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A Shilnikov-type analysis in a system with symmetry
Volume 152, number 1,2
PHYSICS LETTERS A
7 January 1991
A Shilnikov-type analysis in a system with symmetry Phillip Kent and John Elgin Department ofMathematics, Imperial College, 180 Queens Gate, London, SW7 2BZ, UK Received 6 September 1990; accepted for publication 29 October 1990 Communicated by A.P. Fordy
We perform a Shilnikov-type analysis for heteroclinic orbits of a symmetric system, which differs from the usual case in that it involves asymmetry, and there is more than one connecting orbit at the critical parameter value.
1. Introduction Consider the dynamical system 1X2_ ~ — — 2 )(— Y Y—Z ~_c2_ —
‘
‘
(1)
‘
where c is a bifurcation parameter and the dot denotes d/dt. It has been the object of both numerical and analytical study, in part because of its relevance to travelling-wave solutions of the Kuramoto— Sivashinsky equation [1]. Here c2 is the velocity of the travelling wave. The purpose of this Letter is to report on a new variety of “heteroclinic Shilnikov” behaviour, introducing only as much background on (1) as is necessary. A fuller report [2] is in preparation. Ref. [31 describes some other unusual features of (1). First we note some basic properties of (1): It has an invariance under the symmetry 1’. (X, Y, Z, t)—~(—X,Y, —Z,
—t)
.
(2)
There is no Liapunov function most orbits are unbounded and escape to infinity. When c=0 there is a single fixed point at the origin. For c> 0 there are two fixed points F ±= ( ±~ 0, 0). The eigenvalues of the linearised stability about F ± are given by 2~+~±c~=0.Atc=0,)~=0or ±i.Forc>OatF, there is one real positive eigenvalue, and a negativereal conjugate pair. So c=0 is a “degenerate Hopf” bifurcation point in the sense that a simple periodic solution is born there, although the full structure of this bifurcation is more complicated (see ref. [4], §7.4 and ref. [5]). At F— (for forward time) there —
28
is a stable 2-D manifold, and an unstable l-D manifold; vice versa at Ft The periodic orbit born at c=0 has Sf-symmetry. We have numerical orbit-following codes to examine itsused bifurcation behavior as c varies. Some resuits are shown in fig. 1. Not all the bifurcation points of the primary curve are shown; we mark only the period-doubling points A and B, and show how the bifurcated orbits evolve to infinite period. For the full bifurcation structure see refs. [1—3]. The .9°-symmetricperiod-doubled orbits from A and B converge as period —~ooto the same parameter value c=c 0. The oscillatory approach of the orbits is typical of “Shilnikov” behaviour [6,7]. What is different here is that at c0 there exist three closed doubly-heteroclinic connecting orbits. Associated with these are the two observed periodic solutions; their forms at large period are shown in the insets to fig. 1. We will show by the analysis of section 2 why the expected third periodic solution does not exist: it is because of the symmetry /1’. One part of each connecting orbit is a heteroclinic orbit linking the l-D manifolds of F~(both insets of fig. 1). It is known [8] to have the analytic form .
.
.
.
3
X=a(—9tanhfit+lltanh fit), (3) where a= l5~/Tiji~~,fi= ~/ii7i~ and c0= a~/~0.8495.In ref. [9] it is shown that (1) has no monotonic heteroclinic orbits. The orbit (3) is the simplest member of a possibly infinite heteroclinic family [1,2,10]. The other parts of the connecting orbits link the 2-
0375-9601/91/$ 03.50 © 1991 — Elsevier Science Publishers B.V. (North-Holland)
Volume 152, number 1,2
~RIOD
PHYSICS LETTERS A
7 January 1991
~
~ I
C
I
0
I
I
0.5
I
1.0
I
1.5
Fig. 1. Numerical continuation from the periodic orbit born at c= 0, period versus parameter c. The two orbits from period doublings A and B converge to the same c value at infinite period. Insets: X versus t plots for the orbits at large period. The sections CD approximate t(inset B). the orbit 80, eq. (3), and the sections DE the orbits f~°>(inset A) and f~~
D manifolds of P and arise as follows. The rescaling / (X, Y,Z,t)—.(~-j-~—, \
C
i/3\
2~,3 4/3’2h/6C5/3’2h/6) C
toc~ [ orc~ >c>c + pairs of new transversal intersection orbits X~-(t) .
