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Chaos in the 1:2:3 Hamiltonian normal form
Physica D 44 (1990) 397-406 North-Holland
CHAOS IN THE 1 : 2: 3 HAMILTONIAN NORMAL FORM Igor H O V E I J N and Ferdinand V E R H U L S T Department of Mathematics, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands Received 24 October 1989 Revised manuscript received 2 March 1990 Accepted 5 April 1990 Communicated by H. Flaschka
The normal form of the Hamiltonian 1 : 2 : 3 resonance to degree 3 contains seven families of periodic solutions of which one can be complex unstable. Associated with this complex unstable solution is an invariant manifold N on which the dynamics can be characterised completely; one of the ingredients of N is a set of homoclinic orbits. In the normal form to degree 4 the set of homoclinic orbits breaks up, for certain parameter values, into one homoclinic orbit. This enables us to apply Silnikov-Devaney theory to prove, at this stage numerically, the existence of a horseshoe map in the system with the implication of non-integrability and chaos in the normal form.
1. Introduction Hamiltonian systems with two or more degrees of freedom are in general non-integrable. In this paper we shall consider such systems with three degrees of freedom near a non-degenerate critical point of the equations of motion. Supposing that all the eigenvalues of the quadratic part of the Hamiltonian in the neighbourhood of the critical point are distinct and purely imaginary, the Hamiltonian can be expressed as H =
1
2
+ qt) +---
+//3+//4+
+
1
2
+
....
(1)
where H m is a homogeneous polynomial of degree m in the variables p, q. We simplify the Hamiltonian (1) by a canonical near-identity transformation which leaves H2, which represents the quadratic terms, invariant and which removes a large number of terms of H a, H 4, etc. See refs. [4, 9] and for a recent survey ref. [5]. This transformation process is called normalisation and it is carried out to a certain degree m > 3; the system is then truncated from degree 0167-2789/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
m + 1 on and the resulting Hamiltonian system is called the normal form to degree m. A wide-spread idea is that normal forms are important but also, that they are only describing relatively uninteresting, regular parts of the flow like critical points, periodic solutions, invariant tori, etc. We shall show that this idea is not correct by demonstrating the existence of a horseshoe mapping and chaos in a normal form. The frequencies tOl, to2, to3 in (1) satisfy a resonance relation of order k if there exist integers k l , k2, k 3 such that kltO l + k2to 2 4" k3to 3 --- 0, where k = Ikll + [k2J + [k31. The normal form is complicated by the presence of resonant terms, or slightly more precise, the more resonance relations with k < m, the more resonant terms in the normal form to degree m. If the frequencies are all positive, the first-order resonances, involving at least two resonance relations and so two independent combination angles (see ref. [9]) in the normal form for m = 3 are (tOl, toz, to2)= (1, 2, 1), (1, 2, 2), (1, 2, 3) and (1, 2, 4). The normal form to degree m has at least two independent integrals: the normalised Hamiltonian H 2 + K truncated to degree rn and H2; one
398
I. Hoveijn and F. Verhulst / Chaos m the 1 : 2: 3 Hamiltonian normal form
may also use as independent integrals H 2 and K, which contains the terms from degree 3 to degree m. Note that [/42, K] = 0 by normal form construction with [ , ] the Poisson bracket. If a third independent integral exists, it follows from approximation theory that the amount of chaos in the original Hamiltonian, before normalisation and truncation, is small near the equilibrium position corresponding with the critical point. For technical details see refs. [9-11]. At present we know that the 1 : 2 : 1 resonance in normal form to degree 3 is non-integrable [7], the 1 : 2 : 2 resonance in normal form to degree 3 is integrable [21. In this paper we shall demonstrate the nonintegrability of the normal form of the 1 : 2 : 3 resonance. Using the generators at this resonance we shall write down the normal forms to degree 3 and to degree 4. The first case has been analysed by van der Aa [1], who lists seven short-periodic solutions, which are in fact families of periodic solutions parametrised by the value E of H 2. Only one of these periodic solutions, the normal mode in the second degree of freedom, may be complex unstable. In this case we wish to apply the theory of Silnikov, applied by Devaney [6] to Hamiltonian systems, to establish the existence of a horseshoe mapping generated by the phase flow near the complex unstable solution. For this we need the existence of a transversal homoclinic orbit, around which the flow, spiralling inward and outward, is guided back to the periodic solution, thus giving rise to the horseshoe mapping. It turns out that, in the normal form to degree 3, the complex unstable periodic solution is located on an invariant manifold M~ containing a one-parameter family of homoclinic orbits, a homoclinic set. The manifold M~ in its turn is imbedded in the invariant manifold N: H 2 = E, H 3 = 0. On Mj one also finds two heteroclinic orbits of the complex unstable periodic solution, one homoclinic orbit of another periodic solution and a one-parameter family of periodic orbits. There is another invariant manifold M 2 in N intersecting M 1 along the two heteroclinic orbits
and the special homoclinic orbit. The other orbits on M 2 form a one-parameter family of homoclinic orbits of the other periodic solution. The complement of M 1 U M 2 in N consists of periodic orbits only. (M1 and M 2 are separatrix manifolds in N and M 1 N M 2 consists of separatrices on M 1 and M2.) Because of the presence of a homoclinic set the theory of Devaney-Silnikov does not apply to the normal form to degree 3. Considering the normal form to degree 4 we are able to show that of the homoclinic set in M 1 one homoclinic orbit survives which establishes the existence of a horseshoe map. The implication is that an infinite set of unstable periodic solutions exists near the homoclinic orbit and that the phase flow of the normal form to degree 4 is non-integrable. 2. T h e n o r m a l f o r m s o f the 1 : 2 : 3 r e s o n a n c e
The normal form to degree 3 has been written down and analysed by van der Aa [1]. We reformulate some of the results and we extend them to degree 4. A convenient formulation of normal forms uses complex coordinates: u j = p ~ + i q j and v j = p j iq i ( j = 1, 2, 3). Now the generators f of the normal form can be found from the relation: [He, f ] = 0. This was carried out by Fekken [8] for the general resonance case and for the first-order resonances mentioned above in particular. We confine ourselves to listing the generators, ordered by degree k: k = 2,
A 1 -~- UlU1,
A 2 = u2u2,
A 3 = U3U3;
k = 3,
k=4,
B1=
UlU2U3,
B 2 = u1u2u3,
B 3 = u2v2,
B 4 = u2u2;
C l=u3v3,
C 2=v3u3,
C 3 = UlU2U3,
k = 5,
D1
-_ -_ U 23U,2 3,
k=7,
E l=uSv2v3,
C 4 = ulu2u3.
D2 = U2U3, 3 z. F 2=vlSu2u3 .
