 -- e2 2 el2 + e32) < O,
/./'2 > O,
~/'3 --/3-'2 e~22 < O,
then the heteroclinic net is locally asymptotically stable.
7. The interaction of resonant patterns in rapidly rotating Hagen-Poiseuiile flow In this section we consider interactions of resonant patterns in rapidly rotating Hagen-Poiseuille flow. The basic laminar flow has a parabolic velocity profile with maximum speed Wo (= -(r~/4v)OP/Oz, where ~P/~z is the prescribed pressure gradient) on the center-line. We assume that r 0 is the radius of
J. Guckenheimer, A. Mahalov / Resonant triad interactions
300
Table 4 Numerical values for the constants c n in eq. (7.1). n c~
1 4.366
2 5.676
3 6.916
4 8.126
5 9.316
6 10.56
the pipe and O 0 its angular velocity. With r 0 and r~/4v as units of distance and time, the problem depends on a rotational Reynolds number, O, and an axial Reynolds number, R, defined by O = r~ Oo/V, R = roWo/v. In terms of this scaling the base ttow velocity written in cylindrical coordinates is V = (0, Or, R(1 - r2)). The initial point of our search for resonant patterns is the work of Pedley [19]. In his work he obtained approximate expressions for the neutral stability curves of the Hagen-Poiseuille flow. His set of approximate equations is valid in the limit 0 / ~ 0 (0/ is axial wavenumber),/2 ~ + ~ so that a O and R remain finite. The significance of this limit is that the interaction of long waves in a rapidly rotating pipe is considered. It is known by the Taylor-Proudman theory described in ref. [8] that the velocity field becomes independent of the axial variable z in the case of fast rotation. Hence, the fluid system becomes translation and reflection invariant and, therefore, the theory developed in the previous paragraphs can be applied. Cotton and Salwen [5] found that Pedley's asymptotic results are in excellent agreement with numerical data not only in the region of formal validity of the asymptotic expressions but, in fact, they provide good approximation of the neutral stability curves in a much larger region of the parameter space. The expression for the neutral stability curves in Pedley's limit has the form 40/20 2 +
40~nOR + cn = O.
(7.1)
Here 0/ is axial wavenumber, n is azimuthal wavenumber, O is rotation rate and R is the Reynolds number. Numerical values for the first six coefficients c n taken from ref. [19] are given in table 4. We use eq. (7.1) to find the values of parameters controlling the flow for which three neutral waves exp[i(n~b+0/1 z+tolt)]u~(r),
exp[i(n2~b+a2z+to2t)]u2(r),
exp[i(n3~b+0/3 z+to3t)]u3(r )
form a resonant triad. These waves are neutral for given R and O if and only if the following equations are satisfied simultaneously:
40/702 + 4 0 / l n l O R + Cn~ = 0, Using the resonance conditions a I 4a220 2 + 4 0 / 2 n 2 O R + c n : = 0 ,
40/2zO2 + 4 0 / 2 n 2 O R + cn2 = 0, =
40/320 2 + 4 0 / 3 n 3 O R + Cn3 = O.
0/2 d- 0/3 and n 1 = t/2 +//3, we obtain 4a320 2 + 4 a 3 n 3 O R + c n 3 = 0 ,
4 [ ( a 2 + a 3 ) O ] 2 + 4(0/2 + a a ) ( n 2 + n 3 ) O g +Cnz+. 3 = 0.
(7.2)
Now we consider two cases. In the case n 2 = n 3 it is possible to satisfy (7.2) simultaneously if and only if R = ~
1
¢
c~/~2,~.
(7.3)
J. Guckenheimer,A. Mahalov /Resonant triad interactions
301
Table 5 Reynolds numbers corresponding to resonant triad interactions. Here n~ are the azimuthal wavenumbers of resonant patterns. (nl, n2, n 3) R
(2, 1, 1) 91.1
(3, 1, 2) 111.4
(4, 2, 2) 133.8
(4, 1, 3) 138.4
(5, 2, 3) 162.3
(5, 1, 4) 170.3
(6, 3, 3) 192.9
(6, 2, 4) 196.9
(6, 1, 5) 207.8
In t h e case n 2 :~ n 3 eqs. (7.2) c a n b e satisfied if a n d only if t h e R e y n o l d s n u m b e r R is a solution o f the following n o n l i n e a r e q u a t i o n :
~-n29R2-Cn2 -~n~R2-cn3
d- ~(n2q-n3)2R2-Cn2+n3=O.
(7.4)
If n 2 = n 3 t h e n eq. (7.4) r e d u c e s to (7.3). W e n o t e t h a t in b o t h cases t h e c o n d i t i o n to 1 = t o 2 + to 3 is satisfied since t h e f r e q u e n c i e s have a p p r o x i m a t e e x p r e s s i o n to = - n O - aR w h i c h i s l i n e a r in a a n d n. F r o m f o r m u l a (7.3) a n d f r o m t h e n o n l i n e a r e q u a t i o n (7.4) we find t h e values o f R e y n o l d s n u m b e r for which t h r e e n e u t r a l waves f o r m a r e s o n a n t triad. T h e r e a r e infinitely m a n y R e y n o l d s n u m b e r s for which such r e s o n a n c e s occur. T h e n u m e r i c a l values for t h e first nine o f t h e m a r e listed in t a b l e 5. A f t e r R e y n o l d s n u m b e r s c o r r e s p o n d i n g to t h e r e s o n a n t t r i a d s a r e found, we can get axial w a v e n u m b e r s a2, a 3 (a~ = a 2 + a 3) a n d t h e r o t a t i o n r a t e O. In t h e case n 2 = n 3 we o b t a i n
or20=¼(--
C)~2n2-- ~/C2n2 -- 4Cn2 ), Or30=¼(--
C~/-C~n2n2 d- "~/C2n2-- aCn2 ).
(7.5)
In t h e case n 2 ~ n 3 we have two possibilities. In t h e first case we o b t a i n
a20= ½(-n2R- ~/n~R2-cn 2), a30= ½(-n3R + ~/n~R2-cn 3).
(7.6)
0.8-
0.2 ¸
0.2
0.4
0.6
0.8
1
Fig. 8. Imaginary part of the axial velocity component of the neutral mode with n t = 4. Real part is zero.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
302
z'
0.8
0.6'
0.4,
0,2,
I 0.2
0.4
0.6
0.8
Fig. 9. Imaginary part of the axial velocity component of the neutral mode with n 2
I =
1. Real part is zero.
In the s e c o n d case w e get
o~20 (-n2R + ~/n~R2- c,z ), or30 ½(-n3R- ]/n~R 2- c,~3). =
(7.7)
=
W e find that for (7.6) the amplitude equations (1.1) with appropriate scaling of amplitudes have 3'1 = 1, 3"2 = 1, 3'3 = - 1. In the case (7.7) w e obtain 3'1 = - 1, 3,z = 1, 3"3 = - 1. T h e two choices (7.6), (7.7) give us different dynamical behavior. In the first case the third m o d e is unstable, in the s e c o n d case w e find that the s e c o n d m o d e is unstable. 0.7
¸
0.5
¸
0.3
¸
0.2-
,, 0.2
0.4
0.6
0.8
~
Fig. 10. Imaginary part of the axial velocity component of the neutral mode with n 3 = 3. Real part is zero.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
303
0.6-
0.4-
0.2-
l
I
I
0.8
i
-0.2
-0.
Fig. 11. Real part of the azimuthal velocity component of the neutral mode with n I = 4. Imaginary part is zero.
0.1-
I
I
-0.1"
-0.2-
Fig. 12. Real part of the azimuthal velocity component of the neutral mode with n 2 = 1. Imaginary part is zero.
Now we describe the interaction of resonant patterns with azimuthal wavenumbers n 1 = 4, n 2 1 and n 3 = 3 for the case (7.7). This resonance occurs for the value of Reynolds n u m b e r R = 138.4. In the fast rotation limit resonant waves are approximately independent of z and have the form u~. = exp(in~.~) fi~.(r) ( j = 1, 2, 3). However, all three components of the velocity field (radial, axial and azimuthal) are nonzero. The velocity profiles of ~j(r) are shown in figs. 8-16. The streamline patterns of these pure modes projected on the z = const, plane are shown in figs. 17-19. Thus, for each pure m o d e the pipe is divided into a number of cells. Due to the fact that the axial component of the velocity field is nonzero, the fluid particles are engaged in a helical motion in each cell. The equations describing interaction of these pure modes forming a resonant triad have the form (2.1). The coupling coefficients were computed numerically and their values are given in table 6. =
304
J.
Guckenheimer, A. Mahalov / Resonant triad interactions
0.6-
0.4-
0.2-
I
I
-0.2-
-0.
Fig. 13. R e a l part o f the a z i m u t h a l velocity c o m p o n e n t o f the n e u t r a l m o d e with n 3 = 3. I m a g i n a r y part is zero.
0.8'
0.6'
0.4'
0.2'
02
04
O~
Fig. 14. Imaginary part of the radial velocity c o m p o n e n t o f the neutral m o d e with n~ = 4. R e a l part is zero.
Table 6 . : Cubic c o u p l i n g coeffi¢ient~ in the a m p l i t u d e e q u a t i o n s (2.1) describing r e s o n a n t triad interaction o f w a v e s with n l = 4, n 2 = 1 and n 3 = 3 at R e y n o l d s n u m b e r R = 138.4. e l l = -- 18.8 = -3.19
el2 =
-
14.2
e13 =
--81.9
e22 =
- 5.19
e23 =
- 7.96
e3~ =
~e32 =
- 28.1
e33 =
- 83.7
e21
- 33.2
J. Guckenheimer,A. Mahalov/ Resonant triad interactions
305
O. 2 5 -
0.2-
0.15-
0.1-
0.05-
I
I
I
I
0.2
0.4
0.6
0.8
Fig. 15. Imaginary part of the radial velocity component of the neutral mode with n 2 = 1. Real part is zero.
0.8'
0.6"
0.2-
0.2
0.4
0.6
0.8
1
Fig. 16. Imaginary part of the radial velocity component of the neutral mode with n 3 = 3. Real part is zero.
We use theorem 3.1 to find the number of periodic orbits that survive cubic perturbations. For simplicity we restrict ourselves to the c a s e / ~ + ~2 + / z 3 -- 0. We find that the inequality (3.13) cannot be satisfied. Using the data presented in table 6, we obtain -~/AE~ = 0 : 6 9 5 5 2 0 - 0.957519 E2/E r The graphs of the functions F(x), G(x) and L(x) = 0.695520 - 0.957519x (x = E2/E ~) are presented in fig. 20. We note that the graph of L(x) intersects the graph of F(x) at .~ = 0.585129. From theorem 3.1 it follows that for given ~ l a n d / z 2 the number of periodic orbits that survive cubic perturbations equals the number of nondegenerate solutions of the following problem:
fl(E1,E2,1~I,tz2)=O,
f2(E1,E2,1~l,tZ2)=O,
E I > O , E2>O, E2/El <0.585129.
(7.8)
Fig. 17. S t r e a m l i n e p a t t e r n of the m o d e with n 1 = 4 proj e c t e d on the p l a n e z = const.
Fig. 18. S t r e a m l i n e p a t t e r n of the m o d e with r~2 = 1 proj e c t e d on the p l a n e z = const.
Fig. 19. S t r e a m l i n e p a t t e r n of the m o d e with n 3 = 3 proj e c t e d on the p l a n e z = const. 1-
0.8 .
0.6
0.4"
0.2 .
:
I
-0.2
Fig. 20. G r a p h s of the functions F ( x ) , the p o i n t x = 0.585129.
G(x) a n d L(x) (straight line) (0 < x < 1). T h e g r a p h of L(x) intersects the g r a p h of F(x) at
J. Guckenheimer, A. Mahalov / Resonant triad interactions
2
0.5"~" |
1
I -0.5
~.~
! O.S
-0. $'I
307
0
Fig. 21. Regions of the parameter space /zl,/~ 2 with 0, 1 and 2 periodic orbits. Horizontal axis-/z~, vertical axis-/~ 2. The boundaries separating different regions are given by the equations arg(/~l + ip, 2) = 1.5, arg(p,t + ip, 2) = 2.8, a r g ( ~ + i/~ 2) = 3.8, arg(/~ 1 + i/~2) = 5.9.
It can be easily shown that the problem (7.8) is invariant with respect to the transformation E L ~ cE1, E 2 ~ cE2,1~1 ~ clzl, l z 2 - ~ c l ~ 2 where c is an arbitrary positive constant. This implies that boundaries separating regions on the (/~1,/z 2) plane having different number of periodic orbits are straight lines passing through the origin. From (7.8) we find that depending on the values of/~1 and ~2 the number of periodic orbits that survive cubic perturbations is 0, 1 or 2. The regions of the parameter space with 0, 1 and 2 periodic orbits are shown in fig. 21. It follows from theorem 6.1 that the heteroclinic net is locally asymptotically stable (when it exists) in the region of the parameter space/~,/~2,/~3 given by the inequalities /x 1 + tz 3 - 8.15/~ 2 < 0,
/z 2 > 0,
/~3 - 5.41/~ 2 < 0.
The velocity field (streamlines) was reconstructed for a trajectory approaching the heteroclinic net. The corresponding patterns are shown in fig. 22. The initial and the final states are identical except for the direction of rotation. The sequence of patterns shown in fig. 22 provides an example of the recovery of initial state after its breakdown. It can be seen that the initial pattern formed by two eddies breaks down and is replaced by a sequence of asymmetric patterns. Nevertheless, the order is regained and the initial
308
J. Guckenheimer, A. Mahalov / Resonant triad interactions
Fig. 22. Streamline patterns corresponding to the heteroclinic orbit.
state of two eddies (but with rotation in the opposite direction) is recovered. The system is apparently hovering chaotically in time between these two states. The corresponding trajectory in the phase space is shown in fig. 23. We note that Wersinger et al. [23] also provide an example of chaotic behavior in three-wave equations. In their case the chaos is associated with a sequence of period doubling bifurcations.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
309
Fig. 23. Trajectory in the phase space corresoondin.g to the streamline patterns of fig. 22. Horizontal axis-x t, vertical axis-~x 2. The trajectory is hovering (apparently chaotically in time) between two states formed by two eddies. These states have opposite directions of rotation. The parameters are / z ~ = - 1 , /z2~ 1, /~3 = -0.68, e=0.022. The initial conditions are x 1=0.1, y l = 0 . 0 , x 2=0.0, y2=0.0, x 3=0.1, Y3 = 0.0.