/
replaces the right hand side of the Z equation with 2 1
—
X
—
2i~~Y c213
In the limit of c—+cc the last term in Y vanishes, McCord [11] has shown that this reduced system has a “Liapunov-type” function L= YZ+ ~X~—X, L=z2~o.
elson [1] argues that it must exist continuously for all values of c (it has been observed numerically down
(4)
Then, the periodic solutions cannot exist, and a heteroclinic connection of the l-D manifolds of F ± is not possible as it would require L(t—~—cc) < L ( + cc), contradicting (4). So there is a unique, bounded, non-monotonic, symmetric heteroclinic orbit X0 ( t) formed by the transversal intersection of the 2-D manifolds of Ft For large c
.
.
are created in an infinite sequence of tangent bifurcations [1]. These orbits have 2n + 1 zeroes. As c decreases from c, pairs are destroyed by inverse tangent bifurcations. We postulate that near c=c 0 only X0 and X~remain. The periodic orbits of fig. 1 lie close to these intersection orbits. Inset A shows X0, and inset B, Xj~.Xç does not have any associated periodic orbits, see section 2. We do not known whether all the orbits X~n> 1, do not exist near c0, but we have not observed them.
2. Analysis We will analyse the behaviour of periodic orbits near the critical parameter value c0, using the Shilnikov techniques of Glendinning and Sparrow [6] and Wiggins [7], but with the additional ingredient of symmetry. In particular we will show that the 29
Volume 152, number 1,2
PHYSICS LETTERS A
symmetry prevents the existence of periodic orbits associated with the orbit Xj-. Consider c close to c0. Define u= c—c0, where ui ~ 1 in the following. As u~—p0, period —*cc, periodic orbits remain for increasing lengths oftime in the neighbourhoods of F ~ It is appropriate to partition the flow into the four regions shown in fig. 2. Two of these are local and are placed close to Ft The other two are global, where orbits pass from the vicinity of one fixed point to the vicinity of the other. In the local regions, where the flow is linear, new local coordinate systems can be constructed. The eigenvectors of the linear stability analysis at P are not mutually orthogonal but can be made so by an appropriate similarity transformation. Define the local system (x, y, z) so formed at F— such that the 2D stable manifold lies in the plane x=0, and the 1D unstable one lies along the x-axis. At P define (x~, j~,~) related to (x, y, z) by Sf. Denote by ~ the 2D manifolds, and by ®~the l-D manifolds, of Ft At i= 0 the heteroclinic orbit (3) leaves F along —
S
-.—.---.---.,
-
7 January 1991
~ (x-axis) and approaches P along &~call this connection O~.Denote by [‘f~F) the heteroclinic connections formed by the transversal intersections of which exist Vji. F~j°~ forms the orbit X0(t) and ~ the orbits X~(t). We now construct surfaces of section for the flow. At F insert the strip S~such that it lies in the plane y=0, perpendicular to and at a distance from the origin such that each orbit intersects S1 only once as it spirals to F—, and there is only one intersection of Z~with S1. Let ~ =I~nSf. ~ is the axis x=0. ~ will have three distinct branches because of the three transversal intersections and each branch will be assumed to be perpendicular to t to simplify the analysis, but the results hold good otherwise. The intersection ~ n~, the points (x=0, z=z~) in S1, is also the intersection of Ft’>. A second strip S2 is inserted parallel to the plane x=0 at a small distance h above it. e— intersects S2 at (y=0, z=0). At,u=0, e0 will intersect S2 there. Conjugate surfaces S1 and S~are constructed similarly at F~,except that the roles of ~ are interchanged: ~ nS1 is x=0 with ~perpendicular. Now we construct maps between the surfaces of ~,
—
~,
section. Define T1:S1~S2, T2:S2~S2, T1.S1~S2, T2.S1~S.