L Hoveijn and F, Verhulst / Chaos in the I : 2: 3 Hamiltonian normal form
T h e r e exist relations like B I B 2 = A 1 A 2 A 3 between these generators but we will not use them at the moment. F u r t h e r m o r e we shall use only generators up to degree k = 4; the remaining generators have been listed for the sake of completeness. W h e n analysing Hamiltonians near an equilibrium position it is natural to rescale p = •,5, q = • q and accordingly u and v. The Hamiltonian is then divided by •2, we omit the bars and (1) becomes H = H 2 + •H 3 + •2H 4 +
E 3 ....
(2)
We take 0 < • << 1 to indicate that we are considering a small neighbourhood of the equilibrium position, • 2 is a measure for the energy level we are considering. Using the generators the normal forms become H 2 = ½i(A 1 + 2A 2 + 3A3), H 3 = aB 3 -
~B 4 + bB l - bB2,
H 4 = i ( c l A2 + c 2 A2 +C3 A2 + c 4 A 1 A 2 +c5A1A 3 + c6AzA3) + d C 1 - d C 2 + eC 3 - -
eC 4
with a , b , d , e ~ C and cl, c z , . . . , c 6 E• ~. The original Hamiltonian system is real so in complex coordinates the normal form can be taken real or purely imaginary. It is possible to simplify the normal form slightly by rigid rotation on coordinate space and putting a = ~, b = b and either d = d or e = ~. But since it is not obvious which one should be preferred we only put a = and b = b.
3. The normal form to degree 3 3.1. Periodic solutions
The system was studied by van der Aa [1] and we summarise the results. The normal form is
399
H 2 + EH 3 with two independent integrals: H 2 and H 3 . The simplest periodic solutions are those with only one coordinate pair (pj, qj) non-vanishing or the cases of periodic solutions located in an e-neighbourhood of such a state; they are called normal modes and there are two of them. One solution we call 7'/"3 with p~ = ql =1192~-- q2 = 0, q3 = (2/3)1/2 sin(3t), P3 = (2/3)1/2 cos(3t) and one we call zr 2 with p l = q l = p 3 = q 3 = O , q2 = cos(2t), P2 = sin(2t) where E is the value of H 2. T h e r e is no normal m o d e with only ( p l , ql)4: (0, 0), in the generic situation. The next type of periodic solutions consists of those with only one coordinate pair equal to zero; there is only one of this type, which we call ,/14 with P2 = q 2 = 0. These periodic solutions are located on the invariant level set N: H E = E, H 3 = 0. There are four periodic solutions in general position: all coordinate pairs unequal to zero. They cannot be complex unstable [1] and they do not lie on M 1 (see section 3b for a definition). We will call them: ~rs, rr6, ~r 7, -rr8 and they will play no important part in our analysis. The location of the periodic solutions can be depicted on the so-called action simplex in action space, see fig. 1. We determine the stability type of a periodic solution by considering it as a critical point of a reduced system and by determining the stability of this critical point. The stability is determined by the eigenvalues of the linearized differential equation. In our case there are four eigenvalues. A real pair (A, - A ) will be called H(yperbolic), an imaginary pair (iA, - i A ) will be called E(lliptic) and a quadruple ( A , - A , A , - A) will be called C(omplex). So stability type E H means eigenvalues iA, - iA,/x, - / z . O f course the stability type depends generally on the p a r a m e t e r s a and b of H 3. The results are summarised in table 1. From now on we will only consider 0 < a < b, so zr 2 is complex unstable. As we see from table 1 the interesting level set is N. F u r t h e r m o r e since the eigenvalues of zr 3 and zr4 have multiplicity 2 their stability type may, under a perturbation, turn into C.
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L Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
7"3 l
71"3
/
/
71"2
'/t'5, 6
/
7/'7,8
/
/"2
T1
Fig. 1. Action simplex for H 2 = 1. Table 1 Summary of the results for periodic solutions.
and q we obtain the following equations:
Periodic
Stability
Multiplicity of
L e v e l set
solution
type
eigenvalues
( H 2 = 1)
~2
C ( a 2 - b 2 < 0), H H ( a 2 - b E > 0)
1 1
H3= 0 H3= 0
"/7"3 7r4 rr 5 rr 6 ~'7
HH EE EH EH EE
2 2 1 1 1
H H H n n
~'s
EE
1
H 3 = h32 ~ 0
dll = P2( 2aq~ -- bq3), /~1 = P 2 ( - 2aPl - bP3),
3= 3= a = 3= 3=
0 0 h31 ~ 0 h31 q: 0 h32 q: 0
c13 = Pz( bql ), 1~3 =P2( bPl), f~2 = a ( p ~ -
q2) _ b ( p , p 3 + qlq3),
H2(p,q)=l, N o t e that
H3(P,q)=O,
H 3
q2=0.
(3)
is linear in P2 and q2 so:
3.2. Orbits on the invariant manifold N T h e periodic solutions listed in table 1 are relative equilibria o f H = H 2 + E H 3 with respect to H 2. T h e r e f o r e they consist of fixed points of the H a m i l t o n i a n H , = H - r H 2. O n the level set N we have the periodic solutions rr 2, zr 3 and ~r4 for which ~- = 1, so in this case H~ = e n 3. Instead of using reduction (the r e d u c e d phase space has conelike singularities) we consider the flow o f n 3 on the level set N, following the m e t h o d o f D u i s t e r m a a t [7]. In the original coordinates p
0H 3 0H 3 t~2P2 --/92q2 = ~---p-~-2P2 + --~-2 q2 = H 3
and since we consider H 3 = 0 we have Q2/P2 = and q2 move along a straight line, which we choose to be q2 = 0. Next we set x = (qt, P l , q 3 , P3) and f ( x ) = a(x 2 - x 2) + b ( x 2 x 4 +XlX3); then we obtain the differential equations
q2/P2:192
YC=pz.4X,
152=-f(x),
(4)
L Hoveijn and F. Verhulst / Chaos in the 1: 2: 3 Hamiltonian normal form
where A is a 4 × 4 matrix obtained from (3) with eigenvalues
where C1 = c 2 = A e
5:a+(a2-bZ)
1/2
and
0
W e solve (4) formally f~ p 2 ( s ) d s , then
by
putting
z(t) =
(5)
(6)
-p2 = - f ( x ( z ) ) .