8. Conclusions We have demonstrated that expansi~on of the resonant triad equations to third order is essential to capture the qualitati,~e features of thi'ee interacting waves in a dissipative system. The third order terms break degenerac~ ,es,in the second order expansion that result from linear flow in the invariant subspaces correspQoding toia single active mode. In the dissipatfize case, there is linear excitation of the amplitudes of the different modes and the cubic terms serve to damp this excitation at finite amplitudes. The main dynamical features of the triad interactions in the dissipative and conservative systems are quite different. The significance of higher order terms in the analysis of resonant interactions in shear flows was first observed by Usher and Craik [22]. This paper begins the systematic investigation of the qualitative dynamics of the equations describing resonant triad interactions in dissipative systems. We have not discussed secondary or higher order bifurcations in this paper. Also, our explorations of regions of the parameter space with chaotic behavior are rudimentary. A more complete understanding of the dynamics of (1.1) remains an interesting and challenging problem. :.,
Acknowledgements This work was supported by the Air Force Office of Scientific Research under contracts AFOSR-890346 and AFOSR-89-0226. J. Guckenheimer was supported by NSF, AFOSR and the Army Research Office through the Mathematical Sciences Institute at Cornell University. A Mahalov is supported by an IBM Watson Fellowship. We would like to thank Sidney Leibovich for very helpful discussions. References [1] D. Armbruster and P. Chossat, Heteroclinic orbits in a spherically invariant system, Physica D 50 (1991) 155-176. [2] D. Armbruster, J. Guckenheimer and P. Holmes, Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry, Physiea D 29 (1988) 257-282.
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[3] N. Aubry, P. Holmes, J. Lumley and E. Stone, The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Fluid Mech. (1988) 115-173. [4] N. Bioembergen, Nonlinear Optics (Benjamin, New York, 1965). [5] F. Cotton and H. Salwen, Linear stability of rotating Hagen-Poiseuille flow, J. Fluid Mech. 108 (1981) 101-125. [6] A.D.D. Craik, J. Fluid Mech. 50 (1971) 393-413. [7] A.D.D. Craik, Wave Interactions and Fluid Flows (Cambridge Univ. Press, Cambridge, 1985). [8] H.P. Greenspan, The Theory of Rotating Fluids (Cambridge Univ. Press, Cambridge, 1968). [9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin, 1983). [10] J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. Soc. 103 (1988) 189-192. [11] J. Hale, Ordinary Differential Equations (Wiley, New York, 1969). [12] D.J. Kaup, The three-wave interaction- a nondispersive phenomenon, Stud. Appl. Math. 55 (1976) 9-44. [13] D.J. Kaup, A Reyman and A. Bers, Space-time evolution of nonlinear three-wave interactions. Interactions in a homogeneous medium, Rev. Mod. Phys. 51 No. 2. (1979). [14] S. Leibovich and A. Mahalov, Resonant interactions in rotating pipe flow, in preparation (1991). [15] A. Mahalov and S. Leibovich, Resonant Tollmien-Schlichting triad interactions in channels, in preparation (1991). [16] A. Mahalov, E.S. Titi and S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations, Arch. Rat. Mech. Anal. 112 (1990) 193-222. [17] I. Melbourne, Intermittency as a codimension three phenomenon, J. Dynam. Diff. Eq. 1 (1989) 347-367. [18] V.K. Melnikov, On the instability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12 (1963) 1-57. [19] T.J. Pedley, On the instability of viscous flow in a rotating pipe, J. Fluid Mech. 35 (1969) 97-115. [20] M.K. Proctor and C. Jones, The interaction of two spatially resonant patterns in thermal convection. Exact 1 : 2 resonance, J. Fluid Mech. 188 (1988) 301-335. [21] R.Z. Sagdeev and A.Ao Galeev, Nonlinear Plasma Theory (Benjamin, New York, 1969). [22] J.R. Usher and A.D.D. Craik, Nonlinear wave interactions in shear flows. Part 2. Third-order theory, J. Fluid Mech. 70 (1975) 437-461. [23] J.-M. Wersinger, J.M. Finn and E, Ott, Bifurcation and "strange" behavior in instability saturation by nonlinear three-wave mode coupling, Phys. Fluids 23 (1980) 1142-1154. [24] J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas (Pergamon, Oxford, 1977). [25] V.E. Zakharov and S.V. Manakov, Resonant interaction of wave packets in nonlinear media. Sov. Phys. JETP Lett. 18 (1973) 243-245. [26] V.E. Zakharov and S.V. Manakov, The theory of resonance interaction of wave packets in nonlinear media. Soy. Phys. JETP 42 (1976) 842-850.
Resonant triad interactions in symmetric systems John G u c k e n h e i m e r and Alex Mahalov Cornell University, Ithaca, NZ, USA
Received 4 February 1991 Revised manuscript received 9 August 1991 Accepted 27 August 1991 Communicated by A.C. Newell
We analyse resonant triad interactions in symmetricsystems. The case of coupling at quadratic order in amplitudes and the case of coupling at cubic order are considered. We study modulated travellingwaves and heteroclinic cyclesin amplitude equations and we discuss their stability types. We describe possible dynamical regimes and spatial patterns arising from triad interactions of long waves in rapidly rotating Hagen-Poiseuille flow.
1. Introduction Nonlinear wave interactions are an important phenomenon in fluid dynamics [7] as well as in plasma physics [21] and nonlinear optics [4]. We describe the phenomenon in the context of fluids. The Navier-Stokes equations governing fluid motion are nonlinear. If one expands a fluid velocity in a Fourier series, then products of individual modes appear in the equations and these serve to excite other modes and to couple sets of modes. The resulting interactions can be ordered by powers of amplitude. If the initial field has small amplitude that is scaled by a small parameter e, then the solution can be expanded as a formal power series in e. Each interaction occurs with an amplitude proportional to e ~. Here k is called the order of the interaction. The strongest interactions are resonant triads. These are triples of modes whose spatial wave vectors sum to zero and they have order k = 2. For this reason, one suspects that strong nonlinear effects will be observed in systems which are close to marginal stability for a set of modes that form a resonant triad. Fluid examples where this occurs can be found in ref. [7]. ~ The dynamics associated with resonant interactions can be complex and chaotic [23]. There have been several studies of these nonlinear systems with primary emphasis upon the study of conservative systems that include only the effects of linear excitation and quadratic coupling. The Navier-Stokes equations are not conservative, and one of our observations is that third order coupling terms are an essential part of the qualitative features of resonant triad interactions in dissipative systems. The third order expansions of the amplitude equations for resonant triad interactions in shear flows were derived from the Navier-Stokes equations by Usher and Craik [22]. Our approach to understanding the dynamics of resonant triad interactions relies heavily upon considerations of symmetry. In our analysis, we neglect most coupling between the phases of the interacting modes and examine only the effects of the amplitudes of the modes. More precisely, we assume that there is a symmetry on the phase spaces of the systems we study that serves to restrict the phase dependence of the system to a single linear combination of the phases. We also assume that there 0167-2789/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
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J. Guckenheimer, A. Mahaloc / Resonant triad interactions
is an additional reflection symmetry. In applications, these symmetries can be approximate, resulting from coordinate changes that reduce the original system to a "normal form", or they can be a consequence of spatial symmetries in the underlying system of PDE's. The consequence of symmetry upon the observed dynamical behavior of "generic" systems have been intensively investigated during the past several years [2, 3]. Two effects of symmetry are particularly dramatic: (1) symmetry can force the existence of invariant subspaces in the phase space of a system, (2) within the context of symmetric systems, new features of a flow can become structurally stable. Both of these aspects play an important role in the systems studied here. In particular, the subspaces corresponding to an individual nonzero mode are invariant. Since the quadratic interaction terms of the system vanish on these subspaces, the cubic terms of the resonant triad equations are needed to achieve structurally stable behavior within the invariant subspaces of the pure modes. The amplitude equations for resonant triad interactions studied in this paper are dZl
dt =YlZ2Z3WzI( [d'l÷elllZll2WelElzEl2÷el3]z3l
2)
'
dz2
dt = "Y2ZlZ~ + z2( Iz2 + e21[zl [2 + e221z21 z ÷ e23[z3[Z),
dz 3
dt = Y3z1z~ ÷ z3(]'£3 ÷ e31[Zl[2 ÷ e32[z212 + e33[z3[2) '
(1.1)
w h e r e / z j and ekj are real parameters and yj = _+1. These equations are equivariant with respect to the symmetry operations ( ZI, Z2, Z3) ~ ( Z~, Z~, Z~ ), ( Z 1 , 22 , 2 3 ) --'~ ( e x p ( i x D
z~,exp(ix2) z2,exp(ix3) z3),
provided X~ =X2 +X3.
Symmetries of this type are obtained from partial differential equations with spatial translation and reflection invariance. We note that there is a single relation between phases. It implies that the normal form is invariant under symmetry group of a two-dimensional torus T 2. We note that in the special case when two identical waves participate in the formation of a resonant triad, the amplitude equations have an invariant subspace corresponding to 2 : 1 resonance with 0(2) symmetry studied by Armbruster et al. [2], and Proctor and Jones [20]. This paper contains a partial analysis of the qualitative dynamics of the resonant triad equations. We place particular emphasis upon the important case in which there is a further symmetry that interchanges two of the modes in the resonant triad. For example, this occurs when a plane travelling wave interacts with two oblique travelling waves that are reflections of each other with respect to a plane containing the streamwise direction of the initial wave. For this restricted situation, there is an invariant subspace on which the equations of motion are identical to those studied by Armbruster et al. [2]. Much of our analysis for the general resonant triads uses the methods developed there. However, there is a much wider range of possibilities for the dynamical states of resonant triads than is found in the 0(2) symmetric systems studied in ref. [2]. In particular, multiple modulated travelling wave solutions are possible in some parameter ranges. Numerical evidence strongly suggests the presence of chaotic attractors in other parameter ranges. The scaling that is appropriate for the quadratic and cubic terms of the resonant triad equations depends upon the physical problem being studied. Normally, scaling of the variables at small amplitude
J. Guckenheimer, A. Mahalov / Resonant triad interactions
269
makes the cubic coefficients small relative to the quadratic coefficients. However~ there are problems, such as channel flow between fixed plates, in which the quadratic interaction terms of the equations vanish. If one changes the boundary conditions for channel flow to allow sliding of the boundaries relative to each other in the flow direction, then the quadratic interaction terms become nonzero. Thus, varying the relative velocity of the boundaries and the applied pressure gradient driving the channel flow produces a two parameter family of flows in which there will be regions where the quadratic interaction coefficients are small relative to the cubic coefficients. Thus both the limits of cubic coefficients vanishing and quadratic coefficients vanishing are relevant for fluid flows. The case of vanishing quadratic coupling coefficients reduces the triad equations to the normal form for a triple Hopf bifurcation with nonresonant frequencies [17]. We do not discuss in this paper the case of resonant triad interactions that allow spatial modulation of wave amplitudes. An inverse scattering transform for the conservative three-wave equations in time and one spatial dimension has been developed by Zakharov and Manakov [25, 26] and Kaup [12]. We refer to Kaup et al. [13] for a review. Three wave interactions with spatial modulation in dissipative systems are governed by coupled Ginzburg-Landau equations. Third order terms in these equations are essential to capture the qualitative features of the dynamics of resonant interactions. This paper is organized as follows. In section 2 we set up the resonant triad equations and discuss some of their basic properties. In section 3 we study modulated travelling waves. We use the method of averaging to find periodic orbits that are preserved after linear and cubic terms are included in the equation. Section 4 gives a full discussion of the dynamics on the real subspace and subspaces obtained from this subspace by applying elements of the symmetry group. We consider both the weak and the strong resonance case building on the earlier work of Guckenheimer and Holmes, [10], Melbourne [17] and Melnikov [18]. In sections 5 and 6 we are concerned with the existence of heteroclinic nets for the full system of equations and discuss their stability. In section 7 we analyze resonant patterns that occur in fast rotating Hagen-Poiseuille flow. We use the results obtained in the previous sections to describe different kinds of time-dependent behavior that can occur.
2. Preliminary remarks The normal form truncated at third order for an 0(2) equivalent vector field involving three modes forming a resonant triad can be written as
dz 1 dt
= Sl°Z2z3
-b
Zl(ld, 1 -b $111z112'+ Sl2lZ2l 2 + Sl3lZ312),
dz2 dt = s20z1z ~ d- z2(l& 2 q-- s21 [z 112 d- $221z212 d- s231z312), dz3 dt = S30ZlZ ~ + Z3(/~3 + S31 ]Z1[2 + S32[Z2[2 + S33[Z3[2).
(2.1) ,
Here ski are real coefficients and /z 1, ]/,2 and ~l,3 are unfolding parameters. Assuming that sk0 ~ 0 (k = 1, 2,3), we can rescale to obtain (1.1).
J. Guckenheimer, A. Mahalov / Resonant triad interactions
270
Letting z~. = r~ exp(i0y), the system (1.1) may be rewritten as d/1
dt = y~r2r3 cos 4' + r,(l~, + e~lr2~ + e,2r ~ + e,3r~),
d/2
dt = Y2rlr3c°s 4, + r2(/z2 +e2~r? + e22r~ + e23r32) '
dr3
2
2
dt -- 3,3rlr2 cos 4' + r3(/~ 3 + e3~r21+ e32r 2 + e33r 3 ),
d4'
dt = -
(
r2r 3
r~r 3
YIT+Y2-~-2
r~r 2 ~
+Y3T]
(2.2)
sin4,'
with 4' = 0~ - 02 - 03. Thus, in polar coordinates the system is reduced to four real dimensions. In our analysis we also use the real Cartesian form d/1
dt =3,~(x2x3-Y2Y3) + x~(Ix~ + e,r2~ + e~2r ~ +e~3r3~),
dYl dt =3q(x2Y3 +Y2x3) +Yl(/~ +e~lr21 +e~2r~ +e~3r32)' dx 2
dt =3,2(x~x3 +Y~Y3) +x2(/~2 +e2~r? + e22r22 +e23r~),
dY2 dt
= 3,2(-xIY3 +fix3)
dx 3
dt = 3,3(x~x2 + r
+Y2(/'/'2
+e2~r? + e22r~ + e23r32)'
y2) + x3( 3 + e3~r? + e32r~ + e33r32),
dy3 dt = 3'3(-x~y2 + y~x2) + Y3(/z3 + e31r2~+ e32r~ + e33r3~),
(2.3)
where r~2 = x~ + y~2. Reflectional symmetry implies that the purely real system (Yl = 0, Y2 = 0, Y3 = 0) is invariant. Furthermore, any 4,-invariant rotation of it is invariant. For example, the rotation Z l - ' exP(½iar) Zl, z 2 -~ ~ , z 3 --->exP(½i~r) z 3 transforms the real subspace to the Yl, x2, Y3 subspace (x~ = 0, Y2 = 0, x 3 -- 0). The scaling rj = esi, gj = e2vy with appropriate scaling of time transforms the polar system to ds 1
dr = ?~s2s~ cos ~ + es~( Ix~ + e.s2~ + e12s~ + e~s~ ),
ds 2 :
cos
+
+
+
ds~
+
d~" = 3,3s1s2 cos ~ + E83(]3,3 + e315 ~ + e32s ~ + e338 ~ ),
d4,
[
s2s3 + 3,~23 +
d~" = - [3,~-~-~
sts2 ~
3,3-~-3 )sin4,"
J. Guckenheimer, A. Mahalov / Resonant triad interactions
271
The limit e = 0 has three integrals known as Manley-Rowe relations (see, for instance, ref. [24]). These are E l ---- ~ l S l2 -- ~2S2, 2
E 2 ~ Y2s~ - yaS~,
L = SLS2S3 sin
~b.