( / ‘N
//
-.~--
,/
/
/
X
/
—‘-..--‘...
7
At y = 0 there exist three closed doubly-heteroclinic orbits e0 ~j F~, which are fixed points of the map t=T~’o’~T2~T1.
/•~.
,‘
/
~
-.~
./
/‘
/ /
s
‘f..
“ “~
--~
/
“~
2 --
-
~ —
—
-.
,/ /
r~ ~
-
--
Fig. 2. Location of surfaces of section for analysis of periodic orbits. Heteroclinic connection orbits 8~,F~°~ shown schematically.
30
Shilnikov’s theorems [6,7] guarantee the existence of both symmetric and non-symmetric periodic orbits for ~ near 0. These are also fixed points oft. We can simplify things considerably for symmetric periodic orbits by judicious use of the symmetry 5/. We show in the following lemmas that a symmetnc periodic orbit must intersect S~and S2 on certain invariant lines in these planes. We do this by considering the global maps T2 and T2. Lemma 2.1. A symmetric periodic_orbit corresponds to fixed points of both T2 and T2. By a fixed point we mean a point that maps to its conjugate point (the point produced by the action of the symmetry 9”).
Volume 152, number 1,2
PHYSICS LETTERS A
Proof Consider the action of 9’.
7 January 1991
0) to S2 and S2 can be represented by linear Taylor expansions, the result (6) follows.
Lemma 2.2. In S1 fixed points ofT2 lie on the lines t’1 are simple constants independent of (5) (x, where k z, u) and (x=0, z=z~j~)are the coordinates of F 1~)ms~. Proof Consider any branch of o~,and drop the “(i)” superscript. (0, z~)is a fixed point of T2. Assume that the domain ofT2 is a small-enough neighbourhood of the fixed point such that the map can be represented by a first order Taylor expansion about the fixed point. Note that
T2(x, z~)= (0, z—z~) which follow from a consideration of the actions of ,9’ on ~ and o~,respectively. Then, for any point (x, z z0) eS1, —
~—z,~)= T2(x, z—z,~)= (a1 (z—z~),a2x) where ai and a2 are constants. For a fixed point, .~=x and (~—z~) = (z—zR)=~’ala2=1, and the result (5) follows with k=a1. (~,
T2 maps S2 to S~forward in time. For u=0 the origin (y=O, z=0)=80rS2 is a fixed point of T2. Lemma 2.3. For u=0, the fixed points of the map T2 in S2 lie on the line y = mz~i~o( z) for some value of the constant m.
(6)
Proof Again we consider the action of 9’. The orbite0passesthroughthepoint (X=0, Y=Y0,Z=0) at time t=0, where Y0= —9afi. The orbit is invariant under 9” and intersects the planes S2 and 52 at their origins. Nearly trajectories with initial points (0, Y0 + f, 0) for small e are similarly invariant under 5/ and strike S2 and S2 near the origins. The locus of these intersections as ~varies will be a continuous curve in the planes. Finally, if we assume that the global mappings from small regions around (0, Y0,
Lemma 2.4. For ~i ~ 0, the fixed points of T2 in S2 lie on the line y=mz+lp~~(z) (7) ,
where m and / are simple constants, m the same as in lemma 2.3. Proof Consider any point (z0, y0=mz0) on (6). Under T2 with .= 0 this maps to the point (z0=z0, j70=y0). If ~ 0 and assuming linearity, the point (z0+3z, y0+ ~y) (for arbitrary ~y, 6z) will map to ~=Zo +b1jt+b2&+b3~y,
where b1,2,3, d1,2,3 are simple constants. For fixed points we require Z=z0+ &Z, j7=y0+ ~y=’-~y=e.t, Sz=fy for some constants e and f Corollary 2.5. Combining lemma 2.1 with lemmas 2.2—2.4 we have that any symmetric periodic orbit mist intersect one of the lines eS1 and the line i~eS~. The symmetry .9°also ensures that the same orbit intersects conjugate lines in the lines S~and S2. ~
We now consider the local map T1. In the linear region near F-, the flow in the local coordinate system (x, y, z) is given by x= v1x, y= —wz— ~2/’~ z=wy— v2z, where v~,— v2 ± iw are the eigenvalues of the linearised system, and v1, v2>0. Introduce polar coordinates z = r cos 0, y= r sin 0. We will use coordinates (x1, r~)in S~and (z2=r2cos02, y2=r2 sin 02) in S2. Under the map T~,a point (x1, r1)cS~will strike the plane S2 after a time T ~ln-~(8) ~ at the coordinate point
(~) v2/vl
r2 = Ti
a 02
=
— ill
h in
—.