We integrate the right-hand side with respect to z: I d V ( z ) / d z = f ( x ( z ) ) then ~ + 1 d V ( z ) / dz=0. Introducing a new variable w = P2 we can put the equation for z into Hamiltonian form: H ( z , w ) = ~wl 2 + ½V(z). Finally we use z(t)lt= o = 0 and w(t)lt=o -- p2(0) then, after some calculations we find
+ 3q (o) + 3p (o)]
1 so H ( z , w ) = 7. W e cannot explicitly solve z as a function of time but from t = ]~¢')[1 - V(z)] -1/2 d z we have a "period function"
r=
dz
a 2 + b 2 + ia( b 2 - a 2) 1/2 = P ei~, tO ----- ( b 2 - -
where a i are the complex eigenvectors of A and c i are complex "initial values". The next step is to derive an equation for z:
v(o) = [qf(o)
ia,
c3 = ~4 = n e it3,
x ( t ) = c I e x~z a I + ... + c 4 e A~z a4,
e
401
(7)
with z _ < 0 < z + and for z ~ ( z _ , z + ) , V ( z ) < 1, holds. For V ( z ) we find, after substituting the solution (5) into (6) and integrating f ( x ( z ) ) :
a2) 1/2.
Having "solved" the equations of motion (3) on N we will proceed to classify the orbits. From the equations of motion (3) we see that ql = q3----0 and Pl =P3 = 0 are invariant manifolds and are also the stable and unstable manifolds of the fixed point x = 0, P2 = 1. Let us consider p t --P3 = 0. We call this two-dimensional invariant manifold M 1 and from H 2 = 1 we find q2 + 2p22 + 3q2 = 2: M~ is an ellipsoid. Orbits on M~ have initial conditions c 3 = c 4 -~- 0 or B -- 0, then V ( z ) reads V ( z ) = A 2 e2az[4b2 + 2p cos(2~oz + 2 a + ~p)]. We can distinguish four types of orbits on M1: (1) (2) (3) (4)
z _ = -oo, V ( z + ) = 1, V'(z+) # 0, T infinite; z _ = -oo, V ( z + ) = 1, V'(z+) = O, T infinite; V ( z _ ) = 1, V ' ( z _ ) ~ O, V ( z + ) = 1, T finite; V ( z _ ) = 1, V ' ( z _ ) = O, V ( z + ) = 1, T infinite.
Type (3) clearly corresponds with a family of periodic orbits and type (1) with a family of homoclinic orbits. Types (2) and (4) are special orbits: two conditions in z+ or z_, type (2) corresponds with a heteroclinic orbit and type (4) with a homoclinic orbit. We also have fixed points on Ml:
x= +
b2+3a2
(-b,O,a,O),
P2 = 0 corresponding with ~'4, x = + ( 2 / 3 ) t / 2 ( 0 , 0 , 1,0),
V ( z ) ---A 2 e2"Z[4b 2 + 2p cos(2toz + 2or + ~o)]
P2 -- 0 corresponding with *r 3,
+ B 2 e-2"~[4b 2 + 2p cos(2taz - 213 - ~p)],
(8)
x=0, P2 = + 1 corresponding with ~r2.
402
L Hoveijn and F. Verhulst/Chaos in the 1 : 2 : 3 Hamiltonian normal form q3
ql
Fig. 2. Projection of phase curves on the ql, q3 plane (P2 > 0).
To see how these orbits and fixed points are arranged on M~, we first note that all orbits, projected on the ql, q3 plane in the P2 direction are elliptic spirals (see fig. 2). If we intersect M 1 with the cylinder q21 - ( 2 a / b ) q m q 3 + q23 ~ C2, Pl =P3 = 0 (some constant c) then we see that the flow on M~ is outwards on the cylinder when P2 > 0 and inwards when P2 < 0. If we intersect M 1 with the plane Pm = P 2 =P3 = 0 then we have ql = t13 = 0 and/~2 changes sign at the fixed points corresponding with zr 3 and ~r4. So we get the following picture: The fixed points corresponding with zr4 are surrounded with a family of periodic orbits, these families are separated from other orbits by homoclinic orbits of the fixed points corresponding with zr 2 by two heteroclinic orbits. The fixed points corresponding with zr 2 are connected with themselves by a family of homoclinic (strictly speaking in this picture by heteroclinic) orbits (see fig. 3). On M 1 the flow is globally directed from P2 = 1 to P2 = - 1. We could do the same for ql = q3 = 0 then we get an invariant manifold which looks exactly like M~ with the flow globally directed from P2 = - 1 to P2 = 1. We should, however, identify P2 = 1 and P2 = - 1 since both corre-
spond with rr 2. We have in fact a twofold covering of the reduced phase space. Let us next consider general initial conditions c~, c 2, c3, c 4 (c I = ?2, c3 = 74 of course) with A :~ 0 and B 4: 0. Now both z_ and z+ are finite. In general V ' ( z ) :/: 0 and V'(z÷) 4:0 so T is finite: the orbits are periodic. When V'(z_) or V'(z÷) happens to be zero the corresponding orbits are homoclinic to the fixed points corresponding with 7r3, since these are the only unstable fixed points left and the periodic orbits cannot be unstable or asymptotically stable. This means there exists another two-dimensional invariant manifold M 2 filled with homoclinic orbits and two exceptional heteroclinic orbits (V'(z_)= 0 and V'(z÷)= 0). On M 1 and M 2 we have separatrices and M 1 and M 2 intersect transversally along these separatrices. This intersection is transversal since the linear stable and unstable manifolds of the fixed points corresponding with zr 3 are transversal and these points belong to M~ and M 2. The stable and unstable manifolds of 7r2 do not intersect transversally: the conditions in Devaney's theorem are not fulfilled. Indeed there can be no spiralling around this three-dimensional manifold orbits of homoclinic orbits (in the original phase space) since H 2 and H 3 are con-
L Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
403
Fig. 3. Phase curves on M r Projection on x, y plane. Dashed lines indicate negative z coordinate, solid lines indicate positive z coordinate.
served. We cannot conclude that H = H 2 + EH 3 is non-integrable; however, numerical calculations suggest that it is indeed non-integrable.
4. The normal form to degree 4 4.1. Pen'odic solutions and their stability
The position of the normal m o d e ,/72 remains unchanged and since 7T2 is complex unstable it will still be complex unstable. Now '172 is located on the level set: H 2 = E, K = ¢2c2E2 and it has the period: "rr(1 + eEc2E) - l . Also the position of rr 3 remains unchanged. Now 1r3 lies on the level set H 2 = E , K=E2Ca(2E) 2. Again the period changes, the new period is: 2"rr(3 + 4E2c3E)-l. We find the stability type of 7r 3 from the discriminant
We will now turn our attention to the Hamiltonian system H = H 2 + K, K = e H 3 + ~2H4. First let us see what happens to the periodic solutions. The stability type of the periodic solutions with different eigenvalues will not change. On the other hand the stability type of ~'3 and rr4 (with double eigenvalues) will change. To determine their stability we must have a (good enough) approximation of their location in phase space. In the case of 7r4 this involves lengthy calculations, which we want to avoid. Instead we impose conditions on the p a r a m e t e r s in H 4 such that the periodic solutions remain on the position of the normal form to degree 3. As there are 4 conditions and 10 parameters in H 4, this condition is not too restrictive. It turns out that only for ~'4 we need restrictions on the parameters.