Steady solutions or fixed points of (2.3) correspond to fixed points of (2.2) with ~b = 0 or xr. In addition to the trivial solution r~ -- r~ = r 3 0, we have pure modes (solutions for which two amplitudes are zero and one amplitude is nonzero) and mixed modes (solutions for which at least two amplitudes are nonzero). Fixed points of (2.2) with ~b different from zero or ~r correspond to travelling waves, in which the phase difference remains constant. Periodic orbits of (2.2) on the subspace ~b = 0, rr correspond to singly periodic standing waves while periodic orbits with ~b different from 0 and ~r correspond to doubly periodic, modulated travelling waves. A quasiperiodic solution of (2.2) corresponds to a triply periodic modulated travelling wave of (2.3). The origin is always a steady solution with eigenvalues/~1, /z~ and /~3, each of multiplicity two. We have invariant subspaces {z~--0, z 3 --0}, {z~ = 0, z3 = 0} and {z~--0, zz--0}. On these subspaces the system reduces to the equation (j -- 1, 2, 3) =
dzj
dt
--
zy(~y + e~ylz~.l~).
When/zy/e~. < 0, there are equilibria (pure modes) along the circles Iz~l = X/-/~-/e~y. The eigenvalues of these equilibria are 0 and -2/~. within the corresponding invariant subspace. The following proposition is proved as in ref. [2].
Proposition 2.1. If yt, Y2 and 'Y3 are positive (negative) then all solutions of (2.2), (2.3) are asymptotic to the invariant subspace {4, = O} ({~b = ~-}) or start on the invariant subspace {~b = "rr} ({4' = 0}). Proof Let us assume without loss of generality that y~ = Y2 ----')/3 ) 0. We consider the phase equation d~b/dt = - ( 3 q r 2 r 3 / r ~ + y2rtr3/r2 + y3rlr2/r3)sin ~b. Since the quantity in parenthesis is positive off the coordinate planes, we have that dd~/dt < 0 for ~b in the interval (0,at) and ddp/dt > 0 for ~b in the interval (rr, 2~r). Thus, d,(t) ~ 0 = 2~ unless solutions start with ~b(0) = "rr. The proposition is proved.
3. Modulated travelling waves
Let us consider the system ds 1
d~" = -s2s~ cos dp + es~(P~l + e~s~ + e~2s~ + e13532),
ds 2
= sis3 c o s ,
+
s2( 2 +
+
+
ds 3 dq" -~ - s 1 s 2 c o s ~ -+- Es3(/.g3 "+ e31s~ -j- e32522 "~ e33s32),
dck
( s2s3 s s3 slS2)sinck. =
-
-
+
s2
s3
(3.1)
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272
The system (3.1) appears in the rotating pipe flow problem. Integrals of this system in the case e = 0 are E~ = s~ + s~,
E~ = s~ + s~,
E3 = s2~ - s~ = E 1 - E2,
L = s~s2s3 sin 4~.
The phase space is the product of a circle S l (varying ~b) with the positive quadrant of the (s 1, s2, s 3) space. We note that the boundary of the phase space is not invariant under the flow and that the points (0, s~, s3, ~b) and (0, s2, s3, ~b + ~r), (Sl, 0, s 3, ~b) and (sl, 0, s 3, ~b + ~), (sl, sz, 0, ~b) and (Sl, s~, 0, ~ + ~) should be identified so that a trajecto~ exiting the b o u n d a ~ at (0, s2,s3, ~) reenters at the point (0, s~, s3, ~ + ~). The vector field is singular at these boundaries. The intersections of level surfaces of E~, E~ and L are transverse except at the b o u n d a ~ of the phase space given by s~s~s 3 = 0 and the surface {~ = 7~,1 s~s~~ - s~s~ + s~s ~ 3~ -- 0}. The latter surface corresponds to steady states of (3.1). On this surface we have L ~ = g(E1, E~) or L 2 = h(E~, E3) , where g( E , , E 2 )
= ~[-2E~+ 3E~E2+ 3E~E~- 2E~ + 2(E~-E,E 2+ E})3/z],
h( E , , E 3 ) = ~ [ 2 E ~ -
3E~E 3 - 3 E I E ~ + 2E~ + 2 ( E ~ - E , E 3 + E~)3/z].
We note that L ~ ~ g(E~, Ez) throughout the phase space. The case E~ = 0 (E~ = E~) corresponds to 2 : 1 resonance considered in ref. [2]. In this case we have L z ~ g(E~, E~)= ~E~. ~ t p = s~. Then for e = 0 we have do ]2 ~] =4[o(EI-p)(o-E,
+E2) -LZ].
(3.2)
Evolution equations for E~, E z and L for the perturbed solutions then may be written in terms of p, dE~d, = 2 e ( a l + ~10 + ~102),
~dE2 = 2 e ( a 2 + O z P + y 2 p 2 ) ,
~d L = e L ( ~ + Ap),
where O/1 ----"]d, 2 g 1 ÷ ( e 2 2 - e } ~ ) E ~
+ e23E1E2,
f l l = 1"/~1-- /"/'2 ÷ ( e l 2 ÷ e21 +
2e23-
el3 -
2ez2)E1 +
(el3 -
ez3)E2,
"Y1 ----e l l ÷ el3 ÷ e22 - e l 2 - e21 - e23,
a 2 = (/z2 - / x 3 ) E 1 +/x3E ~ + (ez~ + e33 - e~3 - e32)E~l + (ez3 + e32 - 2e33)E1E2 ÷ e33E~, ~ 2 = / ~ 3 - - / ~ z + ( e ~ l + 2 e z 3 + 2 e 3 e --
2e~z -
e31 - 2 e 3 3 ) E ~ + (e31 + 2e33 - ez3 - e 3 ~ ) E ~ ,
T2 = e22 + e31 W e33 - e21 - e23 - e32, ~ = ~1 + ~2 + ~3 + (el2 +
e~
+ e32 - el3 -
e~3 -
e33)E 1 +(el3 .
A=
e l l + e21 + e31 + el3 + e23 + e33 -- e l 2 -- e22 -- e32.
We define P ( p ) = -O 3 + ( 2 E 1 - E 2 ) p 2 + E ~ ( E 2 - e l ) p - L 2.
+ e23 + e 3 3 ) E 2 ,
(3.3)
J. Guckenheimer, A. Mahalov / Resonant triad interactions
273
We use the averaging method in our analysis [2, 9, 11]. For small e > 0 periodic solutions will be found near an unperturbed closed orbit when the following function with three components:
AE~ =
for dE~ dr
'
[ r d___Ez AE2 = J0 d r '
/rdL AL = J0 ~-~
has a nondegenerate zero. Here T is the period along the unperturbed orbit. The three components of this function can be written in terms of elliptic integrals
AEI = fCCtI + /31P + VlP 2 dp,
AE~= f ca~+/3zp+y2p~ dp,
zXL= I
dp,
where b = b(E~, E2, L) and c = c(E1, E2, L) are the middle and the largest roots of the cubic polynomial P(p). Using the identity f~ d ( ~ / f f - ~ ) = 0, we get c
p2
c
~
£ ¢N3S
fbc dp
V P)
Let us define I i = f~pi/lff~(P) do" Then we obtain
AE~
~---
1
2
[a~ + ~yxE~( E2-E~)]Io + [/31+ ~ y l ( E E ~ - E 2 ) ] I ~ ,
A E 2 = [a 2 + ½y2E~(E 2 - E~)]I 0 + [/32 + ~yE(EE~- E2)] Ia,
A L = ~ L I 0 +ALIa.
(3.4)
For nonzero L the condition AE 1 = 0, AE 2 = 0, AL = 0 becomes f~(E1, E2) = I [ % + ½y~EI(E 2 - El) ] - ~[/3~ + ~3q(EE 1 - E2) ] = 0,
(3.5a)
f z ( E , , Ez) = A[a z + ½7zEa(E2 - E,)] - 6[/3z + ]Yz(2E~ - E2)] = 0,
(3.5b)
I~ 6 f 3 ( E " E 2 ' L ) = ~0 + ~- = 0 .
(3.5c)
Eq. (3.5c) gives us the ratio of elliptic integrals I1/I o = - ~ / A . We note that the functions f l and f2 are independent of L. We can solve the system of equations (3.5) in two steps. First, we find positive values of E1 and E 2 satisfying (3.5a) and (3.5b). Second, we substitute this solution in (3.5c) and find L such that I i / I o = - ~ / A . The r a t i o - ~/A does not depend directly on L. Proposition 3.1 shows that I1/I o is an increasing function of L. These two facts allow us to write necessary and sufficient conditions for the solvability of the equation I~/I o = -15/A by checking the endpoints in the variation of the function I1/I o with respect to L.
Proposition 3.1. ~L ~o > 0 .
(3.6)
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J. Guckenheimer, A. Mahalou / Resonant triad interactions
Proof. We have (O/OL)(I~/I o) = [(OI~/~L)I o -l~OIo/OL]/Io z. If we define Ji = (1/L)(OIi/OL), then we obtain
0 (I~1 Jllo-I~Jo a--~ ~o = L Ig
(3.7)
We need the following identities:
-LZJi + E,(E 2 - El)J/+ 1q- (2E, - E2) Ji+ 2 - Ji+3 = li, 3
- ½E,(E 2 - E,)Ji+ l - (2E, - E2)Ji+ 2 + 7Ji+3
~-
(3.8)
- (i + l ) I i,
(3.9a)
3
- ½ E , ( E 2 - E , ) J o - ( 2 E l - Ez)J, + ~J2 = 0.
(3.9b)
The validi~ of these relations is proved by direct evaluation of the left hand sides using the definition of
Jk, P(P, L) and the fact that b(E1, E2, L), c(E,, Ez, L) are roots of the polynomial P(p, L). From (3.8) and (3.9a) with i = 0, I together with (3.9b) we obtain five linear equations for Jo, J,, Jz, J3 and J4 treating I 0 and I, as parameters. These allow us to express J0 and J, as functions of I 0 and I,. ~ e determinant of this linear system is E z '~ - 2 E ~ E ~ + E ~ E ~ - 4 E ~ L Z + 6 E ~ E 2 L ~ + 6 E , E ~ L 2 - 4 E ~ L 2 - 2 7 L 4 ) .
D=-~(E,
In the n e ~ lemma we show that the d e t e ~ i n a n t D is always negative except for the steady states.
Lemma 2.1. D < 0 except for the surface of steady states {~ = ~~ , ~z z - ~
+ s~
= 0t, on w~i~n
D=0.
Pr~f
Since L = p sin ~ ( p = s~szs3), we have
D = - ~' t~~' ~2~
~e~e~ + e?e~)
~(-4E~p~sin~6 + 6E~E~p ~sinZ4 + 6E,E~p ~sin26 - 4E~pZsin~6 -
-H~
27p 4 sin46)
- H z sin~,
where
4~ H , = ~~tt~4~2 = , = 2 _ 2E~E~ + EZE , 21,
H2 = ~(-4E~pZ +6E?Ezp2 +6E,E~pZ-4E~p z - 27p').
If H 2 > 0 then D < - H , = - ( E ~ E 2 - E , E ~ ) z < 0. If H 2 ~ 0 then - H 2 sin2~ ~ - H 2 and, therefore, ~ ~
1 ~2~2 -n, ~= -~(~.,~.~_ ~ _
+
+
~] [(,,+~)~+~]
and we have equali~ only in the case ~ = ~w. 1 From our analysis it follows that D = 0 when ~ = ~, 2 2 2 2 2 2 sxsz -s~s 3 + s2s3 = 0, and D < 0 in all other cases. ~ m m a 3.1 is proved.
and
J. Guckenheimer, A. Mahalov / Resonant triad interactions
275
Now we can conclude our proof of proposition 3.1. If we are not at the steady state, the determinant D is nonzero and we can express Jo and J~ in terms of I o and I v We find that 1 Jo = --~( PlI1 --P210),
1 J~ = - ~ ( P211 +P3Io) '
pl = 2E2x - 2E~E 2 + 2E~,
P2 = 2E~ - 3E~E 2 + E~E~ + 9L 2,
(3.10)
P3 = 4E3~Ez - 2E4~ - 2E~E22- 12E, L2 + 6E~ L2"
From (3.7) and (3.10) we obtain a Riccatti equation for I1/Io:
~-~ ~oo = - ~ -
Pl ~o
- 2p2 ~o
"
With an easy calculation one can show that the discriminant of the quadratic expression for I ~ / I o is 3D. Thus, from lemma 3.1 it follows that the discriminant is always negative except for the steady states. Now it follows that (a/OL)(I1/I o) > 0 since Pl > 0. Proposition 3.1 is proved. Now we find bounds on the ratio I ~ / I o. Fix El, E 2 and let L 2 vary. The variable L 2 can be varied in the interval 0 < L 2 < g ( E 1, E2), where g( E 1 , E 2 ) -- 2~(-2E~3 + 3E2~E~ ÷ 3E~E22 - 2E~ ÷ 2(E2~- E1E 2 ÷ E22)3/2).
We note that D = 0 at L 2 = g(E~, E2). We find that
~-~ = P-~ -- ~(2E1- E~ + ¢ ~ - el~2 + ~ ) Pl
Io
at the right endpoint L z = g(E 1, E2). At the left endpoint (L ~= O) the integrals I 0 and I~ can also be evaluated. We have if E~ <_E2,
/I=2V/~-q-~E
E2
÷
~-~
Io = --~
'K[~/~-~
K
i f E 2 < E 1.
if E~ _
(3.11)
Here K is the complete elliptic integral of the first kind and E is the complete elliptic integral of the
276
J. Guckenheimer, A. Mahalov / Resonant triad interactions
second kind. From (3.11) we obtain the value for I 1 / I 0 a t L ~. = 0:
I1
--'°,, ~
(
I-~ = E2 -- E1 + E 1
if E 1_
Since l , / I o is an increasing function of L and - 8 / h is independent of L, the equation l l / I o = - 8 / h has a solution for L ~ between 0 and g(E1, E 2) if and only if
E2 -El --
I+E
/K E, ) < (~E~)(~E~
~ < ~ I [2 - ~ +E z X~:l
E , I ' - ~ E, V/[I,~-~~]
+I
]
(3.12)
in the case E 2 _
El
E-~ - 1 + E
(E~E~) / K (E~E~) <
(~
I[
A ? , < X [2E~ - 1 +
~//(E1 )2_ NE1 + 1 ] ~22
(3.13)
in the case E l _
F(x)=x- l +
e(v;)
G(x)= ½(2-x +
/x2-x +1),
H(x)= ½(2x- l +
/x2-x + l ),
that appear in (3.12) and (3.13). Graphs of these functions are shown in figs. 1 and 2 respectively. We see that F(x) < G(x) and F(x) < H(x) for x in the interval [0, 1]. The periodic orbit with parameters (El, E 2) and with appropriate L is preserved under cubic perturbations if and only if f~(E1, E 2) = 0, f2(E1, E 2) = 0 and one of the inequalities (3.12) or (3.13) is satisfied. The functions f~(E 1, E 2) and f 2 ( g l , g 2) are quadratic in E 1 and E 2. This implies that no more than four periodic orbits can survive under perturbations. Thus, we have proved the following theorem.