X1
(9)
Fix x1. This fixes the time of passage Tbetween S~ and S2, and also the period of the entire orbit since 31
Volume 152, number 1,2
PHYSICS LETTERS A
7 January 1991
the time taken for the “global” passages is effectively the same for all orbits. On Si, if (xi, ri )E~’~ then
x~given, substituting (10) into (11) gives a linear equation in the unknown y. z2 and y2 are given by
(lemma 2.2)
back-substitution in (10). Why are there only two periodic orbits when there are three intersection orbits? A plausible and simple explanation is shown in fig. 3. Under the map T~’the (locally) planar part CD of the manifold ~ at P becomes the curve CD’ at F. At P bounded periodic orbits can lie only below ~ (hatching). Under T~ the same orbits must lie on the hatched side at F—. But symmetric periodic orbits also have to lie on the invariant lines in S~. In the case of~~ the two conditions contradict. So there are no periodic orbits associated with Xj. The ratio of the eigenvaiue real parts v2/v1 < 1 which implies that periodic orbits will have an oscillatory approach to the heteroclinic [6]. The results of our analysis are therefore consistent with the numerical results of fig. 1.
r~
+z~[ =
~-
expanding Z~I) in ~ to
~
+Ap+R~1)
/
\/
k =
~
/
\V2/~I
cos( ~ ln p-),
z2 =(~-~+A/1+R~)(~-) ix,
,
k linearly in a. By (9), T2 maps points
~h
\V
\/x1\v2h~1
+Ay+ RIS’ ~
X1J
/~ i~ sink— In ~).
~
(10)
Now, by corollary 2.5, a symmetric periodic orbit must have (x1, r1 )e~) and (z2, y2)e~, that is (lemma 2.4), y2=mz2+1~i. (11) Eqs. (10) and (11) define how symmetric periodic orbits map under T~between and 9~. With ~
References ~.
I----
F~
\/
.(~#
—
[l]D. Michelson, Physica D 19 (1986) 89.
T~
[2] P. Kent and J. Elgin, Travelling-wave solutions of the
+
I———-
F
I
—
‘t~°
S
Fig. 3.The action of the mapping tween S1 and S1.
32
on the manifold ~ be-
Kuramoto—Sivashinsky equation, in preparation (1990). [3] P. Kent and J. Elgin, “Noose” bifurcation ofperiodic orbits, submitted to Nonlinearity (1990). [4] J. Guckenheimer and P. Holmes, Appi. Math. Sci., Vol. 42. Nonlinear oscillations, dynamical systems and bifurcations of vector fields (Springer, Berlin, 1983). [5]H.-C.Chang,Chem.Eng.Sci.47 (1987) 515. [6] P. Glendinning and C. Sparrow, J. Stat. Phys. 35 (1984) 645. [7] S. Wiggins, AppI. Math. Sci., Vol. 73. Global bifurcations and chaos — analytical methods (Springer, Berlin, 1988). [8] Y. Kuramoto and T. Tsuzuki, Prog. Theor. Phys. 55 (1976) 356. [9] C.J. Amick and J.B. LcLeod, Arch. Rat. Mech. Anal. 109 (1990) 139. [lOjA.P. Hooper and R. Grimshaw, Wave Motion 10 (1988) [II] C.K. McCord, J. Math. Anal. AppI. 114 (1986) 584.