ot 2
1 4 ,~,C5 2 _ c -- fl = - - E 2 b 2 ( c 5 -- C6 )2 + ~E
)2
of the eigenvalue equation: h 4 + 2 a h 2 +/3 = 0, so 7r3 will be complex unstable. To find rr4 on the same position as in the normal form to degree 3 we need the conditions: d l=d E=O,
c 1= -(aE/2b2)cs,
ca = - (b2/2a2)cs. These conditions are somewhat stronger than necessary as we even kept the period fixed. ~'4 lies on the level set H 2 = E, K = 0 but if we allow for a slightly different period then 7r4 lies on H 2 = E, K = [ 4 ¢ 2 E 2 / ( b 2 + 3 a 2 ) ] ( c l b 4 + caa 4 + 2csa2b2). T h e stability type of zr4 follows from the discriminant a 2 - / 3 . In this c a s e ot 2 - / 3 can be positive or negative so zr4 is stable of type E E or complex unstable.
404
I. Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
4.2. Homoclinic and heteroclinic orbits In general we expect a heteroclinic orbit and a o n e - p a r a m e t e r family of homoclinic orbits to break up under a small perturbation. Indeed the heteroclinic connections between 7r 2 and zr 3 will 4 no longer exist since c 2 ~ ~c 3 so in general 7r 2 and rr 3 are lying on different level sets. We expect the o n e - p a r a m e t e r family of homoclinic orbits to break up in discrete homoclinic orbits. We used numerical analysis to confirm this. Before doing so we had to choose values for the p a r a m e t e r s in H 4. We took values close to the ones obtained from normalising the potential problem:
H(p,q)
= 1~ [ ( P l 2 +q12) + 2(p22 + q Z2) +3(p2 +q2)]
+ e(aq~q 2 + bqlq2q3).
If we use the coordinates y then we see from the normal form to degree 3 that Y2 = 0 intersects the homoclinic orbits perpendicularly and symmetrically. In this system the stable and unstable manifolds of 7r 2 intersect Y2 = 0 along the Y4 axis (Yl =Y3 = 0). For the normal form to degree 4 we took initial values on the stable and unstable manifolds obtained by linearisation near the critical point and we computed the intersection with Y2 = 0. We observed two curves in Y2 = 0 intersecting transversally in one point so for this choice of p a r a m e t e r s only one homoclinic orbit is left. The precision of the numerical calculations is such that the conserved quantities H 2 and K = a n 3 + a2H 4 are conserved to within 10 -6. This makes it plausible that the computed orbits are close to the real orbits: error propagation need not always be perpendicular to grad H 2 and grad K. F u r t h e r m o r e the angle of the intersecting 1 curves is roughly ~radians. Since we now have a transverse homoclinic orbit of a complex unstable
critical point corresponding with the normal mode ~'2, the results of Devaney apply.
5. Discussion We analysed two truncated normal forms H = H 2 + K of the 1 : 2 : 3 resonance, K = EH 3 and K = EH 3 + E2H4 . The normal form to degree 4 is a perturbation of the normal form to degree 3. Both normal forms have two independent integrals, H 2 and K, since the normal forms are constructed such that [H2, K ] = 0. If there exists a third independent integral then the system is completely integrable (modulo some non-elementary integrations). The energy manifold is foliated into submanifolds by the level sets of the additional integrals. For the normal form to degree 3 it is possible to characterise the phase curves completely on the level set H 2 = E, K = E H 3 = 0. First restrict to H 2 = E (reduction), then all orbits outside two invariant manifolds are periodic, on the invariant manifold M 1 we have homoclinic orbits, heteroclinic orbits and again periodic orbits (see fig. 3). It is tempting to consider the period function as a possible third integral, but there is a striking resemblance with the 1 : 2 : 1 resonance with a similar structure of the phase space, where the period function displays infinite branching [7]. Numerical results also suggest that the normal form to degree 3 has no analytic third integral (fig. 4). Because of the existence of the twodimensional invariant manifold M~ there can be no spiralling around the (indeed non-transversal) homoclinic orbits so Devaney's results do not apply. For a geometric interpretation of the results for the normal form to degree 3 see ref. [11]. Though the normal form to degree 3 seems to be non-integrable we cannot conclude this from Devaney's result, we can do so, however, for the normal form to degree 4, where we find a transverse homoclinic orbit. In the latter case we know that the complicated behaviour associated with non-integrability is as
L Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
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complicated as a horseshoe map at least near the homoclinic orbit. In the former case however, we cannot yet characterise the nature of the "complicated behaviour". Another question remaining to be answered for both systems is: how large is the region in phase space with chaotic orbits and also what is the measure of the set of chaotic orbits, and how does this measure depend on the parameters in the normal form? Now the original Hamiltonian system is nonintegrable. Note that the normal form to degree 4 contains a horseshoe map which is a structurally
H =/-I2 + H3. Initial values
are such that
H 2 = 1,
H 2 = 1, H 3 ~ 0.
stable phenomenon: it cannot be perturbed away by adding the higher-order terms of the full Hamiltonian system, except of course in special examples. Concluding we might say that normalisation drastically simplifies Hamiltonian systems but the motion in phase space, induced by Hamiltonian systems in normal form, can be very irregular.
Acknowledgements
During the course of this work we had many stimulating discussions with J.J. Duisterrnaat. We
406
I. Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
w o u l d like to t h a n k t h e r e f e r e e s o f P h y s i c a D f o r their valuable comments.
References [1] E. van der Aa, Celest. Mech. 13 (1983) 163. [2] E. van der Aa and F. Verhulst, SIAM J. Math. Anal. 15 (1984) 890. [3] R. Abraham and J.E. Marsden, Foundations of Mechanics, 2nd Ed. (Benjamin/Cummings, Menlo Park, 1978). [4] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, Berlin, 1988). [5] V.I. Arnol'd, ed., Encyclopaedia of Mathematical Sciences, Vol. 3. Dynamical Systems (Springer, Berlin, 1988).
[6] R.L. Devaney, J. Diff. Eqs. 31 (1976) 431. [7] J.J. Duistermaat, Ergod. Theor. Dynam. Syst. 4 (1984) 553. [8] A. Fekken, On the normal form of a fully resonant Hamiltonian function, preprint Free University Amsterdam, No. 317 (1986). [9] J. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, Vol. 59 (Springer, Berlin, 1985). [10] F. Verhulst, Proc. ICIAM, eds. A.H.P. van der Burgh and R.M.M. Matthey, CWI Tract 36 (1987) 129. [11] F. Verhulst and I. Hoveijn, in: Geometry and Analysis in Nonlinear Dynamics, Pitman Research Notes in Mathematics, eds. H.W. Broer and F. Takens (Pitman, London, 1990).
