Approximate first integrals of the Hénon–Heiles system revisited

Communications in Nonlinear Science

and Numerical Simulation 10 (2005) 73–83

www.elsevier.com/locate/cnsns

Approximate first integrals of the H�enon–Heilessystem revisited

G. €Unal a,*, C.M. Khalique b,*

a Faculty of Sciences and Letters, Istanbul Technical University, Maslak, 80626 Istanbul, Turkeyb Department of Mathematical Sciences, International Institute for Symmetry, Analysis and Mathematical

Modelling, University of North-West, P. Bag X2046, Mmabatho 2735, South Africa

Received 22 January 2003; received in revised form 26 May 2003; accepted 29 May 2003

Available online 12 August 2003

Abstract

Approximate first integrals (conserved quantities) of H�enon–Heiles system have been obtained based on

the approximate Noether symmetries for the resonance (x2 ¼ x1) and off resonance. It has been shown that

system undergoes complicated bifurcations near resonances. Analytical results have been verified by nu-

merically obtained KAM curves on the Poincar�e surface of section.� 2003 Published by Elsevier B.V.

Keywords: Hamiltonian dynamical systems; Approximate Noether symmetries; Resonances; Noether�s theorem

1. Introduction

The regular behaviour (order) observed in the numerical studies of the nearly integrableHamiltonian systems [1,2] led to search for analytical methods on the approximate first integrals.Various perturbative methods have been developed to construct approximate first integrals, e.g.,direct method of Contopoulos [1] and Birkhoff–Gustavson normal form method [3]. A compre-hensive study of these methods and others can be found in [4]. However, none of these methodsresort to the celebrated Noether�s theorem which provides a link between the exact Noether

* Corresponding authors. Tel.: +90-212-285-32-88; fax: +90-212-285-63-86 (G. €Unal), tel.: +27-18-389-2009; fax: +27-18-389-2594 (C.M. Khalique).

E-mail addresses: [email protected] (G. €Unal), [email protected] (C.M. Khalique).

1007-5704/$ - see front matter � 2003 Published by Elsevier B.V.

doi:10.1016/S1007-5704(03)00063-7

mail to: [email protected]

74 G. €Unal, C.M. Khalique / Communications in Nonlinear Science and Numerical Simulation 10 (2005) 73–83

symmetries of the dynamical systems and the exact first integrals. In [5], €Unal has shown that theextension of Noether�s theorem to approximate symmetries yields approximate first integralswhich are in good agreement with numerical results. Approximate symmetry groups of differentialequations has been developed by Baikov et al. [6]. Based on their definition, €Unal has given amethod which incorporates resonances in [5] whose use will be made of in the sequel.Here, we consider a Hamiltonian of the form

H ¼ 1

2ðp21 þ p22 þ Ax21 þ Bx22Þ þ e x21x2

�� x32

3

�; ð1Þ

where pi and xi are canonical momentum and position, respectively, and e is a small positiveparameter. Numerical results given in [2] show that this system exhibits chaotic behaviour. Whenwe set A ¼ x2

1 and B ¼ x22, it becomes the Hamiltonian of two coupled harmonic oscillators,

where e is a coupling parameter. We will consider

x22 ¼ x2

1 þ es;

where the parameter s is detuning parameter. Our goal is to derive the approximate first integralsof the Hamiltonian system

_x1_x2_x3_x4

26664

37775 ¼

0 0 1 0

0 0 0 1

�x21 0 0 0

0 �x21 0 0

26664

37775

x1x2x3x4

26664

37775þ �

0

0

�2x1x2�sx2 þ x22 � x21

26664

37775: ð2Þ

Here x3 ¼ p1 and x4 ¼ p2 and the term involving s will be considered as unfolding vector field.Approximate first integrals can be employed to obtain approximate solutions. Here, we will makeuse of them in obtaining KAM curves (intersections of the KAM tori with a surface of section) ofHamiltonian system given in (2). It has been shown that system under study undergoes complexbifurcations when the detuning parameter has been varied. Furthermore, analytical results havebeen verified by numerical ones.