PHYSICS LETTERS A
7 January 1991
A Shilnikov-type analysis in a system with symmetry Phillip Kent and John Elgin Department ofMathematics, Imperial College, 180 Queens Gate, London, SW7 2BZ, UK Received 6 September 1990; accepted for publication 29 October 1990 Communicated by A.P. Fordy
We perform a Shilnikov-type analysis for heteroclinic orbits of a symmetric system, which differs from the usual case in that it involves asymmetry, and there is more than one connecting orbit at the critical parameter value.
1. Introduction Consider the dynamical system 1X2_ ~ — — 2 )(— Y Y—Z ~_c2_ —
‘
‘
(1)
‘
where c is a bifurcation parameter and the dot denotes d/dt. It has been the object of both numerical and analytical study, in part because of its relevance to travelling-wave solutions of the Kuramoto— Sivashinsky equation [1]. Here c2 is the velocity of the travelling wave. The purpose of this Letter is to report on a new variety of “heteroclinic Shilnikov” behaviour, introducing only as much background on (1) as is necessary. A fuller report [2] is in preparation. Ref. [31 describes some other unusual features of (1). First we note some basic properties of (1): It has an invariance under the symmetry 1’. (X, Y, Z, t)—~(—X,Y, —Z,
—t)
.
(2)
There is no Liapunov function most orbits are unbounded and escape to infinity. When c=0 there is a single fixed point at the origin. For c> 0 there are two fixed points F ±= ( ±~ 0, 0). The eigenvalues of the linearised stability about F ± are given by 2~+~±c~=0.Atc=0,)~=0or ±i.Forc>OatF, there is one real positive eigenvalue, and a negativereal conjugate pair. So c=0 is a “degenerate Hopf” bifurcation point in the sense that a simple periodic solution is born there, although the full structure of this bifurcation is more complicated (see ref. [4], §7.4 and ref. [5]). At F— (for forward time) there —
28
is a stable 2-D manifold, and an unstable l-D manifold; vice versa at Ft The periodic orbit born at c=0 has Sf-symmetry. We have numerical orbit-following codes to examine itsused bifurcation behavior as c varies. Some resuits are shown in fig. 1. Not all the bifurcation points of the primary curve are shown; we mark only the period-doubling points A and B, and show how the bifurcated orbits evolve to infinite period. For the full bifurcation structure see refs. [1—3]. The .9°-symmetricperiod-doubled orbits from A and B converge as period —~ooto the same parameter value c=c 0. The oscillatory approach of the orbits is typical of “Shilnikov” behaviour [6,7]. What is different here is that at c0 there exist three closed doubly-heteroclinic connecting orbits. Associated with these are the two observed periodic solutions; their forms at large period are shown in the insets to fig. 1. We will show by the analysis of section 2 why the expected third periodic solution does not exist: it is because of the symmetry /1’. One part of each connecting orbit is a heteroclinic orbit linking the l-D manifolds of F~(both insets of fig. 1). It is known [8] to have the analytic form .
.
.
.
3
X=a(—9tanhfit+lltanh fit), (3) where a= l5~/Tiji~~,fi= ~/ii7i~ and c0= a~/~0.8495.In ref. [9] it is shown that (1) has no monotonic heteroclinic orbits. The orbit (3) is the simplest member of a possibly infinite heteroclinic family [1,2,10]. The other parts of the connecting orbits link the 2-
0375-9601/91/$ 03.50 © 1991 — Elsevier Science Publishers B.V. (North-Holland)
Volume 152, number 1,2
~RIOD
PHYSICS LETTERS A
7 January 1991
~
~ I
C
I
0
I
I
0.5
I
1.0
I
1.5
Fig. 1. Numerical continuation from the periodic orbit born at c= 0, period versus parameter c. The two orbits from period doublings A and B converge to the same c value at infinite period. Insets: X versus t plots for the orbits at large period. The sections CD approximate t(inset B). the orbit 80, eq. (3), and the sections DE the orbits f~°>(inset A) and f~~
D manifolds of P and arise as follows. The rescaling / (X, Y,Z,t)—.(~-j-~—, \
C
i/3\
2~,3 4/3’2h/6C5/3’2h/6) C
toc~ [ orc~ >c>c + pairs of new transversal intersection orbits X~-(t) .