CHAOS IN THE 1 : 2: 3 HAMILTONIAN NORMAL FORM Igor H O V E I J N and Ferdinand V E R H U L S T Department of Mathematics, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands Received 24 October 1989 Revised manuscript received 2 March 1990 Accepted 5 April 1990 Communicated by H. Flaschka
The normal form of the Hamiltonian 1 : 2 : 3 resonance to degree 3 contains seven families of periodic solutions of which one can be complex unstable. Associated with this complex unstable solution is an invariant manifold N on which the dynamics can be characterised completely; one of the ingredients of N is a set of homoclinic orbits. In the normal form to degree 4 the set of homoclinic orbits breaks up, for certain parameter values, into one homoclinic orbit. This enables us to apply Silnikov-Devaney theory to prove, at this stage numerically, the existence of a horseshoe map in the system with the implication of non-integrability and chaos in the normal form.
1. Introduction Hamiltonian systems with two or more degrees of freedom are in general non-integrable. In this paper we shall consider such systems with three degrees of freedom near a non-degenerate critical point of the equations of motion. Supposing that all the eigenvalues of the quadratic part of the Hamiltonian in the neighbourhood of the critical point are distinct and purely imaginary, the Hamiltonian can be expressed as H =
1
2
+ qt) +---
+//3+//4+
+
1
2
+
....
(1)
where H m is a homogeneous polynomial of degree m in the variables p, q. We simplify the Hamiltonian (1) by a canonical near-identity transformation which leaves H2, which represents the quadratic terms, invariant and which removes a large number of terms of H a, H 4, etc. See refs. [4, 9] and for a recent survey ref. [5]. This transformation process is called normalisation and it is carried out to a certain degree m > 3; the system is then truncated from degree 0167-2789/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
m + 1 on and the resulting Hamiltonian system is called the normal form to degree m. A wide-spread idea is that normal forms are important but also, that they are only describing relatively uninteresting, regular parts of the flow like critical points, periodic solutions, invariant tori, etc. We shall show that this idea is not correct by demonstrating the existence of a horseshoe mapping and chaos in a normal form. The frequencies tOl, to2, to3 in (1) satisfy a resonance relation of order k if there exist integers k l , k2, k 3 such that kltO l + k2to 2 4" k3to 3 --- 0, where k = Ikll + [k2J + [k31. The normal form is complicated by the presence of resonant terms, or slightly more precise, the more resonance relations with k < m, the more resonant terms in the normal form to degree m. If the frequencies are all positive, the first-order resonances, involving at least two resonance relations and so two independent combination angles (see ref. [9]) in the normal form for m = 3 are (tOl, toz, to2)= (1, 2, 1), (1, 2, 2), (1, 2, 3) and (1, 2, 4). The normal form to degree m has at least two independent integrals: the normalised Hamiltonian H 2 + K truncated to degree rn and H2; one
398
I. Hoveijn and F. Verhulst / Chaos m the 1 : 2: 3 Hamiltonian normal form
may also use as independent integrals H 2 and K, which contains the terms from degree 3 to degree m. Note that [/42, K] = 0 by normal form construction with [ , ] the Poisson bracket. If a third independent integral exists, it follows from approximation theory that the amount of chaos in the original Hamiltonian, before normalisation and truncation, is small near the equilibrium position corresponding with the critical point. For technical details see refs. [9-11]. At present we know that the 1 : 2 : 1 resonance in normal form to degree 3 is non-integrable [7], the 1 : 2 : 2 resonance in normal form to degree 3 is integrable [21. In this paper we shall demonstrate the nonintegrability of the normal form of the 1 : 2 : 3 resonance. Using the generators at this resonance we shall write down the normal forms to degree 3 and to degree 4. The first case has been analysed by van der Aa [1], who lists seven short-periodic solutions, which are in fact families of periodic solutions parametrised by the value E of H 2. Only one of these periodic solutions, the normal mode in the second degree of freedom, may be complex unstable. In this case we wish to apply the theory of Silnikov, applied by Devaney [6] to Hamiltonian systems, to establish the existence of a horseshoe mapping generated by the phase flow near the complex unstable solution. For this we need the existence of a transversal homoclinic orbit, around which the flow, spiralling inward and outward, is guided back to the periodic solution, thus giving rise to the horseshoe mapping. It turns out that, in the normal form to degree 3, the complex unstable periodic solution is located on an invariant manifold M~ containing a one-parameter family of homoclinic orbits, a homoclinic set. The manifold M~ in its turn is imbedded in the invariant manifold N: H 2 = E, H 3 = 0. On Mj one also finds two heteroclinic orbits of the complex unstable periodic solution, one homoclinic orbit of another periodic solution and a one-parameter family of periodic orbits. There is another invariant manifold M 2 in N intersecting M 1 along the two heteroclinic orbits
and the special homoclinic orbit. The other orbits on M 2 form a one-parameter family of homoclinic orbits of the other periodic solution. The complement of M 1 U M 2 in N consists of periodic orbits only. (M1 and M 2 are separatrix manifolds in N and M 1 N M 2 consists of separatrices on M 1 and M2.) Because of the presence of a homoclinic set the theory of Devaney-Silnikov does not apply to the normal form to degree 3. Considering the normal form to degree 4 we are able to show that of the homoclinic set in M 1 one homoclinic orbit survives which establishes the existence of a horseshoe map. The implication is that an infinite set of unstable periodic solutions exists near the homoclinic orbit and that the phase flow of the normal form to degree 4 is non-integrable. 2. T h e n o r m a l f o r m s o f the 1 : 2 : 3 r e s o n a n c e
The normal form to degree 3 has been written down and analysed by van der Aa [1]. We reformulate some of the results and we extend them to degree 4. A convenient formulation of normal forms uses complex coordinates: u j = p ~ + i q j and v j = p j iq i ( j = 1, 2, 3). Now the generators f of the normal form can be found from the relation: [He, f ] = 0. This was carried out by Fekken [8] for the general resonance case and for the first-order resonances mentioned above in particular. We confine ourselves to listing the generators, ordered by degree k: k = 2,
A 1 -~- UlU1,
A 2 = u2u2,
A 3 = U3U3;
k = 3,
k=4,
B1=
UlU2U3,
B 2 = u1u2u3,
B 3 = u2v2,
B 4 = u2u2;
C l=u3v3,
C 2=v3u3,
C 3 = UlU2U3,
k = 5,
D1
-_ -_ U 23U,2 3,
k=7,
E l=uSv2v3,
C 4 = ulu2u3.
D2 = U2U3, 3 z. F 2=vlSu2u3 .