Theorem 3.1. Suppose that the second order curves given by the equations fl(El,
E2) = ~[o/1 -]- ½ T , E , ( E 2 - E,)]
- ~[~I -]- ] T I ( 2 E I - E2)] = 0,
f 2 ( E l , E2) = ~[0/2 -~" }~2EI(E2 - El)] - ~[~2 ~- ~T2(2El -
E2)] =
0
(3.14)
are in generic position (i.e. have nondegenerate intersection). Suppose that ( E 1, E 2) is a solution of (3.14) satisfying (3.12) in the case 0 _
J. Guckenheimer, A. Mahalov / Resonant triad interactions
277
11
0.8 ¸
0.6
0.4-
0.2-
Fig. 1. Graphs of the f u n c t i o n s F ( x ) and G(x). Note that F(x) < G(x) for x between 0 and 1 (0 < x < 1).
0.50.4-
0.30.20".i~
,
,
0.4
0.6
0.8
1
Fig. 2. Graphs of the functions F(x) and H(x). Note that F(x) < H(x) for x between 0 and 1 (0 < x < 1).
Let us consider the special case studied in ref. [2]. The system of equations (3.1) has an invariant subspace defined by s 1 = s 3 if/~i =/z3, et~ + el3 = e3r + e33 , e~2 = e32. Let us denote e~.l + e~3 = e31 + e33 by e~l and e2~ + e23 by e2v Then the system of equations restricted to this invariant subspace becomes dsl
d z = -s~s2 cos ~b + es~( l~x + ellS~ + et2s~ ),
ds 2
dz = s12 cos d' + es2(/z2 + e21s21+ e22s22),
d~b
d---~ = -
1
sa / - 2 s 2 + s 2]/ sin ~b.
(3.15)
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J. Guckenheiraer, A. Mahalov / Resonant triad interactions
In this case we have fl(E1, E 2) = fE(El, E2) and we define Solutions of the equation f(E) = 0 are E =
- 3 ( 2 / ~ 1 d- /22) 4ell + 2e12 + 2e21 + e22 '
f(E) = f l ( E l , E 2) = f2(El, E 2) at E = E l = E 2.
g ~- /22 - / 2 1 el2 - e22 •
From theorem 3.1 it follows that the periodic orbit on the energy surface E survive perturbations if and only if ~
2
0 < - ~-~ < 7.
(3.16)
In the case E = -3(2/21 + / 2 2 ) / ( 4 e l l + 2e12 + 2e21 + e22) we find that - ~ / A E -- 3. Thus, in this case the inequality (3.16) cannot be satisfied. In the case E = (/22 - / 2 1 ) / ( e 1 2 - e22) the inequality (3.16) becomes 2 0 < (/21e22 - ~2e12)/,~(/2 2 - / 2 1 ) < ~. W e also note t h a t E = (/22 - / 2 1 ) / ( e 1 2 - e22) s h o u l d b e positive. Thus, the perturbed system of equations (3.15) possesses one periodic orbit if and only if 0 < /21e22- - /22e12 < ~-, 'h'(/22 --/21)
//,2--/1,1
> 0.
(3.17)
e12 -- e22
The inequalities (3.17) can be easily analyzed and we obtain the following
Theorem 3.2. Suppose that one of the conditions listed in the left column of table 1 is satisfied. Then the perturbed equation (3.15) possesses a periodic orbit in the region of the parameter space defined by the inequalities given in the right column of table 1.
4. The system of equations (2.1) on the real subspace The real subspace of (2.1) and subspaces obtained from the real subspace by applying elements of the symmetry group play a very important role in the overall dynamics. In this paragraph we discuss the dynamics of (2.1) restricted to the real subspace. In some cases symmetries in the problem result in the fact that second order coupling coefficients in the amplitude equations are identically zero. For example, symmetry with respect to the midplane requires quadratic coupling coefficients to vanish identically in the equations describing resonant triad interactions of the Tollmien-Schlichting waves in channel Poiseuille flow [7, 15]. In other cases, quadratic coupling coefficients are small compared with cubic coupling coefficients and, therefore, quadratic terms can be treated as a small perturbation. In the case of channel flow the symm.etry can be broken in a natural way by sliding one boundary of the channel with respect to the other with a small constant velocity. Then the base flow is called plane Couette-Poiseuille flow. In the case of small symmetry-breaking perturbations or in the case when quadratic coupling coefficients are small compared with the cubic coupling coefficients, the quadratic terms can be treated as a perturbation of a system with
279
J. Guckenheimer, A. Mahalov / Resonant triad interactions Table 1 Regions of existence of periodic orbits in the parameter space.
A > O, e12 < O, {3. + el2 > O, el2-- e22 > 0
e22 ~.~ > O, P-2 > e~--~Zl
A > O, el2 < O, {A + e~2 < O, et2 - e22 > 0
//'1 > O, e22 < ~2 < e22 + {A e~---~/-~ e~2 + ~ 1
A <0,
e12<0,
e12-e22>O
~1 > O,
A
e12>O ,
{A+eI2
~1 >0,
A>O,
e12
{A+el2>O,
e12-e22
~x>O, ~2< e ~
A >0,
e12<0,
{A+el2
e12--e22
~1 > O, e22 + {A e22 e~x+{A ~t <~2 < ~e12 ~
A <0,
e12
e12-e22
A <0,
e12>O,
{A +e12
A>O,
e12>O,
e~2-e22>O
A>O,
e12
{A+e~2>O,
A <0,
e12>O,
{A+e12
A >0,
e~2>O,
e~2--e22
A > 0,
e12 < 0,
{A + e12 > 0,
e12- e22 < 0
~1 <0,
A <0,
e12>0,
{A+e12>0,
e12-e22<0
~1 < O,
e22 + ~2~1 < ~2 < e2~ el2 + el 2 ~1
~<0,
e2~ ~2< et2~ ~
e22 + ~ ~ e~2 + {A ~ <~2 < ~ e22 + ~A ~2 > el 2 + { ~ 1 e22
~>0,
e2~ <~2 < ~e22 + {A e~2~l e~2+ {A ~
~1 >0,
e22 + {A ~2 < et 2 + ~ ~
~<0,
e22 + ~A e22 e~2+~A~l <~2 < ~e~2 ~
e~2-e22>O
~<0,
~2 > ~ ~12 + {A ~1
e12-e22>O
~ < O, ~2 > e2~ e12 ~1
e12-e22
~22 ~1 < O, ~ 1
A < O, e~2 > O, {A + e~2 < O, e~2- e22 < 0
~ e22 + ~A ~ ~2 < el 2 + { ~ 1
e22 + {A ~2 < el 2 + { ~ 1
J. Guckenheiraer,A. Mahalov / Resonant triad interactions
280
cubic terms only. In the case when quadratic coupling coefficients are not small we use the rescaling of amplitudes zi -~ ezy and unfolding parameters /xy ~ evy to get a system of equations in which the odd degree terms can be considered as a small perturbation. In both cases the unperturbed case can be analyzed and used to understand the dynamics of the full system. The general case lies between these two systems in which the quadratic terms are small compared with the odd degree terms and vice versa. We find modulated travelling waves and heteroclinic cycles in perturbations of the system with only quadratic terms, and we find heteroclinic cycles in perturbations of the system with only cubic terms. We conjecture that in the general case both modulated travelling waves and heteroclinic cycles are possible.
4.1. Weak resonance We assume that symmetry in a problem that forces quadratic coupling coefficients to be identically zero is broken. Then the system of amplitude equations describing resonant triad interactions has the following form: dz 1
dt
---//'lZ1 q- es~°z2z3 + SllZ1 [ZI [2 + s12z11z212 + s~3z~ [z312,
dz 2
dt = Iz:z2 + es2°z~z~ + s21z2lz112 + s22z2[z212 q- S23Z21Z312'
dz3 dt ---/d,3z3 q- Es30z1z ~ --ks31z 3 Iz I 12 + $32z3 Iz212 + s33z31z312. Here e is a small parameter in the problem. In the case of channel Couette-Poiseuille flow e is proportional to the size of a symmetry breaking perturbation (sliding). This problem is studied in ref. [15]. We note that the coupling coefficients in this problem are complex. Nevertheless, for e = 0 the phases are decoupled from the amplitudes and we obtain a three-dimensional system of ordinary differential equations with real coefficients for the amplitudes r~. = Iz~l, The results obtained in this paragraph can be applied to this system of equations. For e = 0 the system of equations with 0(2) symmetry restricted to the real subspace becomes dx 1
dt
= Izlx~ + SllX31 "b Sl2X1X ~ + s13xlx~,
dX2dt =b~2x2 +s21x2x~ +s22x~ +s23x2x~' dx 3
dt
=/z3x3 + s31x3x~ -[-s32x3x~ -[-$33x33"
(4.1)
It turns out that this system of equations has the same form as the system of equations studied in refs. [10, 17]. The system of equations has two-dimensional invariant subspaces {x 3 = 0}, {x 2 = 0} and {x~ = 0}, and is equivariant with respect to the transformation x~--> -x~. Solutions remain in an octant, and it suffices to consider this system of equations in the domain {x~ > 0, x 2 > 0, x 3 > 0}. The axes x~, x 2, x 3 are invariant under the flow. Provided that ~1/s11 < 0, /z2/s22 < 0, /~,3/$33 < 0, we have three steady
J. Guckenheimer, A. Mahalor' / Resonant triad interactions
281
Fig. 3. Heteroclinic cycle between the steady states A, B and C (schematic).
states
.=(0, lying on the invariant lines x 1, x 2 and x 3. We give necessary and sufficient conditions for the existence of a heteroclinic cycle between the steady states .4, B and C with the property t h a t each heteroclinic connection lies in the corresponding two-dimensional invariant subspace as shown in fig. 3. We also investigate stability of heteroclinic cycles.
Lemrna 4.1. A heteroclinic cycle between the steady states A, B and C can exist only if all the coefficients sij (i, j = 1, 2, 3) have the same sign.
Proof. Guckenheimer and Holmes [9] gives a full description for the dynamics of the system of equations (4.1) restricted to the two-dimensional invariant subspaces {x 3 = 0}, {x 2 = 0} and {x I = 0}. From their results it follows that heteroclinic orbits connecting steady states in the two-dimensional invariant subspaces exist only in the three cases la, lb and 5a in their classification. We will call a heteroclinic connection corresponding to their case la a type 1 connection; a heteroclinic connection corresponding to their case lb a type 2 connection; and a heteroclinic connection corresponding to their case 5a a type 3 connection. In type 1 and type 2 connection, all the coefficients have the same sign (either positive or negative). First, we show that if a heteroclinic cycle connects A, B and C, then in the two-dimensional invariant subspaces {x I = 0}, {x 2 = 0} and {x 3 = 0} heteroclinic connections of type 1 or type 2 but not type 3 occur. The proof is by contradiction. Suppose that a type 3 connection occurs in at least one two-dimensional invariant subspace. Then the following two cases are possible: (a) Type 3 connection occurs in at least two subspaces. (b) Type 3 connection occurs in exactly one subspace. Suppose that a type 3 connection occurs in at least two subspaces. Then there are subspaces with the type 3 connection that are adjacent to each other. By inspection of the type 3 connection we find that the heteroclinic orbit always goes in the direction from the steady state that is unstable on the invariant axis on which it sits to the steady state that is stable on the invariant axis on which it sits. Let us now consider the steady state lying on the invariant line separating two adjacent two-dimensional subspaces where the heteroclinic connection is of type 3. Since we have a heteroclinic orbit in one of the subspaces, the heteroclinic orbit should enter this steady state. Therefore, this steady state must be stable on the corresponding invariant line. On the other hand, the heteroclinic orbit in the other subspace is leaving this steady state. Therefore, since the heteroclinic connection in this subspace is of type 3, the steady
282
J.
Guckenheimer,A. Mahalov / Resonant triad interactions
state must be unstable on the invariant line. It cannot be stable and unstable at the same time, so a contradiction is obtained. Therefore, case (a) is impossible. Now we suppose that case (b) is realized. Without loss of generality we assume that a heterodinic connection of type 3 is in the two-dimensional invariant subspace {x3 = 0} and heteroclinic connections either of type 1 or of type 2 occur in the subspaces {xt = 0} and {x 2 --- 0}. Inspecting the type 3 connection, we find that the unfolding parameters /£1 and /£2 always have opposite signs in the case of a heteroclinic connection. On the other hand, inspecting the type 1 and the type 2 connections we find that the unfolding parameters /£1 and /£3, /£2 and /£3 always have the same sign implying that /£1 and /z 2 are of the same sign. Thus, we have a contradiction in the case (b) as well. We conclude that each two-dimensional invariant subspace has either a type 1 or a type 2 connection. It remains to show that the signs of all the coefficients are the same. So far we have shown that signs of the coefficients in each two-dimensional invariant subspaces are the same. Without loss of generality let us assume that all the coefficients in the invariant subspace {x 3 = 0} are positive. Then we have a 1 > 0, b 2 > 0. Since the signs of the coefficients in the subspace {x 2 = 0} are the same, then c 3 has the same sign as a 1 and, therefore, c 3 > 0. Thus, a 1 > 0, b 2 > 0, c 3 > 0. Now it follows immediately that all the coefficients a~., b~ and c~. are positive. The lemma is proved. Thus, in our derivation of conditions for the existence of a heteroclinic cycle we can restrict ourselves to the case when all the coefficients sii have the same sign. Assume that all the coefficients are negative. Necessary and sufficient conditions for the case when the coefficients are positive are obtained from this case by reversing time t ~ - t and the unfolding parameters/£1 " ~ ' --/£1, /£'/'2" ~ --/£2, /£3 - - ~ --/£3" Let a I ----- - - S l l , b I =
-s12
, c 1 =
-s13
a 3 =
-$32
~c3 =
-$33.
-$31
, b 3 =
~
a 2 =
-$21
, b 2 =
-$22
~c2 =
-$23
,
Then we have the following system of equations: dXl d"~- = / £ 1 X l
_ a2x
~ -
b ~ x l x ~ - ClXlX ~,
dx2 dt =/£2x2 _ a2x2x21 - b2x32 - c2x2x~, dx3 dt
=/£3x3
_ a3x3x ~ - b3x3x ~ - c3x ~.
Theorem 4.1. Suppose that a j, b~. and c~ are positive numbers. There exists a heteroclinic cycle between the steady states A--(/£~1
,0,0),
B=(0,
/£~2-~2,0), a n d C = ( 0 , 0 , / £ ~ 3 3 )
if one of the sets of conditions listed in the middle column of table 2 is satisfied. A heteroclinic cycle exists in the region of the parameter space defined by the inequalities given in the right column of table 2.