2. Approximate Noether symmetries

Lie considered a differential equation as a surface in a certain space which he called as a frame(manifold) [7]. He looked for transformations which leave the frame form invariant in the spacespanned by the transformed coordinates. This is to say that if one takes a picture of the newframe, one would observe the frame which has the same shape with the picture of the frame takenin the original space. This led him to show that not only these transformations form a continuousgroup (called Lie symmetry group) but also they transform solutions of a differential equation tonew solutions. Unfortunately, many differential equations arising in mathematical physics do notadmit nontrivial exact symmetries. Is it possible to relax the exact invariance condition? A positiveanswer has been provided by Baikov et al. [6]. These transformations are called approximatesymmetries of the differential equation.

G. €Unal, C.M. Khalique / Communications in Nonlinear Science and Numerical Simulation 10 (2005) 73–83 75

Following Ibragimov [7] and Baikov et al. [6], first-order approximate invariance condition ofthe Hamiltonian system (2) reads as

2664

½X;F� ¼ Oðe3Þ; ð3Þ
where ½; � is the Lie bracket and,
X ¼ X0 þ eX1 þ e2X2; Xb ¼X4l¼1

glbðxÞ

o

oxl;

F ¼ F0 þ eF1; Fb ¼X4l¼1

f lb ðxÞ

o

oxlðb ¼ 0; 1; 2Þ:

Here X is the first-order approximate symmetry vector field, F0 and F1 are the vector fields cor-responding to the linear and nonlinear part of (2), respectively. Since dynamical system (2) isautonomous, we confine ourselves to time-independent symmetries. Determining system ofequations for the symmetries can be obtained by evaluating (3) in ascending order of e.
X4
j¼1gj0

of k0

oxj

�� f j

0

ogk0

oxj

�¼ 0;

X4j¼1

gj1

of k0

oxj

�� f j

0

ogk1

oxj

�¼X4j¼1

f j1

ogk0

oxj

�� gj

0

of k1

oxj

�;

X4j¼1

gj2

of k0

oxj

�� f j

0

ogk2

oxj

�¼X4j¼1

f j1

ogk1

oxj

�� gj

1

of k1

oxj

�:

ð4Þ

As it is seen from the first set of partial differential equations in (4), X0 is the exact symmetryvector field of the linear part of the system corresponding to F0. This set of equations can besolved by the method of characteristics. It can be verified that it forms an infinite dimensional Liealgebra.In order to cope with the normal form theory [5], we now follow a different path. Introducing

the transformation

x ¼ Sz; ð5Þ
where
S ¼

ix1

�ix1

0 0

0 0 ix1

�ix1

�1 �1 0 0

0 0 �1 �1

2664

3775

to (2) yields

_z1_z2_z3_z4

3775 ¼

ix1 0 0 0

0 �ix1 0 0

0 0 ix1 0

0 0 0 �ix1

2664

3775

z1z2z3z4

2664

3775þ e

2x21

�2ðz1 � z2Þðz3 � z4Þ�2ðz1 � z2Þðz3 � z4Þ

�ðz1 � z2Þ2 þ ðz3 � z4Þ2 þ isx1ðz3 � z4Þ�ðz1 � z2Þ2 þ ðz3 � z4Þ2 þ isx1ðz3 � z4Þ

2664

3775:ð6Þ


Determining system of equations for (6) becomes

X4j¼1

kjzjo�gk

0

ozj� kk�g

k0 ¼ 0; ð7Þ

X4j¼1

kjzjo�gk

1

ozj� kk�g

k1 ¼

X4j¼1

� �f j

1

o�gk0

ozjþ �gj

0

o�f k1

ozj

!; ð8Þ

X4j¼1

kjzjo�gk

2

ozj� kk�g

k2 ¼

X4j¼1

� �f j

1

o�gk1

ozjþ �gj

1

o�f k1

ozj

!; ð8aÞ

where k1 ¼ ix1, k2 ¼ �ix1, k3 ¼ ix1, k4 ¼ �ix1. Solutions of (4) can be transformed to solutionsof (8) via [5]

gk0 ¼

X4l¼1

Skl�gl0ðS

�1x; tÞ and gk1 ¼

X4l¼1

Sklgl1ðS

�1x; tÞ: ð9Þ

Set of partial differential equations (7) is called homological equation in the context of normalform theory [8]. Its solutions in the space of homogenous polynomials read as [9]:

�gj0 ¼

Xs1k1þ��þs4k4�kj¼0

Cj0s1s2s3s4

zs11 zs22 z

s33 z

s44 ; ð10Þ

where C0s1��s4 are the group parameters, and the sum in (10) will be taken over all the resonantmonomials which satisfy the resonance conditions s1k1 þ � � � þ s4k4 � kj ¼ 0.Third-order resonant monomials can be obtained by solving the algebraic equations

s1k1 þ s2k2 þ s3k3 þ s4k4 � kj ¼ 0; s1 þ s2 þ s3 þ s4 ¼ 3: ð11Þ

Rendering back the solutions of (11) to (10) we obtain

X0 ¼ ða1z1z2z3 þ a2z1z3z4 þ a3z21z2 þ a4z21z4 þ a5z2z23 þ a6z23z4Þo

oz1þ ðb1z1z2z4 þ b2z2z3z4

þ b3z1z22 þ b4z1z24 þ b5z22z3 þ b6z3z24Þo

oz2þ ðc1z1z2z3 þ c2z1z3z4 þ c3z21z2 þ c4z21z4 þ c5z2z23

þ c6z23z4Þo

oz3þ ðd1z1z2z4 þ d2z2z3z4 þ d3z1z22 þ d4z1z24 þ d5z22z3 þ d6z3z24Þ

o

oz4; ð12Þ

where a1; . . . ; d6 are complex coefficients. Notice that X0 is written as a linear combination of 24independent symmetries. In order for a symmetry vector field to be a Noether symmetry, it mustleave the Hamiltonian invariant and it must be divergence free (conservative) [5]. These conditionsfor X0 read as

X4j¼1

�gj0

oH 0

ozj¼ 0 and

X4j¼1

o�gj0

ozj¼ 0; ð13Þ


where H 0 ¼ 2ðz1z2 þ z3z4Þ. Imposing the Noether symmetry conditions on X0 in (12) we find

a1 þ b5 þ d3 ¼ 0; a2 þ b2 þ c1 þ d1 ¼ 0; a3 þ b3 ¼ 0;

a4 þ b1 þ c3 ¼ 0; a5 þ d5 ¼ 0; a6 þ c5 þ d2 ¼ 0;

b4 þ c4 ¼ 0; b6 þ c2 þ d4 ¼ 0; c6 þ d6 ¼ 0;

a1 þ 2b5 þ 2c5 þ d2 ¼ 0; a2 þ b2 þ 2c6 þ 2d6 ¼ 0;

2a3 þ 2b3 þ c1 þ d1 ¼ 0; 2a4 þ b1 þ c2 þ 2d4 ¼ 0:

ð14Þ

Furthermore, solutions of (14) must yield real gj0 when they are transformed according to (9), i.e.,

g10 ¼i

x1

ð�g10 � �g20Þ; g20 ¼i

x1

ð�g30 � �g40Þ; g30 ¼ ��g10 � �g20; g40 ¼ ��g30 � �g40

must be functions in real space. This leads to

�g10 ¼ �g20 ; �g30 ¼ �g40 :

In order for these relations to be valid, we must have:

a1 ¼ b1; a2 ¼ b2; a3 ¼ b3; a4 ¼ b5; a5 ¼ b4; a6 ¼ b6;

c1 ¼ d1 ; c2 ¼ d

2 ; c3 ¼ d3 ; c4 ¼ d

5 ; c5 ¼ d4 ; c6 ¼ d

6 ;ð15Þ

where ð Þ stands for the complex conjugate. Solving the set of algebraic equations in (14) underthe conditions given in (15) leads to following Noether symmetries (see also [10]).

X1

0 ¼ iz1z3z4o

oz1� iz2z3z4

o

oz2; X

2

0 ¼ iz1z2z3o

oz3� iz1z2z4

o

oz4;

X3

0 ¼ iz21z2o

oz1� iz1z22

o

oz2; X

4

0 ¼ iz23z4o

oz3� iz3z24

o

oz4;

X5

0 ¼ iz2z23o

oz1� iz1z24

o

oz2þ iz21z4

o

oz3� iz22z3

o

oz4;

X6

0 ¼ z2z23o

oz1þ z1z24

o

oz2� z21z4

o

oz3� z22z3

o

oz4;