L Hoveijn and F, Verhulst / Chaos in the I : 2: 3 Hamiltonian normal form
T h e r e exist relations like B I B 2 = A 1 A 2 A 3 between these generators but we will not use them at the moment. F u r t h e r m o r e we shall use only generators up to degree k = 4; the remaining generators have been listed for the sake of completeness. W h e n analysing Hamiltonians near an equilibrium position it is natural to rescale p = •,5, q = • q and accordingly u and v. The Hamiltonian is then divided by •2, we omit the bars and (1) becomes H = H 2 + •H 3 + •2H 4 +
E 3 ....
(2)
We take 0 < • << 1 to indicate that we are considering a small neighbourhood of the equilibrium position, • 2 is a measure for the energy level we are considering. Using the generators the normal forms become H 2 = ½i(A 1 + 2A 2 + 3A3), H 3 = aB 3 -
~B 4 + bB l - bB2,
H 4 = i ( c l A2 + c 2 A2 +C3 A2 + c 4 A 1 A 2 +c5A1A 3 + c6AzA3) + d C 1 - d C 2 + eC 3 - -
eC 4
with a , b , d , e ~ C and cl, c z , . . . , c 6 E• ~. The original Hamiltonian system is real so in complex coordinates the normal form can be taken real or purely imaginary. It is possible to simplify the normal form slightly by rigid rotation on coordinate space and putting a = ~, b = b and either d = d or e = ~. But since it is not obvious which one should be preferred we only put a = and b = b.
3. The normal form to degree 3 3.1. Periodic solutions
The system was studied by van der Aa [1] and we summarise the results. The normal form is
399
H 2 + EH 3 with two independent integrals: H 2 and H 3 . The simplest periodic solutions are those with only one coordinate pair (pj, qj) non-vanishing or the cases of periodic solutions located in an e-neighbourhood of such a state; they are called normal modes and there are two of them. One solution we call 7'/"3 with p~ = ql =1192~-- q2 = 0, q3 = (2/3)1/2 sin(3t), P3 = (2/3)1/2 cos(3t) and one we call zr 2 with p l = q l = p 3 = q 3 = O , q2 = cos(2t), P2 = sin(2t) where E is the value of H 2. T h e r e is no normal m o d e with only ( p l , ql)4: (0, 0), in the generic situation. The next type of periodic solutions consists of those with only one coordinate pair equal to zero; there is only one of this type, which we call ,/14 with P2 = q 2 = 0. These periodic solutions are located on the invariant level set N: H E = E, H 3 = 0. There are four periodic solutions in general position: all coordinate pairs unequal to zero. They cannot be complex unstable [1] and they do not lie on M 1 (see section 3b for a definition). We will call them: ~rs, rr6, ~r 7, -rr8 and they will play no important part in our analysis. The location of the periodic solutions can be depicted on the so-called action simplex in action space, see fig. 1. We determine the stability type of a periodic solution by considering it as a critical point of a reduced system and by determining the stability of this critical point. The stability is determined by the eigenvalues of the linearized differential equation. In our case there are four eigenvalues. A real pair (A, - A ) will be called H(yperbolic), an imaginary pair (iA, - i A ) will be called E(lliptic) and a quadruple ( A , - A , A , - A) will be called C(omplex). So stability type E H means eigenvalues iA, - iA,/x, - / z . O f course the stability type depends generally on the p a r a m e t e r s a and b of H 3. The results are summarised in table 1. From now on we will only consider 0 < a < b, so zr 2 is complex unstable. As we see from table 1 the interesting level set is N. F u r t h e r m o r e since the eigenvalues of zr 3 and zr4 have multiplicity 2 their stability type may, under a perturbation, turn into C.
400
L Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
7"3 l
71"3
/
/
71"2
'/t'5, 6
/
7/'7,8
/
/"2
T1
Fig. 1. Action simplex for H 2 = 1. Table 1 Summary of the results for periodic solutions.
and q we obtain the following equations:
Periodic
Stability
Multiplicity of
L e v e l set
solution
type
eigenvalues
( H 2 = 1)
~2
C ( a 2 - b 2 < 0), H H ( a 2 - b E > 0)
1 1
H3= 0 H3= 0
"/7"3 7r4 rr 5 rr 6 ~'7
HH EE EH EH EE
2 2 1 1 1
H H H n n
~'s
EE
1
H 3 = h32 ~ 0
dll = P2( 2aq~ -- bq3), /~1 = P 2 ( - 2aPl - bP3),
3= 3= a = 3= 3=
0 0 h31 ~ 0 h31 q: 0 h32 q: 0
c13 = Pz( bql ), 1~3 =P2( bPl), f~2 = a ( p ~ -
q2) _ b ( p , p 3 + qlq3),
H2(p,q)=l, N o t e that
H3(P,q)=O,
H 3
q2=0.
(3)
is linear in P2 and q2 so:
3.2. Orbits on the invariant manifold N T h e periodic solutions listed in table 1 are relative equilibria o f H = H 2 + E H 3 with respect to H 2. T h e r e f o r e they consist of fixed points of the H a m i l t o n i a n H , = H - r H 2. O n the level set N we have the periodic solutions rr 2, zr 3 and ~r4 for which ~- = 1, so in this case H~ = e n 3. Instead of using reduction (the r e d u c e d phase space has conelike singularities) we consider the flow o f n 3 on the level set N, following the m e t h o d o f D u i s t e r m a a t [7]. In the original coordinates p
0H 3 0H 3 t~2P2 --/92q2 = ~---p-~-2P2 + --~-2 q2 = H 3
and since we consider H 3 = 0 we have Q2/P2 = and q2 move along a straight line, which we choose to be q2 = 0. Next we set x = (qt, P l , q 3 , P3) and f ( x ) = a(x 2 - x 2) + b ( x 2 x 4 +XlX3); then we obtain the differential equations
q2/P2:192
YC=pz.4X,
152=-f(x),
(4)
L Hoveijn and F. Verhulst / Chaos in the 1: 2: 3 Hamiltonian normal form
where A is a 4 × 4 matrix obtained from (3) with eigenvalues
where C1 = c 2 = A e
5:a+(a2-bZ)
1/2
and
0
W e solve (4) formally f~ p 2 ( s ) d s , then
by
putting
z(t) =
(5)
(6)
-p2 = - f ( x ( z ) ) .