Proof. The eigenvalues corresponding to the steady state A = ( ~ l V / - ~ - ~ , 0 , 0 ) are -2/£1,/£ 2 (a2/al)/£1,/£ 3 - ( a 3 / a l ) / £ 1 , the eigenvalues corresponding to the steady state B = (0, V/--~z/b2,0) are
J. Guckenheimer,A. Mahalov / Resonant triad interactions
283
Table 2 Regions of existence of heteroclinic cycles in the parameter space. la
a2
b-~a-~>l,
cl a3 ~-3~-i > 1 ,
c2 b3 ~-~2>1,
a2 ct b3 al c3b2 < 1
g,Z>~-lp,,,
~ > b~z~z, ~ > ~
lb
b~ a 2
b~ a~ > l, cS a~ > l,
c~ a~
c2 b3 ~ > 1 ,
a 3 b~ c 2
a~ b~ c~ < l
a3 ~3>~1,
b~ cz ~ > b-~u,z, u,2 > ~3~3
2a
bl a2
c~ a~
c 2 b~ ~ < 1 ,
a2 c3 < 1
~2 > ~~2,
2b
3a
3b
bt a2
b2a~
>1,
b~ a 2
bS~>x,
c3a~ < 1 , cI a 3
c2 b 3
~aC<~,
E~
b~ a 2 b~a~
c~ ~ a3 a, > 1 ,
EE
bl a2 b~a~
c~a~>l,
Cl a3
c2 b3 ~<1,
c2 b3
4a
b~ a 2 cI a3 bS.S<~, cSaS
4b
b~ a 2 b2 al < 1,
5a
5b
6a
6b
cI a3 ~c~ ~, < 1,
b~ a 2 cI a3 b~a~
b2atb~a~< 1 , b~ a 2
bS~S
cI a3
b~ c 3
b~
~,>E~,
Ec~<~ b 2 c~
<1
EE
b2 a3 ~ < 1
c 2 b3
b2 g2>~g3,
a~ c 2
c3
E(<~
~>l, c2 b3
b2
~> ~ '
~>~,,
a~ b 3 > 1,
E ~ c2 b3 ~ < 1 ,
at <
E E
~
a t b 2 c3
a~b~c~
c 2 b3 ~ < 1 ,
c2 b3
b 2 at c3
~bta3c~ <1 b3 cI
~>1,
~>~,
E~<~
~, > ~ ' al ~1>~2,
~2>~,
b3
~>~,
C3
~ > ~'~ c3
bz /.I,2 > ~3~3, C1
~ > ~,
hi'3 > ~1~1 C3
~ > ~2t~
a3
~ > ~(~,
a1
~ > ~2~
a! C2 ~ > ~,~, ~ > ~,~
b3
~ > ~2~,
b~ "
t~ > ~ - ~
b~
c~
u-~ > b~3~,~, ~ > ~ al /.t 1 > ~3/./,3, b:,
c3 /.1,3 ). ~2~/,2 c1
~2> ~~1,
/~1 > ~3~3 ~ c2 /.,I,2)" ~3#,3
cS aS > L E ~ > ~, ~ ~ < ~
Cl a3
c2 b3
a3 c2
~ > ~,,
al ~./'1 ~> ~2#'2,
7a
bt a 2 bEal >1,
cI a3 -c3 - ~al <1,
c2 b3 ~>1,
b~ c 2 ~c ~1 < 1
b~ ~ > ~g2,
~2 > ~3~3, ,3 > ( ~
7b
b~ a 2 b2a~ > 1 ,
cI a3 ~ < 1 ,
c2 b3 ~>1,
a 2 b3 ~ < 1
a2 ~2>~1,
~>~3,
~3>~
8a
bl a2
b~a~>l,
~c~ ~al> 1 ,
~
Cl //~1 ~> ~33~3,
C3 ~3 > ~2/'t2
bl a 2 bz a 1 > 1,
cI a3 ~ ~ > 1,
c~ a~
c2 b3
c2 b 3
~ E
a2 cl
<1,
~E
< 1,
~ E
a 3 bl
a3
al
m > ~3,
c1
b~ ~
~
bt a~ < 1,
c~a~C~a3 <1,
<1,
a~c~
b3
a2
<1
~ > a~"l,
< 1
~ > ~,
-
-
a3
172
at
bl
t~t > ~ ' 2 ,
C3
b3
b2
~2 > ~ 3
284
J. Guckenheimer, A. Mahalov /
Resonant triad interactions
I.Xl-(bl/b2)Iz2,
- 2 / z 2, Iz3-(b3/b2)tz2, the eigenvalues corresponding to the steady state C = (0,0, #3VC#-----3~)are ~i -(c~/c3)ix3, ~ 2 - (c2/c3)P.3, --2/z3- It is clear that the steady states A, B and C are stable on the corresponding invariant lines since the eigenvalues in these directions are -2/.~1, -2/x 2 and - - 2 g , 3. The remaining two eigenvalues are responsible for the stability in the two dimensional invariant subspaces. We have shown that if a heteroclinic cycle exists then heteroclinic connections in two-dimensional invariant subspaces are either of type 1 or of type 2. There are the following possibilities: (1) The heteroclinic connection in each invariant subspace is of the type 1. (2) The heteroclinic connection in one invariant subspace is of type 2 and heteroclinic connections in two other invariant subspaces are of type 1. (3) The heteroclinic connections in all three two-dimensional invariant subspaces are of type 1. (4) The heteroclinic connections in two invariant subspaces are of the type 2 and the heteroclinic connection in one invariant subspace is of type 1. We consider case (1): all heteroclinic connections are of type 2. Then the coefficients satisfy [9]
b_~_la__22 b2 al > 1 ,
¢1 a3 c--~a-~-> 1 '
c2 b3 c3 b2 > 1 .
Analyzing the type 2 connection we find that in case (1) a necessary and sufficient condition for the existence of a heteroclinic cycle is that each steady state A, B and C has at least one unstable eigenvalue in each two-dimensional invariant subspace. This is equivalent tO the condition that the following logical expression is true: AND
, We find that this logical expression is true if and only if one of the following logical expressions is true:
a3
bl
[(#3 > ~ll g . , ) A N D (/.Ll > "b-~2/.d.2)AND (~2
> C2 ~3)]"
The first of these expressions is valid if and only if (a2/alXCl/c3Xb3/b 2) < 1. The second expression is valid if and only if (a3/alXbl/,b2Xc2/c 3) < 1. Thus, we proved the statements la and lb of the theorem. Now we consider case (2): in one two-dimensional invariant subspace the heteroclinic connection is of type 2 and in two other two-dimensional invariant subspaces heteroclinic connections are of type 1. Without loss of generality we assume that the heteroclinic connection in the subspace {x 3 = 0} is of type 2 and the heteroclinic connections in the subspaces {x 2 = 0} and {x 1 = 0} are of type 1. Then we have (bl/b2)(a2/a l) > 1, (Cl/C3Xa3/al) < 1, (c2/c3)(b3/b2) < 1. Necessary and sufficient conditions for the existence of a heteroclinic cycle are that there be a heteroclinic connection in each subspace and that each steady state A, B, C has at least one unstable eigenvalue. The condition that we are in the region of existence of a heteroclinic orbit in the first subspace is [/z 2 > (a2/al)izl] OR [/x 2 < (b2/bl)~z~]. The
J. Guckenheiraer,A. Mahalov/Resonant triadinteractions
285
condition that we are in the region of existence of a heteroclinic orbit in the second~ subspace is [/x3 >(c3/c~)lx~] OR [/x 3 <(a3/a~)laq]. The condition that we are in the region of existence of a heteroclinic orbit in the third subspace is [/~3 > (c3/c2)l-~2] OR [/~3 < (b3/b2)tx2]. Now we can easily find conditions guaranteeing that at least one eigenvalue is unstable for each of the steady states A, B, C. The condition that the steady state A has at least one unstable eigenvalue is [~2>(a2/al)l~] OR [/z3> (a3/a~)pq]. The condition that the steady state B has at least one unstable eigenvalue is [l~ > (bl/b2)~2] OR [~3 > (b3/b2)lx2]. The condition that the steady state C has at least one unstable eigenvalue is [/x~ > (¢1/c3)]A,3] OR []'/'2 > (C2/¢3)]'/'3 ]" Then a necessary and sufficient condition for the existence of a heteroclinic cycle in the case (2) is that the following logical expression is true:
~,,)] ~o ~,)o~(~,> ~,~,)]~o
~--~/Zl) O R ( / x 3 < a 3 > C3
b3
> bl
b3
a3
<
C2
~.~to~(.~ ~.~tl.
It turns out that this logical expression is true if the following logical expression is true:
ANo(~3>~2)]o. ~ / x l ) AND a~ I(.~~°~ (", > ~"~t c3
b~ b2 > ¢3 [(/~1> ~2/z2) AND (/z2> ~3~3 ) A N D (~/,3 t The fiLst alternative holds if and only if (a2/a3Xc3/c2) < l. The second logical expression holds if and only if (bl/b3Xc3/c 1) < 1. This proves statement 2 of the theorem. Statements 3 and 4 are proved with similar arguments. Now we consider case (3): all three heteroclinic connections are of type 1. This implies that b___~a__~2 b2 al <1,
cl a 3 c-~a~-< 1 '
¢2 b3 c3 b2 <1.
We have a hctcroclinic cycle if each steady state A, B, C has at least one unstable eigcnvaluc and we arc in regions of the parameter space (/z l,/z 2,/~a) where heteroclinic connections in each two-dimensional invariant subspaccs occur. This is equivalent to the following logical expression: a2
~2/z2) OR bl
b2
> ca b3
AND [(~,~ ~,)°' O. (~, > ~,)]c, ~D
~O[(~. >a2~,)O. (~.> ~,)]~O a, ~o[(~,>c~~)o~ (~>~)]. ~
2%
J. Guckenheimer,
A. Mahalov /Resonant
triad interactions
This logical expression is equivalent to
From the first expression we get the condition (a,/a,Xb,/b,Xc,/c,) < 1. From the second expression we get the condition (b,/b,Xa,/a,Xc,/c,) < 1. Thus, we have proved the statements 5a and 5b of the theorem. Now we consider case (4): two invariant subspaces have heteroclinic connections of type 2 and one subspace has a heteroclinic connection of type 1. Without loss of generality we assume that the subspace {x3 = 0) has a heteroclinic connection of type 1. The coefficients satisfy the following relations: b,
a2
iqiy
c’fL>l, Cl,
c3
??>l.
a,
There is a heteroclinic
3
2
cycle if
(lL3>%1”2)AND(1”2>~P~)AND(~~‘~~3)oR P3 > $,
AND )
PI>
$2
AND )
(
(
From the first expression we find that the coefficients must satisfy (b,/b,Xc,/c,) < 1. From the second expression we find that the coefficients must satisfy (a,/a,)(c,/c,) < 1. Thus, we proved the statements 6a and 6b of the theorem. Statements 7 and 8 can be proved in a similar fashion. Theorem 4.1 is proved. The paper of Guckenheimer and Holmes [lo] contains the first example of structurally stable heteroclinic cycles. In our notation, their sufficient conditions for the existence of a heteroclinic cycle are a,=b2=c3=a,b,=c2=a3=b,c,=a2=b,=c,~,=~2=~~=1,c>a>b.Wefindthatifbc/a2>1 then condition lb of the theorem 4.1 is satisfied, if bc/a2 < 1 condition 5a of theorem 4.1 for the existence of a heteroclinic cycle is satisfied. But one of these conditions is always satisfied. Thus, we have a heteroclinic cycle. Melbourne [17] also gives sufficient conditions for the existence of a heteroclinic cycle. In our notation, his sufficient conditions are a, = 1, b, = 1, c3 = 1, b, < 1, c2 < 1, a3 < 1, a2 + b, > 2, c2 + b3 > 2, c, + a3 > 2. From these inequalities it follows that a2 > 1, b, > 1, c, > 1. Then we obtain a,b,c, < 1, bl/b3
a3/a2
<
b3,
c2/cI
<
a2,
(l/a2Xl/b3Xl/cl)
< 1, a3/a2 < l/c,,
b, < cJcZ, ~3 < b,/b,. Thus,
the fourth condition in lb, 2b 3b, 4a, 5a, 6b, 7a and 8b of theorem 4.1 is satisfied. Therefore, conditions of theorem 4.1 is necessarily satisfied and we have a heteroclinic cycle. Now we discuss stability of heteroclinic cycles. We have
one of the
Theorem 4.2. Suppose that one of the conditions of theorem 4.1 for the existence of heteroclinic cycles is satisfied. Then the regions of the parameter space where these heteroclinic cycles are attracting (repelling) are given in the right column of table 3.
Z Guckenheimer, A. M a h a l o v / R e s o n a n t
triad interactions
287
Table 3 Regions of stability of heteroclinic cycles in the p a r a m e t e r space. la
Attracting (fig. 4, la)
lb
Attracting (fig. 4, lb)
2a
Attracting: inside the cone S (fig. 4, 2a). Repelling: inside the cone U (fig. 4, 2a).
•: (~,- ~)(~-~)(~-~) - (~- ~,/(~- ~)(~,- ~/~0 2b
t? I
bl
c2
Attracting: inside the cone S (fig. 4, 2b). Repelling: inside the cone U (fig. 4, 2b). (~12.ttt
•: 3a
b3
''b 3
'[c,
-,~#tr~,~-l)t
_p.,)
~
a3
_
('-~,~,t(~, ~ ) ( ~ - ~/~°
Attracting: inside the cone S (fig. 4, 3a). Repelling: inside the cone U (fig. 4, 3a).
•: (~,- ~)(~-~,)(~-~1 - (~- :+,/(~- ~)(~,- ~t ~0 3b
Attracting: inside the cone S (fig. 4, 3b). Repelling: inside the cone U (fig. 4, 3b). a2 b3
(c~-~,t
4a
a3
b3
c1
b~
c2
Attracting: inside the cone S (fig. 4, 4a). Repelling: inside the cone U (fig. 4, 4a). a2 b3
a3 _ b~ c2 = 0 (~-~,/ -(~-~,~,/(", ~"~)(-~-~t
4b
Attracting: inside the cone S (fig. 4, 4b). Repelling: inside the cone U (fig. 4, 4b). a3 c2
,(~,-~t(~-~,)(~-~1-(~-~,1(~ 5a
_ b.3
cl
~t(~,-~1 ~°
Repelling (~g. 4, 5a).
5b
Re~elling (fig. 4, 5b).
6a
Attracting: inside the cone S (fig. 4, 6a). Re~elling: inside ~he cone U (fig. 4, 6a).