/
replaces the right hand side of the Z equation with 2 1
—
X
—
2i~~Y c213
In the limit of c—+cc the last term in Y vanishes, McCord [11] has shown that this reduced system has a “Liapunov-type” function L= YZ+ ~X~—X, L=z2~o.
elson [1] argues that it must exist continuously for all values of c (it has been observed numerically down
(4)
Then, the periodic solutions cannot exist, and a heteroclinic connection of the l-D manifolds of F ± is not possible as it would require L(t—~—cc) < L ( + cc), contradicting (4). So there is a unique, bounded, non-monotonic, symmetric heteroclinic orbit X0 ( t) formed by the transversal intersection of the 2-D manifolds of Ft For large c
.
.
are created in an infinite sequence of tangent bifurcations [1]. These orbits have 2n + 1 zeroes. As c decreases from c, pairs are destroyed by inverse tangent bifurcations. We postulate that near c=c 0 only X0 and X~remain. The periodic orbits of fig. 1 lie close to these intersection orbits. Inset A shows X0, and inset B, Xj~.Xç does not have any associated periodic orbits, see section 2. We do not known whether all the orbits X~n> 1, do not exist near c0, but we have not observed them.
2. Analysis We will analyse the behaviour of periodic orbits near the critical parameter value c0, using the Shilnikov techniques of Glendinning and Sparrow [6] and Wiggins [7], but with the additional ingredient of symmetry. In particular we will show that the 29
Volume 152, number 1,2
PHYSICS LETTERS A
symmetry prevents the existence of periodic orbits associated with the orbit Xj-. Consider c close to c0. Define u= c—c0, where ui ~ 1 in the following. As u~—p0, period —*cc, periodic orbits remain for increasing lengths oftime in the neighbourhoods of F ~ It is appropriate to partition the flow into the four regions shown in fig. 2. Two of these are local and are placed close to Ft The other two are global, where orbits pass from the vicinity of one fixed point to the vicinity of the other. In the local regions, where the flow is linear, new local coordinate systems can be constructed. The eigenvectors of the linear stability analysis at P are not mutually orthogonal but can be made so by an appropriate similarity transformation. Define the local system (x, y, z) so formed at F— such that the 2D stable manifold lies in the plane x=0, and the 1D unstable one lies along the x-axis. At P define (x~, j~,~) related to (x, y, z) by Sf. Denote by ~ the 2D manifolds, and by ®~the l-D manifolds, of Ft At i= 0 the heteroclinic orbit (3) leaves F along —
S
-.—.---.---.,
-
7 January 1991
~ (x-axis) and approaches P along &~call this connection O~.Denote by [‘f~F) the heteroclinic connections formed by the transversal intersections of which exist Vji. F~j°~ forms the orbit X0(t) and ~ the orbits X~(t). We now construct surfaces of section for the flow. At F insert the strip S~such that it lies in the plane y=0, perpendicular to and at a distance from the origin such that each orbit intersects S1 only once as it spirals to F—, and there is only one intersection of Z~with S1. Let ~ =I~nSf. ~ is the axis x=0. ~ will have three distinct branches because of the three transversal intersections and each branch will be assumed to be perpendicular to t to simplify the analysis, but the results hold good otherwise. The intersection ~ n~, the points (x=0, z=z~) in S1, is also the intersection of Ft’>. A second strip S2 is inserted parallel to the plane x=0 at a small distance h above it. e— intersects S2 at (y=0, z=0). At,u=0, e0 will intersect S2 there. Conjugate surfaces S1 and S~are constructed similarly at F~,except that the roles of ~ are interchanged: ~ nS1 is x=0 with ~perpendicular. Now we construct maps between the surfaces of ~,
—
~,
section. Define T1:S1~S2, T2:S2~S2, T1.S1~S2, T2.S1~S.