We integrate the right-hand side with respect to z: I d V ( z ) / d z = f ( x ( z ) ) then ~ + 1 d V ( z ) / dz=0. Introducing a new variable w = P2 we can put the equation for z into Hamiltonian form: H ( z , w ) = ~wl 2 + ½V(z). Finally we use z(t)lt= o = 0 and w(t)lt=o -- p2(0) then, after some calculations we find
+ 3q (o) + 3p (o)]
1 so H ( z , w ) = 7. W e cannot explicitly solve z as a function of time but from t = ]~¢')[1 - V(z)] -1/2 d z we have a "period function"
r=
dz
a 2 + b 2 + ia( b 2 - a 2) 1/2 = P ei~, tO ----- ( b 2 - -
where a i are the complex eigenvectors of A and c i are complex "initial values". The next step is to derive an equation for z:
v(o) = [qf(o)
ia,
c3 = ~4 = n e it3,
x ( t ) = c I e x~z a I + ... + c 4 e A~z a4,
e
401
(7)
with z _ < 0 < z + and for z ~ ( z _ , z + ) , V ( z ) < 1, holds. For V ( z ) we find, after substituting the solution (5) into (6) and integrating f ( x ( z ) ) :
a2) 1/2.
Having "solved" the equations of motion (3) on N we will proceed to classify the orbits. From the equations of motion (3) we see that ql = q3----0 and Pl =P3 = 0 are invariant manifolds and are also the stable and unstable manifolds of the fixed point x = 0, P2 = 1. Let us consider p t --P3 = 0. We call this two-dimensional invariant manifold M 1 and from H 2 = 1 we find q2 + 2p22 + 3q2 = 2: M~ is an ellipsoid. Orbits on M~ have initial conditions c 3 = c 4 -~- 0 or B -- 0, then V ( z ) reads V ( z ) = A 2 e2az[4b2 + 2p cos(2~oz + 2 a + ~p)]. We can distinguish four types of orbits on M1: (1) (2) (3) (4)
z _ = -oo, V ( z + ) = 1, V'(z+) # 0, T infinite; z _ = -oo, V ( z + ) = 1, V'(z+) = O, T infinite; V ( z _ ) = 1, V ' ( z _ ) ~ O, V ( z + ) = 1, T finite; V ( z _ ) = 1, V ' ( z _ ) = O, V ( z + ) = 1, T infinite.
Type (3) clearly corresponds with a family of periodic orbits and type (1) with a family of homoclinic orbits. Types (2) and (4) are special orbits: two conditions in z+ or z_, type (2) corresponds with a heteroclinic orbit and type (4) with a homoclinic orbit. We also have fixed points on Ml:
x= +
b2+3a2
(-b,O,a,O),
P2 = 0 corresponding with ~'4, x = + ( 2 / 3 ) t / 2 ( 0 , 0 , 1,0),
V ( z ) ---A 2 e2"Z[4b 2 + 2p cos(2toz + 2or + ~o)]
P2 -- 0 corresponding with *r 3,
+ B 2 e-2"~[4b 2 + 2p cos(2taz - 213 - ~p)],
(8)
x=0, P2 = + 1 corresponding with ~r2.
402
L Hoveijn and F. Verhulst/Chaos in the 1 : 2 : 3 Hamiltonian normal form q3
ql
Fig. 2. Projection of phase curves on the ql, q3 plane (P2 > 0).
To see how these orbits and fixed points are arranged on M~, we first note that all orbits, projected on the ql, q3 plane in the P2 direction are elliptic spirals (see fig. 2). If we intersect M 1 with the cylinder q21 - ( 2 a / b ) q m q 3 + q23 ~ C2, Pl =P3 = 0 (some constant c) then we see that the flow on M~ is outwards on the cylinder when P2 > 0 and inwards when P2 < 0. If we intersect M 1 with the plane Pm = P 2 =P3 = 0 then we have ql = t13 = 0 and/~2 changes sign at the fixed points corresponding with zr 3 and ~r4. So we get the following picture: The fixed points corresponding with zr4 are surrounded with a family of periodic orbits, these families are separated from other orbits by homoclinic orbits of the fixed points corresponding with zr 2 by two heteroclinic orbits. The fixed points corresponding with zr 2 are connected with themselves by a family of homoclinic (strictly speaking in this picture by heteroclinic) orbits (see fig. 3). On M 1 the flow is globally directed from P2 = 1 to P2 = - 1. We could do the same for ql = q3 = 0 then we get an invariant manifold which looks exactly like M~ with the flow globally directed from P2 = - 1 to P2 = 1. We should, however, identify P2 = 1 and P2 = - 1 since both corre-
spond with rr 2. We have in fact a twofold covering of the reduced phase space. Let us next consider general initial conditions c~, c 2, c3, c 4 (c I = ?2, c3 = 74 of course) with A :~ 0 and B 4: 0. Now both z_ and z+ are finite. In general V ' ( z ) :/: 0 and V'(z÷) 4:0 so T is finite: the orbits are periodic. When V'(z_) or V'(z÷) happens to be zero the corresponding orbits are homoclinic to the fixed points corresponding with 7r3, since these are the only unstable fixed points left and the periodic orbits cannot be unstable or asymptotically stable. This means there exists another two-dimensional invariant manifold M 2 filled with homoclinic orbits and two exceptional heteroclinic orbits (V'(z_)= 0 and V'(z÷)= 0). On M 1 and M 2 we have separatrices and M 1 and M 2 intersect transversally along these separatrices. This intersection is transversal since the linear stable and unstable manifolds of the fixed points corresponding with zr 3 are transversal and these points belong to M~ and M 2. The stable and unstable manifolds of 7r2 do not intersect transversally: the conditions in Devaney's theorem are not fulfilled. Indeed there can be no spiralling around this three-dimensional manifold orbits of homoclinic orbits (in the original phase space) since H 2 and H 3 are con-
L Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
403
Fig. 3. Phase curves on M r Projection on x, y plane. Dashed lines indicate negative z coordinate, solid lines indicate positive z coordinate.
served. We cannot conclude that H = H 2 + EH 3 is non-integrable; however, numerical calculations suggest that it is indeed non-integrable.
4. The normal form to degree 4 4.1. Pen'odic solutions and their stability
The position of the normal m o d e ,/72 remains unchanged and since 7T2 is complex unstable it will still be complex unstable. Now '172 is located on the level set: H 2 = E, K = ¢2c2E2 and it has the period: "rr(1 + eEc2E) - l . Also the position of rr 3 remains unchanged. Now 1r3 lies on the level set H 2 = E , K=E2Ca(2E) 2. Again the period changes, the new period is: 2"rr(3 + 4E2c3E)-l. We find the stability type of 7r 3 from the discriminant
We will now turn our attention to the Hamiltonian system H = H 2 + K, K = e H 3 + ~2H4. First let us see what happens to the periodic solutions. The stability type of the periodic solutions with different eigenvalues will not change. On the other hand the stability type of ~'3 and rr4 (with double eigenvalues) will change. To determine their stability we must have a (good enough) approximation of their location in phase space. In the case of 7r4 this involves lengthy calculations, which we want to avoid. Instead we impose conditions on the p a r a m e t e r s in H 4 such that the periodic solutions remain on the position of the normal form to degree 3. As there are 4 conditions and 10 parameters in H 4, this condition is not too restrictive. It turns out that only for ~'4 we need restrictions on the parameters.