,: (~,_~(~_~,~(~_~_(~_ ~,~(1 ~(~,_~0 a3
6b
c2
-- b 3
Attracting: inside the cone S (fig. 4, 6a). Re~elling: inside ~he ~ n e U (fig. 4, 6b). a2 b3
.: (~,-~(~.~-,)(~-~,~-(~-~.,~(~, 7a
7b
Attracting: inside the cone S (fig. 4, 7a). Repelling, inside the cone U (fig. 4, 7a). [ a2 ~[ b3
.:t~,-~t~-~)(~ -~,)- (~- ~,)(~,-
b~ ~)(~_ ~c~, ~o
_~/_(~_:+~/(,_~~ ) t ~,,, - ~ ) - 0~,,
Attracting: inside the ~ n e S (fig. 4, 8a). Repelling: inside the ~ n e U (fig. 4, 8a). [a 3 ,[b, )(c 2
.: t ~ , - ~ ) t ~ - ~ , 8b
c~ s~)(.~_~0
_ b~
Attracting: inside the cone S (fig. 4, 7b). Repelling: inside the cone U (fig. 4, 7b). [ a3 ~[ bI "~[ c 2
•:tz,.,- :)t~.2-.,jt~ 8a
c1
~
)~
~)~
- ~)t~,-~)~o
Attracting: inside the cone S (fig. 4, 8b). Repelling: inside the cone U (fig. 4, 8b).
~, ~0 .: (~.,-.~(~~-~)(~ _.,/_(~_~.,~(.,_~,~ ) t ~. ~ - ~)
288
J. Guckenheimer, A. Mahalov / Resonant triad interactions
lb
la a2 P'2=~1 P'I
~2
/-,~~1/
P'2
--',
I."/'
b2 P2= ~11pI
,
gl
gl
~
I
Cl
al b2
b 1 c2
a~
c3
a2 b 3
b2 c 3
a3
2a
2b a2 " 2 = ~ 1 P'I
~2
o~~ "
" gl a1 _
b2 P'2=~1 "1
"2
| / ~- ~ 1
,
~
I
~
a 1 c2
_
a3
a2 c 3
bl
Cl
-
-
-
b3
3a
gl
-
c3
3b b2 P-2=~1"1P'I
"2
a2 P'2=~1"1pI
"2 -
•
°
i
~
c1 -
/ ~°~ ~"! :,,,~,
b 1 c2
al b 2
a.~_l
b2 c 3
a2 b 3
a3
-
c3
Fig. 4. Regions of the parameter space where the heteroclinic cycle is attracting (repelling).
J. Guckenheimer, .4. Mahalov /Resonant triad interactions
289
4b
4a a2 P-2=~1 P-1
I~2
b2 p.2 = ~--~1.u.1
P-2 "
S
~
/
~"
PI a 1 c2
c1
~
•
I
a1
bl
a3
b3
_ _
- -
--r-
a2 c 3
c3
5b
5a a2 P2=~1 P'I
P2
b2 P'2
P2=~11 PI
i .
.
~
/ I
g~
g~ a 1 b2
c1
a1
a 2 b3
c3
a3
b 1 c2 b2 c 3
- -
6b
6a b2 P'2~11 P'I
~2
a2 P.2=a~- P-1
I~2
I ~'1
PI c1
b1
a1 c2
c3
b3
a2 c 3
- -
- -
al
- -
a3
F i g . 4. ( C o n t i n u e d . ) 4
290
J. Guckenheimer, A. Mahalov / Resonant triad interactions 7a
7b b2 .2=~11 "1
~2
I ', us,,
_ _~~ ~
a2 ~2=~1 ~1
~2
.
~ bl c2
~ ¢1
b2 c 3
c3
gl
8a
al
gl al b2
a3
a2 b 3
8b a2 P-2=~1 P-1
P'2
b2 P2 =~1-1P1
P2
,,
i"i "
gl
~'
~'1
c1
a 1 c2
b1
al
~
~
c3
a2 c3
b3
a3
Fig. 4. (Continued.)
Proof. Guckenheimer and Holmes [10], and Melbourne [17] have shown that a sufficient condition for the stability of a heteroclinic cycle is that the product of unstable eigenvalues is less than the absolute value of the product of stable eigenvalues. In case la the unstable eigenvalues are /x2 -(a2/al)/~l, /~3- (b3/b2)/'~2, ~ 1 - (Cl/C3)~t~3 and the stable eigenvalues are /-~3- (a3/al)tzl, Ixl- (b~/b2)lz2, 1~2- (c2/c3)lz3. Thus, the heteroclinic cycle is stable if f(/xl,/z2,/~3) > 0 where c2
a2
b3
Cl
~,,~,~,~(~,-~t( ~' -~1t(~-~/-(~ ~,/(~ ~4(~, ~4 ~
- -
We have
~
a2 ct b3 a 1 a~ c 3 b2 (~2/z2-/a'l)( b2
(-~/.~
--/'~2) c3
_
_
.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
291
From the inequalities b~ a 2 b-~a-~- > 1 ,
cI a3
c2 b3
a2 c 1 b3
c-~a-~ > 1 ,
c3b2 > 1 ,
a~cab 2
<1
(theorem 4.1, la) it follows that f(]/'l,/'/'2, ~'/'3) > 0 and, therefore, the heteroclinic cycle is attracting. Case lb can be considered using a similar argument. Now we turn our attention to case 2a. In this case the unstable eigenvalues are /z 2 - ( a 2 / a ~ ) l x ~, l z 3 - (b3/b2)l.~2, I - ~ - (q/c3)!~3 and the stable eigenvalues are / ~ 3 - (a3/a~)Iz~, l z ~ - ( b l / b 2 ) ~ 2 , ]2,2 (c2/c3)1x 3. Thus, the heteroclinic cycle is stable if f ( ~ l , Ix2,/z3) > 0 where -
-
a3 bl ¢2 a2 b3 c~ f(/zl,/Z2,/z3) = (~11/zl- ]-~3)(~2]-/,2- ]-/q)(~33]-~3- ]/,2 ) - ( ] ~ 2 - ~ 1 ~/'1)(]-~3 - ~2]d'2)(~t~l - ~33]~3) • The function f(//"l, ]d'2, ~U'3) is homogeneous in the variables /-h, ~2,/z3. Therefore, solutions of the inequality f(/zl,/~2,/x3) > 0 form a cone. Without loss of generality we can assume that /~3 = 1. The function f(/x~,/~2,/z 3) is defined in the interior of the triangle bounded by the lines ~u,1 = a l / a 3, ix 2 = c 2 / c 3 and ~2 = a2/a~lx~ (see fig. 4, 2a). It takes positive values on the line ~ = (a2/al)iz~ and negative values on the line ~ = a~/a 3 a n d / z 2 = c2/c 3. The function is zero at the points (a~/a 3, a2/a 3) and ( ( c 2 / c 3 X a l / a 2 ) , c 2 / c 3 ) . The curve f(pq,/z2,/x 3) = 0 passes through these points and divides the interior of the triangle into two regions. In the region S the function is positive and the heteroclinic cycle is stable. In the region U the function is negative and the heteroclinic cycle is unstable (see fig. 4, 2a). The remaining statements of theorem 4.2 are proved using similar arguments. •
4.2. Strong resonance Now we consider the system restricted to the real subspace without making the assumption that the quadratic terms are small. Then we have
dx 1
d)" = Y ' x z x ~ + x , ( l ~ , + e,,x~ + e,2x ~ + e13x~),
dx 2 dt = ~2x~x3 + x2(/~2 + e2~x~ + e22x~ + e23x~)'
dx3 dt = "~3XlX2 + X3(IA'3 -b e3~x2~ + e32x 22 + e33x 3). 2
(4.2)
Here we consider the system with "~1 < 0, ~/2 > 0, ~/3 < 0 and the system with 3q > 0, Y2 > 0, Y3 > 0. These systems appear in the rotating pipe flow problem. The scaling x I = ex, x 2 = ey, x 3 - ez, l,~j = e2vj with appropriate scaling of time transforms the system (4.2) to
dx ~ = TlYZ + dy
~
dz
= ~2 XZ +
ex(I.~l +
ellx 2 + el2Y 2 + el3z2),
eY(/~'2 + e21 X2 + e22Y 2 + e2322),
d--'~ = y3xy + e z ( Ix3 + e31x2 + e32y2 + e33z2)"
(4.3)
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J. Guckenheimer, A. Mahalov / Resonant triad interactions
We treat this system as a perturbation of the system with quadratic terms only. The limit e = 0 has the form dx d--~ = T l Y Z '
dy ~
=
"~2XZ,
dz ~
=
(4.4)
~/3XY.
L e m m a 4.2. If two of the coefficients y~. in (4.4) have different signs then the system of equation~ (4.14) is
equivalent to the system of equations describing the dynamics of a rigid body. Proof. First we show that via x ~ ax, y -~ by, z ~ cz the system of equations (4.4) can be transformed to
the system of equations dx d--~=pyz,
dy
-d-~z = q x z ,
dz
-d-~z = rxy
(4.5)
w i t h p + q + r = O.
We have p = y l b c / a , q = y z a c / b , r = y 3 a b / c . The condition p +.q + r = 0 b e c o m e s 'Yl(bc) 2 + "y2(gC)2 + y3(ab) 2 = 0. Let us assume without loss of generality that Yl < 0, Y2 > 0, ~3 < 0. If we define a = c = 1 and b = ~/Y2/([Yl[ + [Y3[) then the condition p + q + r = 0 is satisfied. The equations describing the dynamics of a rigid body are dml d'c
12- I3
dm 2
= I-~-~3 mEm3,
~
I3 - I 1 = I--~l
dm3 m3m~'
dr
IlI 2 ~m~m2"
(4.6)
In order to show that (4.5) can be written in the form (4.6) it remains to find positive I1, 12 and 13 SO that 1 ia
1 12 = p ,
1 11
1 13 = q ,
1 12
--
_
1 11
--
.~_
r.
•
(4.7)
Let us again assume without loss of generality that p < 0, q > 0, r < 0. If 11 = 2 / 3 q , 12 = 2/(21pl + q), then (4.7) is satisfied. Thus, the system (4.4) is equivalent to the system (4.6) describing the dynamics of a rigid body. Lemma 4.2 is proved.
I a = 2/q,
Now we can describe the dynamics of (4.4). The phase space of (4.4) is three-dimensional. Consider a level surface of an integral of the motion. (a) The case "Y1 < 0, "Y2> 0, 3/3 < 0. The phase portrait has the form shown in fig. 5~ We have heteroclinic orbits connecting two opposite poles of the ellipsoid belonging to t h e y axis. Four heteroclinic orbits divide the ellipsoid into four regions. Each region is filled with a faiiiily'of periodic , orbits. : ~' ; :
Fig. 5. Phase Pot.trait of (4.4). correspon.ding to the case "Yl dO, ~/2 > O, "Y3<~0.
Z Guckenheimer,A. Mahalov/Resonant triad interactions
293
(b) The case Yl > 0, Y2 > 0, "Y3 < 0. This case is similar to the case (a) but now we have a heteroclinic connection between points lying on the z axis. In the m~. variables the system of equations (4.3) becomes
dml 12 -- 13 dr = ~ mEm3 + e :ml(I£1+ c11m~ + c~2m~ + c l 3 m ~ ) ' din2 = ~
dr
din3
dr =
~
m3ml + em2(/~2 +c21m~ + c22m~ + c23m~)'
mlm 2 + em3(/x 3 +c31m~2 +c32m~ +c33m~) ,
(4.8)
t~
where 11 = 2b/3y2, I 2 = 2 b / ( y 2 + 213qlb2), I 3 = 2b/y2, b = Cy2/(13ql + 13'31) • We also have cij= eij( j --/:2), Ci2 ~- bEei2 . Without loss of generality we assume that 11 > I 2 > I 3. We are interested in the heteroclinic orbit of the unperturbed equations connecting the point A = (0, ¢ - - ~ 2 / e 2 2 , 0 ) and the point B = (0, -- ¢ - / . t E / e E 2 , 0) lying on the m 2 axis. This heteroclinic orbit is given by
qb~(t)=l
-Pq s e c h ( - v / ~ l t ) ,
q~(t)=ltanh(-f~-~lt),
q~(t)=l
- rq s e c h ( - ¢ - ~ l t ) , (4.9)
w h e r e 1 = ¢ - g,2/e22, p = (I2 - I3)/I213, q = (I3 -
11)/1311, r
= (I1 - I2)/Ili2"
We write (4.8)jn the form dm
dt' = f ( m ) + eg(m), ,,
(4.10)
,
where d m / d t = f ( m ) is the equation describing the dynamics of a three dimensional rigid body. Thus,
[ Pm2m3 I~ f(m)=
[qm3ml k ITl'llgl'12
p=
i2_i3 13 --11 11 -12 1213 , q = 1311 , r = 1112
A and B are steady states of (4.10). They lie on the sphere of radius ¢ - ~ 2 / e 2 2
(fig. 6). The
A
B
Fig. 6. Heteroclinic orbit between the steady states A = ' ( 0 , - ¢ - ~ / e ~ , 0 ) and B = ( 0 , - ~ , 0 ) Of the per~rbed system. A and 8 lie on the sphere of radius ¥/"~/~2/e22.
z Guckenheimer,A. Mahalov / Resonant triad interactions
294
i
/ ' / Fig. 7. Splitting of the heteroclinic orbit. The splitting distance is measured in the plane P. This plane is spanned by the vectors VHt(q°(t )) and VHz(q°(t )).
unperturbed equation (e -- 0) has two integrals of motion:
H,( m ) = m~ + m2~ + m~,
ml2 m2~ m~ H2(m) = -~l + ~ + I-~-"
We define L(m) = m l m 2 m 3 and we find that f ( m ) = I VL where I = diag{p, q, r} is a diagonal matrix. We study the splitting of the heteroclinic orbit q°(t) given by (4.9) under the perturbation g(tn) using the technique originally proposed by Melnikov for planar systems [9, 18]. For the perturbed system we have
a:(t) = a°(t) + ~a~(t) + : ( ~ )
a : ( t ) = a ° ( t ) + ~a~(t) + : ( ~ )
t e [o, +~o),
t~(-~,o], (4.11)
where q~(t) and q~(t) satisfy dq~ dt = D f ( q ° ( t ) ) q ~ ( t ) + g ( q ° ( t ) ) t > - O '
da~ dt = D f ( q ° ( t ) ) q ~ ( t ) + g ( q ° ( t ) ) '
t<_O.
At each point q°(t) of the heteroclinic orbit we define a plane spanned by the vectors VHt(q°(t)) and VH2(q°(t)). The heteroclinic orbit q°(t) splits into two parts: q~(t) and q~(t). We measure the splitting in the plane (see fig. 7) span[ V H , ( q ° ( t ) ), VH2(q°(t) )] using the following functions: At = (q~ _ q : , V H 1 ) ,
A2 = (qS~ _ q : , V H 2 ) .
Using (4.11), we obtain
At = (qS~ _qU~,VHt) = e ( q ~ - q ~ , V g 1) + ~'(e2),
d2
=
(qS~-qu~,vg2) = e ( q ~ - q ~ , V H 2) + ~'(e2).