( / ‘N
//
-.~--
,/
/
/
X
/
—‘-..--‘...
7
At y = 0 there exist three closed doubly-heteroclinic orbits e0 ~j F~, which are fixed points of the map t=T~’o’~T2~T1.
/•~.
,‘
/
~
-.~
./
/‘
/ /
s
‘f..
“ “~
--~
/
“~
2 --
-
~ —
—
-.
,/ /
r~ ~
-
--
Fig. 2. Location of surfaces of section for analysis of periodic orbits. Heteroclinic connection orbits 8~,F~°~ shown schematically.
30
Shilnikov’s theorems [6,7] guarantee the existence of both symmetric and non-symmetric periodic orbits for ~ near 0. These are also fixed points oft. We can simplify things considerably for symmetric periodic orbits by judicious use of the symmetry 5/. We show in the following lemmas that a symmetnc periodic orbit must intersect S~and S2 on certain invariant lines in these planes. We do this by considering the global maps T2 and T2. Lemma 2.1. A symmetric periodic_orbit corresponds to fixed points of both T2 and T2. By a fixed point we mean a point that maps to its conjugate point (the point produced by the action of the symmetry 9”).
Volume 152, number 1,2
PHYSICS LETTERS A
Proof Consider the action of 9’.
7 January 1991
0) to S2 and S2 can be represented by linear Taylor expansions, the result (6) follows.
Lemma 2.2. In S1 fixed points ofT2 lie on the lines t’1 are simple constants independent of (5) (x, where k z, u) and (x=0, z=z~j~)are the coordinates of F 1~)ms~. Proof Consider any branch of o~,and drop the “(i)” superscript. (0, z~)is a fixed point of T2. Assume that the domain ofT2 is a small-enough neighbourhood of the fixed point such that the map can be represented by a first order Taylor expansion about the fixed point. Note that
T2(x, z~)= (0, z—z~) which follow from a consideration of the actions of ,9’ on ~ and o~,respectively. Then, for any point (x, z z0) eS1, —
~—z,~)= T2(x, z—z,~)= (a1 (z—z~),a2x) where ai and a2 are constants. For a fixed point, .~=x and (~—z~) = (z—zR)=~’ala2=1, and the result (5) follows with k=a1. (~,
T2 maps S2 to S~forward in time. For u=0 the origin (y=O, z=0)=80rS2 is a fixed point of T2. Lemma 2.3. For u=0, the fixed points of the map T2 in S2 lie on the line y = mz~i~o( z) for some value of the constant m.
(6)
Proof Again we consider the action of 9’. The orbite0passesthroughthepoint (X=0, Y=Y0,Z=0) at time t=0, where Y0= —9afi. The orbit is invariant under 9” and intersects the planes S2 and 52 at their origins. Nearly trajectories with initial points (0, Y0 + f, 0) for small e are similarly invariant under 5/ and strike S2 and S2 near the origins. The locus of these intersections as ~varies will be a continuous curve in the planes. Finally, if we assume that the global mappings from small regions around (0, Y0,
Lemma 2.4. For ~i ~ 0, the fixed points of T2 in S2 lie on the line y=mz+lp~~(z) (7) ,
where m and / are simple constants, m the same as in lemma 2.3. Proof Consider any point (z0, y0=mz0) on (6). Under T2 with .= 0 this maps to the point (z0=z0, j70=y0). If ~ 0 and assuming linearity, the point (z0+3z, y0+ ~y) (for arbitrary ~y, 6z) will map to ~=Zo +b1jt+b2&+b3~y,
where b1,2,3, d1,2,3 are simple constants. For fixed points we require Z=z0+ &Z, j7=y0+ ~y=’-~y=e.t, Sz=fy for some constants e and f Corollary 2.5. Combining lemma 2.1 with lemmas 2.2—2.4 we have that any symmetric periodic orbit mist intersect one of the lines eS1 and the line i~eS~. The symmetry .9°also ensures that the same orbit intersects conjugate lines in the lines S~and S2. ~
We now consider the local map T1. In the linear region near F-, the flow in the local coordinate system (x, y, z) is given by x= v1x, y= —wz— ~2/’~ z=wy— v2z, where v~,— v2 ± iw are the eigenvalues of the linearised system, and v1, v2>0. Introduce polar coordinates z = r cos 0, y= r sin 0. We will use coordinates (x1, r~)in S~and (z2=r2cos02, y2=r2 sin 02) in S2. Under the map T~,a point (x1, r1)cS~will strike the plane S2 after a time T ~ln-~(8) ~ at the coordinate point
(~) v2/vl
r2 = Ti
a 02
=
— ill
h in
—.