ot 2
1 4 ,~,C5 2 _ c -- fl = - - E 2 b 2 ( c 5 -- C6 )2 + ~E
)2
of the eigenvalue equation: h 4 + 2 a h 2 +/3 = 0, so 7r3 will be complex unstable. To find rr4 on the same position as in the normal form to degree 3 we need the conditions: d l=d E=O,
c 1= -(aE/2b2)cs,
ca = - (b2/2a2)cs. These conditions are somewhat stronger than necessary as we even kept the period fixed. ~'4 lies on the level set H 2 = E, K = 0 but if we allow for a slightly different period then 7r4 lies on H 2 = E, K = [ 4 ¢ 2 E 2 / ( b 2 + 3 a 2 ) ] ( c l b 4 + caa 4 + 2csa2b2). T h e stability type of zr4 follows from the discriminant a 2 - / 3 . In this c a s e ot 2 - / 3 can be positive or negative so zr4 is stable of type E E or complex unstable.
404
I. Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
4.2. Homoclinic and heteroclinic orbits In general we expect a heteroclinic orbit and a o n e - p a r a m e t e r family of homoclinic orbits to break up under a small perturbation. Indeed the heteroclinic connections between 7r 2 and zr 3 will 4 no longer exist since c 2 ~ ~c 3 so in general 7r 2 and rr 3 are lying on different level sets. We expect the o n e - p a r a m e t e r family of homoclinic orbits to break up in discrete homoclinic orbits. We used numerical analysis to confirm this. Before doing so we had to choose values for the p a r a m e t e r s in H 4. We took values close to the ones obtained from normalising the potential problem:
H(p,q)
= 1~ [ ( P l 2 +q12) + 2(p22 + q Z2) +3(p2 +q2)]
+ e(aq~q 2 + bqlq2q3).
If we use the coordinates y then we see from the normal form to degree 3 that Y2 = 0 intersects the homoclinic orbits perpendicularly and symmetrically. In this system the stable and unstable manifolds of 7r 2 intersect Y2 = 0 along the Y4 axis (Yl =Y3 = 0). For the normal form to degree 4 we took initial values on the stable and unstable manifolds obtained by linearisation near the critical point and we computed the intersection with Y2 = 0. We observed two curves in Y2 = 0 intersecting transversally in one point so for this choice of p a r a m e t e r s only one homoclinic orbit is left. The precision of the numerical calculations is such that the conserved quantities H 2 and K = a n 3 + a2H 4 are conserved to within 10 -6. This makes it plausible that the computed orbits are close to the real orbits: error propagation need not always be perpendicular to grad H 2 and grad K. F u r t h e r m o r e the angle of the intersecting 1 curves is roughly ~radians. Since we now have a transverse homoclinic orbit of a complex unstable
critical point corresponding with the normal mode ~'2, the results of Devaney apply.
5. Discussion We analysed two truncated normal forms H = H 2 + K of the 1 : 2 : 3 resonance, K = EH 3 and K = EH 3 + E2H4 . The normal form to degree 4 is a perturbation of the normal form to degree 3. Both normal forms have two independent integrals, H 2 and K, since the normal forms are constructed such that [H2, K ] = 0. If there exists a third independent integral then the system is completely integrable (modulo some non-elementary integrations). The energy manifold is foliated into submanifolds by the level sets of the additional integrals. For the normal form to degree 3 it is possible to characterise the phase curves completely on the level set H 2 = E, K = E H 3 = 0. First restrict to H 2 = E (reduction), then all orbits outside two invariant manifolds are periodic, on the invariant manifold M 1 we have homoclinic orbits, heteroclinic orbits and again periodic orbits (see fig. 3). It is tempting to consider the period function as a possible third integral, but there is a striking resemblance with the 1 : 2 : 1 resonance with a similar structure of the phase space, where the period function displays infinite branching [7]. Numerical results also suggest that the normal form to degree 3 has no analytic third integral (fig. 4). Because of the existence of the twodimensional invariant manifold M~ there can be no spiralling around the (indeed non-transversal) homoclinic orbits so Devaney's results do not apply. For a geometric interpretation of the results for the normal form to degree 3 see ref. [11]. Though the normal form to degree 3 seems to be non-integrable we cannot conclude this from Devaney's result, we can do so, however, for the normal form to degree 4, where we find a transverse homoclinic orbit. In the latter case we know that the complicated behaviour associated with non-integrability is as
L Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
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;
°•
H 3 = 0. ( b ) S a m e
.."."-. ..
."
':'%" "~ :'~''~::: ";'' ~,k." : ,~': :
:
:."
,
(q3 = 0) for system
as in (a) now with initial values such that
complicated as a horseshoe map at least near the homoclinic orbit. In the former case however, we cannot yet characterise the nature of the "complicated behaviour". Another question remaining to be answered for both systems is: how large is the region in phase space with chaotic orbits and also what is the measure of the set of chaotic orbits, and how does this measure depend on the parameters in the normal form? Now the original Hamiltonian system is nonintegrable. Note that the normal form to degree 4 contains a horseshoe map which is a structurally
H =/-I2 + H3. Initial values
are such that
H 2 = 1,
H 2 = 1, H 3 ~ 0.
stable phenomenon: it cannot be perturbed away by adding the higher-order terms of the full Hamiltonian system, except of course in special examples. Concluding we might say that normalisation drastically simplifies Hamiltonian systems but the motion in phase space, induced by Hamiltonian systems in normal form, can be very irregular.
Acknowledgements
During the course of this work we had many stimulating discussions with J.J. Duisterrnaat. We
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I. Hoveijn and F. Verhulst / Chaos in the 1 : 2: 3 Hamiltonian normal form
w o u l d like to t h a n k t h e r e f e r e e s o f P h y s i c a D f o r their valuable comments.
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[6] R.L. Devaney, J. Diff. Eqs. 31 (1976) 431. [7] J.J. Duistermaat, Ergod. Theor. Dynam. Syst. 4 (1984) 553. [8] A. Fekken, On the normal form of a fully resonant Hamiltonian function, preprint Free University Amsterdam, No. 317 (1986). [9] J. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, Vol. 59 (Springer, Berlin, 1985). [10] F. Verhulst, Proc. ICIAM, eds. A.H.P. van der Burgh and R.M.M. Matthey, CWI Tract 36 (1987) 129. [11] F. Verhulst and I. Hoveijn, in: Geometry and Analysis in Nonlinear Dynamics, Pitman Research Notes in Mathematics, eds. H.W. Broer and F. Takens (Pitman, London, 1990).