We define d t = (q~ - q[, VHt), d 2 = (q~ - q~, VH2). These functions measure splitting of the heteroclinlc orbit. We have d t = d~ - d~, d 2 = d~ - d~ where d~ -- (q[,VHt), d~ = (q~,VH1), d~ = (q~,VH2), d~= (q~,VH2). Now we find differential equations for d~,d~,d~,d~. Without loss of generality we
J. Guckenheimer, A. Mahalov / Resonant triad interactions
295
consider d~ and d[. We have d
=
d
=(vn,(q°(t)),g(q°(t,))) +(vn,(q°(t)),Df(q°(t))q~(t))
Now we show that
(VHl(q°(t)), Df(q°(t)) q~(t, to)) +(d2H,(q°(t)) f(q°(t)), q~(t)) = O. We have f = I VL, D r = I dZL (d2L is the Hessian matrix). We note that I and d~L are symmetric matrices. Therefore, we have
(VH~(q°(t)), Df(q°(t)) q~(t))
= ( d 2 L I VH~, q~),
f(qO( t)),q~(t))
=(
Lemma 4.3. d2L I VH 1 ÷ d2H1 1 VL =
0.
Proof.
mlm2),
( d2H~(q°(t))
We have VL = (mzm 3, mlm3, dEHl = 2E (E is the identity matrix),
d2L
--
(0 m 3
0
m 2
m 1
1
d2H~I VL,q~).
VH~ = 2(m~, m 2, m3), I = diag{p, q, r}. Then we obtain
•
Substituting these expressions for the matrices, we get
d2LIVH~+d2HIlVL=2
m3 ,m 2
0 mi m~ 0
/qm /+2
qmlm31=O ,
~ rm3 ]
rmlm2 ]
since p + q + r = 0. Lemma 4.3 is proved. Using lemma 4.3, we obtain get (d/dt)(d~)= (VH~(q°(t)),
d
d'-}"(d~) = ( V H 2 ( q ° ( t ) ) ,
(d/dt)(d~)= (VHl(q°(t)), g(q°(t))). We can apply a similar g(q°(t))). The function d~ satisfies the following equation: g( q°( t ) )) +(VH2(q°(t)), Df ( q°( t ) ) q~( t ))
+( d2H2( q°( t ) ) f ( q°( t ) ), q~( t )). Lemma 4.4. d2LI
VH 2 ÷ d2H2 I VL = 0.
argument to
296
J. Guckenheimer, A. Mahalov / Resonant triad interactions
Proof.
We use the expressions for VL, d2L, I obtained in the proof of lemma 4.3. Here
VH 2 = 2
(1
, i2 , i3
, d2H2--2diag i 1 , i 2 , i 3
and
pm2m3 I d2LIVH 2+d2H2IVL=2
~-~ + ~ + ~3 t~'~ 1m 2 ]
since
,
~ = (I~ -I~)/I~Ia, q = (I~ -I1)/I311, r = (I~ -I~)/I~I~.
~ m m a 4.4 is proved.
Using lemma 4.4, we obtain
~(d~)=(VH2(q°(t)),g(o°(t))),
d
~(d~)=(VH2(q°(t),g(q°(t))).
Thus, the functions d[, d[, d~ and d~ satis~ the following differential equations: -ff-7(d[) = ( V H , ( q ° ( t ) ) ,
g(q°(t))),
d ~'~(d~) = ( VHI(q°(t)),
g(q°(t))),
ff-Tt(d-~) = ( VH2(q°(t) ),
g(q°(t) )),
d ~7 (d~) = ( 7H2(q°(t),
g(qO(t))).
Integrating the first equation from 0 to + 0% we get +~
d~( +~)-d~(O) = fo (VHl(q°(t))'g(q°(t)))dt Integrating the second equation from -oo to O, we get
d~(O)-d~(-oo) = fo ( Hl(qO(t))g(qO(t)))dt" -oo
We note that d~( + ~) = d~( - ~) = O. Thus, dl(0)
= d~(0)
- d~(0)
~ f_+~(VHl(q°(t))
g(
q°(t ), t )1 dt.
Similarly, d2(0 ) = d ~ ( 0 ) - d ~ ( O ) - -
f_~VH~(q°(t)),g(q°(t))~
dr.
J. Guckenheimer,A. Mahalov / Resonant triad interactions
297
Therefore, the following two functions measure the splitting of a heteroclinic orbit: df(/z~,~2,/z3) = L+~(VH~(q°(t)) g(q°(t))) dt, ,
d2(/z~2,~3) =
(4.12)
VH2(q°(t)) g(q°(t))) dt. _
:
We have
VH~(qO(t)) = 2(q~O(t), q 2o( t ) , q 3o( t ) ) ,
0 1 o 1 o ) VH2(q°(t)) = 2 ( 1-~ql(t),~2q2(t),7~3q3(t) •
(4.13)
We substitute (4.9) and (4.13) in (4.12). Using the identities (a > 0)
f + ~ s e c h 2 ( - a t ) dt = -~, f7 s e c h 2 ( - a t ) t a n h 2 ( - a t ) dt = -~-d' ~ - -
o~
- -
we evaluate the integrals (4.12). We find that the condition following: ~
f +? sech4( - a t )
dt = 43a
--
d1(~1,/~2,~3)=0 is
equivalent to the
3pqe22/x 1 + [~eHp 2 - q2e22 + 2e33r2 - p q ( e l 2 + e21) + 2pr(e~3 + e31) - qr(e23 + e32)]tz2
"
+3rqez21z3 = O. (,
The condition d2(/Xl, tz2, ~3) --- 0 can be written in the form
3pqe22
:
r 2en/92
[
r,
q2e22 2e33 r2 - - ;2 + 13
[ e~2 e21] ,, [ e13 e31] _ q r [ e23 e321] PO~ i1 + i2 } q- zPr~ 7-~ + i3 ) ~ i2 + ~
1]~1"2
3rqe22 + - - - ~ 3 tx3=O. Thus, we proved the following theorem.
Theorem 4.3. For e > 0 sufficiently small bifurcations with a heteroclinic orbit are given approximately by the equations dl(/xl,/x2, ~3) = 0, d 2 ( ~ , ~2,/~3)--0. These two conditions are satisfied on a straight line passing through the origin in the parameter space (/x~, ~2, ~3). This line is defined as the intersection of two planes '. II1:
3pqe2zN 1 + [2e~lp 2 --q2e2z + 2e33 r2 - p q ( e12 + e2, ) + 2pr( e13 + %,) -qr(e23 + e32)]/x2 + 3rqe221~3 -= O, 112: ,,
3pqe22 [ 2enp2 i1 ~1 + i1 3qe22 + ---1-~3/x3 = 0.
q 2e22 2e33r2 _ p q ( e12 e2, ( e13 e3,1 ~I1 + T 2 ) + 2 p r ~ I 1 + ia]-qr( I2 + I3
e23
e32
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5. Existence of heteroclinic cycles Reflection symmetry of (2.3) implies that the purely real subspace is invariant as is any ~b-invariant rotation of it. Thus, the following are invariant subspaces of (2.3): (Xl, x2, x3), (Yl, x2, Y3), (Xl, Y2, Y3), (Y~, Y2, x3). In order to show the existence of a heteroclinic cycle it is enough to show the existence of connecting orbits in the real subspace. Then the heteroclinic cycles in the (Yl, x2, Y3) subspace are obtained by applying the element of the symmetry group. The system of equations (2.3) restricted to the real subspace is dx 1 dt = - x 2 x 3 -]-Xl(/~l + e1'x21 +e12x~2 +e13x32)' dx 2 dt = XlX3 + x2(/x2 + e21x~ + e22x22 + e23x~) ' dx 3 dt = - g i g 2 +xa(/z3 + e31x~ + e32x~ + e33x~)"
(5.1)
We want to establish the existence of heteroclinic connections between the steady states (0, ~ , 0 ) and (0, - ~ ,0) lying on the x 2 axis. If ~z~ =/~3, ell + e~3 = e3~ + e33, e~2 = e32 then the system of equations (5.1) has two-dimensional invariant subspaces x a = x I and x 3 = - x r Let us assume that the conditions of proposition 5.1 of ref. [2] are satisfied. Then the heteroclinic cycles in the two-dimensional invariant subspaces exist and they are locally asymptotically stable in the real subspace. Thus, we have proved the following. Theorem 5.1. Suppose that the conditions of proposition 5.1 of ref. [2] are satisfied and ~t~1 1~3' ell + e l 3 =e31 +e33 , el2 =e32. Then there is a heteroclinic net for the system of equations (2.3) connecting two pair of points on the torus of pure modes. This net consist of four heteroclinic cycles in the real subspace and four heteroclinic cycles in the (y~, x 2, Y3) subspace which is conjugate to the real subspace under symmetry transformation. =
6. Stability of heteroclinic nets In this section we discuss stability of heteroclinic nets (cycles). Heteroclinic cycles for the unperturbed system exist in the case ~b = 0, xr with E 1 = E 2. Eq. (3.2) with L = 0 (~b = 0, xr) and E 1 = E 2 becomes
~dp -g!
- 4p2(E_p) •
(6.1)
The solution of this equation corresponding to heteroclinic cycles is p(~-) = E sechZ(v~- ~').
(6.2)
We want to establish conditions that guarantee local asymptotic stability of the heteroclinic net lying on
J. Guckenheimer, A. Mahalov / Resonant triad interactions
299
the level set El = _ /~2 + Fl '
E2 = _/~_~_2+ F2"
e22
After substitution of
U = eL
/d,1 "~- /'g3
(6.3)
e22
p(r) --
from (6.2), eq. (3.3) for L then becomes
/z_~2( /~2(e e22 el2 + e32) -- --e22, 11 + eEl + e3~ + e13 + e23 + e3a -- e12 -- e22 -- e32)
X s e c h 2 ( ¢ - ~2/e22 ~)) + ~ ( L F I , L F 2 ) .
(6.4)
Since s e c h 2 ( ~ - ~ 2 / e 2 2 7 ) exponentially converges to zero at infiniff, the behavior of L(r) at infini~ is ~ n t r o l l e d by the sign of the term ~l + ~3 -- (~2/e22~e12 + e32)- If this t e ~ is negative then L(¢) ~ 0 e ~ o n e n t i a l l y when • ~ + ~. From (3.3) we also obtain equations for F~ and F 2. These equations are
~ [' F~=ze[[~2 ~e 2 3 - ~ 2 ) F I - ~ 2 ~ F 2 ]
+
H~,
~2 F '~ = 2 e [ ( - ~ z - ~ 3 + e2~ ~2 (e23 +ea2))Fl+(~3_e22
(e23 +e32 ) ) F2 ] + H 2 "
~ e t e ~ s H~ and H 2 have s e c h Z ( ~ - ~ 2 / e 2 2 ~) as a factor and, therefore, they converge to zero at infinity at an e ~ o n e n t i a l rate. Then the behavior of F~ and F 2 at infini~ is controlled by the linear t e ~ s . The m a t r ~ corresponding to the linear terms is
-~
- ~ + (~/e~z)(e~3 + e~)
~ - (~/e~)(e~
+ e3~ ) "
The eigenvalues of the matrN M are - ~ and ~3 - ~ e 3 a / e ~ • F~(r),F~(~) ~ 0 at an exponential rate when r ~ + ~ if and only if both eigenvalues are negative. Combining these results with our results on the behavior of L(~), we obtain the following
Theorem 6.1. If a heteroclinic net formed by heteroclinic cycles exists for the scaled equations, e ~: 0 is sufficiently small and /'~'1 @ //'3 -- e2 2 el2 + e32) < O,
/./'2 > O,
~/'3 --/3-'2 e~22 < O,
then the heteroclinic net is locally asymptotically stable.
7. The interaction of resonant patterns in rapidly rotating Hagen-Poiseuiile flow In this section we consider interactions of resonant patterns in rapidly rotating Hagen-Poiseuille flow. The basic laminar flow has a parabolic velocity profile with maximum speed Wo (= -(r~/4v)OP/Oz, where ~P/~z is the prescribed pressure gradient) on the center-line. We assume that r 0 is the radius of
J. Guckenheimer, A. Mahalov / Resonant triad interactions
300
Table 4 Numerical values for the constants c n in eq. (7.1). n c~
1 4.366
2 5.676
3 6.916
4 8.126
5 9.316
6 10.56
the pipe and O 0 its angular velocity. With r 0 and r~/4v as units of distance and time, the problem depends on a rotational Reynolds number, O, and an axial Reynolds number, R, defined by O = r~ Oo/V, R = roWo/v. In terms of this scaling the base ttow velocity written in cylindrical coordinates is V = (0, Or, R(1 - r2)). The initial point of our search for resonant patterns is the work of Pedley [19]. In his work he obtained approximate expressions for the neutral stability curves of the Hagen-Poiseuille flow. His set of approximate equations is valid in the limit 0 / ~ 0 (0/ is axial wavenumber),/2 ~ + ~ so that a O and R remain finite. The significance of this limit is that the interaction of long waves in a rapidly rotating pipe is considered. It is known by the Taylor-Proudman theory described in ref. [8] that the velocity field becomes independent of the axial variable z in the case of fast rotation. Hence, the fluid system becomes translation and reflection invariant and, therefore, the theory developed in the previous paragraphs can be applied. Cotton and Salwen [5] found that Pedley's asymptotic results are in excellent agreement with numerical data not only in the region of formal validity of the asymptotic expressions but, in fact, they provide good approximation of the neutral stability curves in a much larger region of the parameter space. The expression for the neutral stability curves in Pedley's limit has the form 40/20 2 +
40~nOR + cn = O.
(7.1)
Here 0/ is axial wavenumber, n is azimuthal wavenumber, O is rotation rate and R is the Reynolds number. Numerical values for the first six coefficients c n taken from ref. [19] are given in table 4. We use eq. (7.1) to find the values of parameters controlling the flow for which three neutral waves exp[i(n~b+0/1 z+tolt)]u~(r),
exp[i(n2~b+a2z+to2t)]u2(r),
exp[i(n3~b+0/3 z+to3t)]u3(r )
form a resonant triad. These waves are neutral for given R and O if and only if the following equations are satisfied simultaneously:
40/702 + 4 0 / l n l O R + Cn~ = 0, Using the resonance conditions a I 4a220 2 + 4 0 / 2 n 2 O R + c n : = 0 ,
40/2zO2 + 4 0 / 2 n 2 O R + cn2 = 0, =
40/320 2 + 4 0 / 3 n 3 O R + Cn3 = O.
0/2 d- 0/3 and n 1 = t/2 +//3, we obtain 4a320 2 + 4 a 3 n 3 O R + c n 3 = 0 ,
4 [ ( a 2 + a 3 ) O ] 2 + 4(0/2 + a a ) ( n 2 + n 3 ) O g +Cnz+. 3 = 0.
(7.2)
Now we consider two cases. In the case n 2 = n 3 it is possible to satisfy (7.2) simultaneously if and only if R = ~
1
¢
c~/~2,~.
(7.3)
J. Guckenheimer,A. Mahalov /Resonant triad interactions
301
Table 5 Reynolds numbers corresponding to resonant triad interactions. Here n~ are the azimuthal wavenumbers of resonant patterns. (nl, n2, n 3) R
(2, 1, 1) 91.1
(3, 1, 2) 111.4
(4, 2, 2) 133.8
(4, 1, 3) 138.4
(5, 2, 3) 162.3
(5, 1, 4) 170.3
(6, 3, 3) 192.9
(6, 2, 4) 196.9
(6, 1, 5) 207.8
In t h e case n 2 :~ n 3 eqs. (7.2) c a n b e satisfied if a n d only if t h e R e y n o l d s n u m b e r R is a solution o f the following n o n l i n e a r e q u a t i o n :
~-n29R2-Cn2 -~n~R2-cn3
d- ~(n2q-n3)2R2-Cn2+n3=O.