X1
(9)
Fix x1. This fixes the time of passage Tbetween S~ and S2, and also the period of the entire orbit since 31
Volume 152, number 1,2
PHYSICS LETTERS A
7 January 1991
the time taken for the “global” passages is effectively the same for all orbits. On Si, if (xi, ri )E~’~ then
x~given, substituting (10) into (11) gives a linear equation in the unknown y. z2 and y2 are given by
(lemma 2.2)
back-substitution in (10). Why are there only two periodic orbits when there are three intersection orbits? A plausible and simple explanation is shown in fig. 3. Under the map T~’the (locally) planar part CD of the manifold ~ at P becomes the curve CD’ at F. At P bounded periodic orbits can lie only below ~ (hatching). Under T~ the same orbits must lie on the hatched side at F—. But symmetric periodic orbits also have to lie on the invariant lines in S~. In the case of~~ the two conditions contradict. So there are no periodic orbits associated with Xj. The ratio of the eigenvaiue real parts v2/v1 < 1 which implies that periodic orbits will have an oscillatory approach to the heteroclinic [6]. The results of our analysis are therefore consistent with the numerical results of fig. 1.
r~
+z~[ =
~-
expanding Z~I) in ~ to
~
+Ap+R~1)
/
\/
k =
~
/
\V2/~I
cos( ~ ln p-),
z2 =(~-~+A/1+R~)(~-) ix,
,
k linearly in a. By (9), T2 maps points
~h
\V
\/x1\v2h~1
+Ay+ RIS’ ~
X1J
/~ i~ sink— In ~).
~
(10)
Now, by corollary 2.5, a symmetric periodic orbit must have (x1, r1 )e~) and (z2, y2)e~, that is (lemma 2.4), y2=mz2+1~i. (11) Eqs. (10) and (11) define how symmetric periodic orbits map under T~between and 9~. With ~
References ~.
I----
F~
\/
.(~#
—
[l]D. Michelson, Physica D 19 (1986) 89.
T~
[2] P. Kent and J. Elgin, Travelling-wave solutions of the
+
I———-
F
I
—
‘t~°
S
Fig. 3.The action of the mapping tween S1 and S1.
32
on the manifold ~ be-
Kuramoto—Sivashinsky equation, in preparation (1990). [3] P. Kent and J. Elgin, “Noose” bifurcation ofperiodic orbits, submitted to Nonlinearity (1990). [4] J. Guckenheimer and P. Holmes, Appi. Math. Sci., Vol. 42. Nonlinear oscillations, dynamical systems and bifurcations of vector fields (Springer, Berlin, 1983). [5]H.-C.Chang,Chem.Eng.Sci.47 (1987) 515. [6] P. Glendinning and C. Sparrow, J. Stat. Phys. 35 (1984) 645. [7] S. Wiggins, AppI. Math. Sci., Vol. 73. Global bifurcations and chaos — analytical methods (Springer, Berlin, 1988). [8] Y. Kuramoto and T. Tsuzuki, Prog. Theor. Phys. 55 (1976) 356. [9] C.J. Amick and J.B. LcLeod, Arch. Rat. Mech. Anal. 109 (1990) 139. [lOjA.P. Hooper and R. Grimshaw, Wave Motion 10 (1988) [II] C.K. McCord, J. Math. Anal. AppI. 114 (1986) 584.