(7.4)
If n 2 = n 3 t h e n eq. (7.4) r e d u c e s to (7.3). W e n o t e t h a t in b o t h cases t h e c o n d i t i o n to 1 = t o 2 + to 3 is satisfied since t h e f r e q u e n c i e s have a p p r o x i m a t e e x p r e s s i o n to = - n O - aR w h i c h i s l i n e a r in a a n d n. F r o m f o r m u l a (7.3) a n d f r o m t h e n o n l i n e a r e q u a t i o n (7.4) we find t h e values o f R e y n o l d s n u m b e r for which t h r e e n e u t r a l waves f o r m a r e s o n a n t triad. T h e r e a r e infinitely m a n y R e y n o l d s n u m b e r s for which such r e s o n a n c e s occur. T h e n u m e r i c a l values for t h e first nine o f t h e m a r e listed in t a b l e 5. A f t e r R e y n o l d s n u m b e r s c o r r e s p o n d i n g to t h e r e s o n a n t t r i a d s a r e found, we can get axial w a v e n u m b e r s a2, a 3 (a~ = a 2 + a 3) a n d t h e r o t a t i o n r a t e O. In t h e case n 2 = n 3 we o b t a i n
or20=¼(--
C)~2n2-- ~/C2n2 -- 4Cn2 ), Or30=¼(--
C~/-C~n2n2 d- "~/C2n2-- aCn2 ).
(7.5)
In t h e case n 2 ~ n 3 we have two possibilities. In t h e first case we o b t a i n
a20= ½(-n2R- ~/n~R2-cn 2), a30= ½(-n3R + ~/n~R2-cn 3).
(7.6)
0.8-
0.2 ¸
0.2
0.4
0.6
0.8
1
Fig. 8. Imaginary part of the axial velocity component of the neutral mode with n t = 4. Real part is zero.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
302
z'
0.8
0.6'
0.4,
0,2,
I 0.2
0.4
0.6
0.8
Fig. 9. Imaginary part of the axial velocity component of the neutral mode with n 2
I =
1. Real part is zero.
In the s e c o n d case w e get
o~20 (-n2R + ~/n~R2- c,z ), or30 ½(-n3R- ]/n~R 2- c,~3). =
(7.7)
=
W e find that for (7.6) the amplitude equations (1.1) with appropriate scaling of amplitudes have 3'1 = 1, 3"2 = 1, 3'3 = - 1. In the case (7.7) w e obtain 3'1 = - 1, 3,z = 1, 3"3 = - 1. T h e two choices (7.6), (7.7) give us different dynamical behavior. In the first case the third m o d e is unstable, in the s e c o n d case w e find that the s e c o n d m o d e is unstable. 0.7
¸
0.5
¸
0.3
¸
0.2-
,, 0.2
0.4
0.6
0.8
~
Fig. 10. Imaginary part of the axial velocity component of the neutral mode with n 3 = 3. Real part is zero.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
303
0.6-
0.4-
0.2-
l
I
I
0.8
i
-0.2
-0.
Fig. 11. Real part of the azimuthal velocity component of the neutral mode with n I = 4. Imaginary part is zero.
0.1-
I
I
-0.1"
-0.2-
Fig. 12. Real part of the azimuthal velocity component of the neutral mode with n 2 = 1. Imaginary part is zero.
Now we describe the interaction of resonant patterns with azimuthal wavenumbers n 1 = 4, n 2 1 and n 3 = 3 for the case (7.7). This resonance occurs for the value of Reynolds n u m b e r R = 138.4. In the fast rotation limit resonant waves are approximately independent of z and have the form u~. = exp(in~.~) fi~.(r) ( j = 1, 2, 3). However, all three components of the velocity field (radial, axial and azimuthal) are nonzero. The velocity profiles of ~j(r) are shown in figs. 8-16. The streamline patterns of these pure modes projected on the z = const, plane are shown in figs. 17-19. Thus, for each pure m o d e the pipe is divided into a number of cells. Due to the fact that the axial component of the velocity field is nonzero, the fluid particles are engaged in a helical motion in each cell. The equations describing interaction of these pure modes forming a resonant triad have the form (2.1). The coupling coefficients were computed numerically and their values are given in table 6. =
304
J.
Guckenheimer, A. Mahalov / Resonant triad interactions
0.6-
0.4-
0.2-
I
I
-0.2-
-0.
Fig. 13. R e a l part o f the a z i m u t h a l velocity c o m p o n e n t o f the n e u t r a l m o d e with n 3 = 3. I m a g i n a r y part is zero.
0.8'
0.6'
0.4'
0.2'
02
04
O~
Fig. 14. Imaginary part of the radial velocity c o m p o n e n t o f the neutral m o d e with n~ = 4. R e a l part is zero.
Table 6 . : Cubic c o u p l i n g coeffi¢ient~ in the a m p l i t u d e e q u a t i o n s (2.1) describing r e s o n a n t triad interaction o f w a v e s with n l = 4, n 2 = 1 and n 3 = 3 at R e y n o l d s n u m b e r R = 138.4. e l l = -- 18.8 = -3.19
el2 =
-
14.2
e13 =
--81.9
e22 =
- 5.19
e23 =
- 7.96
e3~ =
~e32 =
- 28.1
e33 =
- 83.7
e21
- 33.2
J. Guckenheimer,A. Mahalov/ Resonant triad interactions
305
O. 2 5 -
0.2-
0.15-
0.1-
0.05-
I
I
I
I
0.2
0.4
0.6
0.8
Fig. 15. Imaginary part of the radial velocity component of the neutral mode with n 2 = 1. Real part is zero.
0.8'
0.6"
0.2-
0.2
0.4
0.6
0.8
1
Fig. 16. Imaginary part of the radial velocity component of the neutral mode with n 3 = 3. Real part is zero.
We use theorem 3.1 to find the number of periodic orbits that survive cubic perturbations. For simplicity we restrict ourselves to the c a s e / ~ + ~2 + / z 3 -- 0. We find that the inequality (3.13) cannot be satisfied. Using the data presented in table 6, we obtain -~/AE~ = 0 : 6 9 5 5 2 0 - 0.957519 E2/E r The graphs of the functions F(x), G(x) and L(x) = 0.695520 - 0.957519x (x = E2/E ~) are presented in fig. 20. We note that the graph of L(x) intersects the graph of F(x) at .~ = 0.585129. From theorem 3.1 it follows that for given ~ l a n d / z 2 the number of periodic orbits that survive cubic perturbations equals the number of nondegenerate solutions of the following problem:
fl(E1,E2,1~I,tz2)=O,
f2(E1,E2,1~l,tZ2)=O,
E I > O , E2>O, E2/El <0.585129.
(7.8)
Fig. 17. S t r e a m l i n e p a t t e r n of the m o d e with n 1 = 4 proj e c t e d on the p l a n e z = const.
Fig. 18. S t r e a m l i n e p a t t e r n of the m o d e with r~2 = 1 proj e c t e d on the p l a n e z = const.
Fig. 19. S t r e a m l i n e p a t t e r n of the m o d e with n 3 = 3 proj e c t e d on the p l a n e z = const. 1-
0.8 .
0.6
0.4"
0.2 .
:
I
-0.2
Fig. 20. G r a p h s of the functions F ( x ) , the p o i n t x = 0.585129.
G(x) a n d L(x) (straight line) (0 < x < 1). T h e g r a p h of L(x) intersects the g r a p h of F(x) at
J. Guckenheimer, A. Mahalov / Resonant triad interactions
2
0.5"~" |
1
I -0.5
~.~
! O.S
-0. $'I
307
0
Fig. 21. Regions of the parameter space /zl,/~ 2 with 0, 1 and 2 periodic orbits. Horizontal axis-/z~, vertical axis-/~ 2. The boundaries separating different regions are given by the equations arg(/~l + ip, 2) = 1.5, arg(p,t + ip, 2) = 2.8, a r g ( ~ + i/~ 2) = 3.8, arg(/~ 1 + i/~2) = 5.9.
It can be easily shown that the problem (7.8) is invariant with respect to the transformation E L ~ cE1, E 2 ~ cE2,1~1 ~ clzl, l z 2 - ~ c l ~ 2 where c is an arbitrary positive constant. This implies that boundaries separating regions on the (/~1,/z 2) plane having different number of periodic orbits are straight lines passing through the origin. From (7.8) we find that depending on the values of/~1 and ~2 the number of periodic orbits that survive cubic perturbations is 0, 1 or 2. The regions of the parameter space with 0, 1 and 2 periodic orbits are shown in fig. 21. It follows from theorem 6.1 that the heteroclinic net is locally asymptotically stable (when it exists) in the region of the parameter space/~,/~2,/~3 given by the inequalities /x 1 + tz 3 - 8.15/~ 2 < 0,
/z 2 > 0,
/~3 - 5.41/~ 2 < 0.
The velocity field (streamlines) was reconstructed for a trajectory approaching the heteroclinic net. The corresponding patterns are shown in fig. 22. The initial and the final states are identical except for the direction of rotation. The sequence of patterns shown in fig. 22 provides an example of the recovery of initial state after its breakdown. It can be seen that the initial pattern formed by two eddies breaks down and is replaced by a sequence of asymmetric patterns. Nevertheless, the order is regained and the initial
308
J. Guckenheimer, A. Mahalov / Resonant triad interactions
Fig. 22. Streamline patterns corresponding to the heteroclinic orbit.
state of two eddies (but with rotation in the opposite direction) is recovered. The system is apparently hovering chaotically in time between these two states. The corresponding trajectory in the phase space is shown in fig. 23. We note that Wersinger et al. [23] also provide an example of chaotic behavior in three-wave equations. In their case the chaos is associated with a sequence of period doubling bifurcations.
J. Guckenheimer, A. Mahalov / Resonant triad interactions
309
Fig. 23. Trajectory in the phase space corresoondin.g to the streamline patterns of fig. 22. Horizontal axis-x t, vertical axis-~x 2. The trajectory is hovering (apparently chaotically in time) between two states formed by two eddies. These states have opposite directions of rotation. The parameters are / z ~ = - 1 , /z2~ 1, /~3 = -0.68, e=0.022. The initial conditions are x 1=0.1, y l = 0 . 0 , x 2=0.0, y2=0.0, x 3=0.1, Y3 = 0.0.
8. Conclusions We have demonstrated that expansi~on of the resonant triad equations to third order is essential to capture the qualitati,~e features of thi'ee interacting waves in a dissipative system. The third order terms break degenerac~ ,es,in the second order expansion that result from linear flow in the invariant subspaces correspQoding toia single active mode. In the dissipatfize case, there is linear excitation of the amplitudes of the different modes and the cubic terms serve to damp this excitation at finite amplitudes. The main dynamical features of the triad interactions in the dissipative and conservative systems are quite different. The significance of higher order terms in the analysis of resonant interactions in shear flows was first observed by Usher and Craik [22]. This paper begins the systematic investigation of the qualitative dynamics of the equations describing resonant triad interactions in dissipative systems. We have not discussed secondary or higher order bifurcations in this paper. Also, our explorations of regions of the parameter space with chaotic behavior are rudimentary. A more complete understanding of the dynamics of (1.1) remains an interesting and challenging problem. :.,
Acknowledgements This work was supported by the Air Force Office of Scientific Research under contracts AFOSR-890346 and AFOSR-89-0226. J. Guckenheimer was supported by NSF, AFOSR and the Army Research Office through the Mathematical Sciences Institute at Cornell University. A Mahalov is supported by an IBM Watson Fellowship. We would like to thank Sidney Leibovich for very helpful discussions. References [1] D. Armbruster and P. Chossat, Heteroclinic orbits in a spherically invariant system, Physica D 50 (1991) 155-176. [2] D. Armbruster, J. Guckenheimer and P. Holmes, Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry, Physiea D 29 (1988) 257-282.
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[3] N. Aubry, P. Holmes, J. Lumley and E. Stone, The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Fluid Mech. (1988) 115-173. [4] N. Bioembergen, Nonlinear Optics (Benjamin, New York, 1965). [5] F. Cotton and H. Salwen, Linear stability of rotating Hagen-Poiseuille flow, J. Fluid Mech. 108 (1981) 101-125. [6] A.D.D. Craik, J. Fluid Mech. 50 (1971) 393-413. [7] A.D.D. Craik, Wave Interactions and Fluid Flows (Cambridge Univ. Press, Cambridge, 1985). [8] H.P. Greenspan, The Theory of Rotating Fluids (Cambridge Univ. Press, Cambridge, 1968). [9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin, 1983). [10] J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. Soc. 103 (1988) 189-192. [11] J. Hale, Ordinary Differential Equations (Wiley, New York, 1969). [12] D.J. Kaup, The three-wave interaction- a nondispersive phenomenon, Stud. Appl. Math. 55 (1976) 9-44. [13] D.J. Kaup, A Reyman and A. Bers, Space-time evolution of nonlinear three-wave interactions. Interactions in a homogeneous medium, Rev. Mod. Phys. 51 No. 2. (1979). [14] S. Leibovich and A. Mahalov, Resonant interactions in rotating pipe flow, in preparation (1991). [15] A. Mahalov and S. Leibovich, Resonant Tollmien-Schlichting triad interactions in channels, in preparation (1991). [16] A. Mahalov, E.S. Titi and S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations, Arch. Rat. Mech. Anal. 112 (1990) 193-222. [17] I. Melbourne, Intermittency as a codimension three phenomenon, J. Dynam. Diff. Eq. 1 (1989) 347-367. [18] V.K. Melnikov, On the instability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12 (1963) 1-57. [19] T.J. Pedley, On the instability of viscous flow in a rotating pipe, J. Fluid Mech. 35 (1969) 97-115. [20] M.K. Proctor and C. Jones, The interaction of two spatially resonant patterns in thermal convection. Exact 1 : 2 resonance, J. Fluid Mech. 188 (1988) 301-335. [21] R.Z. Sagdeev and A.Ao Galeev, Nonlinear Plasma Theory (Benjamin, New York, 1969). [22] J.R. Usher and A.D.D. Craik, Nonlinear wave interactions in shear flows. Part 2. Third-order theory, J. Fluid Mech. 70 (1975) 437-461. [23] J.-M. Wersinger, J.M. Finn and E, Ott, Bifurcation and "strange" behavior in instability saturation by nonlinear three-wave mode coupling, Phys. Fluids 23 (1980) 1142-1154. [24] J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas (Pergamon, Oxford, 1977). [25] V.E. Zakharov and S.V. Manakov, Resonant interaction of wave packets in nonlinear media. Sov. Phys. JETP Lett. 18 (1973) 243-245. [26] V.E. Zakharov and S.V. Manakov, The theory of resonance interaction of wave packets in nonlinear media. Soy. Phys. JETP 42 (1976) 842-850.