X7

0 ¼ ðz1z2z3 � z21z4Þo

oz1þ ðz1z2z4 � z22z3Þ

o

oz2þ ð�z1z3z4 þ z2z23Þ

o

oz3þ ð�z2z3z4 þ z1z24Þ

o

oz4;

X8

0 ¼ iðz1z2z3 þ z21z4Þo

oz1� iðz1z2z4 þ z22z3Þ

o

oz2þ iðz1z3z4 þ z2z23Þ

o

oz3� iðz2z3z4 þ z1z24Þ

o

oz4;

X9

0 ¼ �z21z4o

oz1� z22z3

o

oz2þ ð�2z1z3z4 þ z21z2 þ 2z2z23Þ

o

oz3þ ð�2z2z3z4 þ z1z22 þ 2z1z24Þ

o

oz4;

X10

0 ¼ iz21z4o

oz1� iz22z3

o

oz2þ ið2z1z3z4 � z21z2 þ 2z2z23Þ

o

oz3þ ið�2z2z3z4 þ z1z22 � 2z1z24Þ

o

oz4;

X11

0 ¼ z23z4o

oz1þ z3z24

o

oz2þ ð�2z1z3z4 þ z2z23Þ

o

oz3þ ð�2z2z3z4 þ z1z24Þ

o

oz4;

X12

0 ¼ iz23z4o

oz1� iz3z24

o

oz2þ ið2z1z3z4 þ z2z23Þ

o

oz3� ið2z2z3z4 þ z1z24Þ

o

oz4:

ð16Þ


We first form the linear combination of the Noether symmetries given in (16), i.e.,

X0 ¼X12k¼1

qkXk0

and then render it back to the right-hand side of Eq. (8) to check resonance conditions on eachmonomials. No resonant monomials have been detected at the first order of approximation. Wehave solved Eq. (8) but for the sake of brevity will not be displayed here. We next substitute thissolution into the right-hand side of Eq. (8a). We have detected following set of resonant mono-mials.

r21 ¼ �28z31z2z4q8 � 14z31z24q1 þ 14z31z

24q3 � 14z31z

24q5 � ð14iÞz31z24q6 þ 28z21z

22z3q8 � 28z21z

22z3q10

þ ð56iÞz21z2z3z4q6 þ 28z21z3z24q8 þ ð14iÞz21z3z24q9 þ 14z21z3z

24q10 þ ð14iÞz21z3z24q11 � 42z21z3z

24q12

þ 14z1z22z23q1 � 42z1z22z

23q3 þ 42z1z22z

23q5 � ð42iÞz1z22z23q6 þ 28z1z22z

23q2 � 28z1z2z23z4q8

þ ð28iÞz1z2z23z4q9 þ 28z1z2z23z4q10 þ ð28iÞz1z2z23z4q11 þ 84z1z2z23z4q12 � ð28iÞz1z23z24q6� ð42iÞz22z33q9 þ 42z22z

33q10 � ð42iÞz22z33q11 þ 42z22z

33q12 � 28z2z33z4q1 � 28z2z33z4q5

þ ð28iÞz2z33z4q6 þ 28z2z33z4q4 � 28z33z24q12

r23 ¼ 28z31z22q10 þ 28z31z2z4q3 � 28z31z2z4q5 � ð28iÞz31z2z4q6 � 28z31z2z4q2 � ð42iÞz31z24q9 � 42z31z

24q10

� ð42iÞz31z24q11 � 42z31z24q12 þ ð28iÞz21z22z3q6 � 28z21z2z3z4q8 þ ð28iÞz21z2z3z4q9

� 140z21z2z3z4q10 þ ð28iÞz21z2z3z4q11 � 84z21z2z3z4q12 þ 28z21z3z24q1 þ 42z21z3z

24q5

þ ð42iÞz21z3z24q6 þ 14z21z3z24q2 � 42z21z3z

24q4 þ 28z1z22z

23q8 þ ð14iÞz1z22z23q9 þ 98z1z22z

23q10

þ ð14iÞz1z22z23q11 þ 42z1z22z23q12 � ð56iÞz1z2z23z4q6 þ 28z1z23z

24q8 þ 56z1z23z

24q10 þ 84z1z23z

24q12

� 14z22z33q5 þ ð14iÞz22z33q6 � 14z22z

33q2 þ 14z22z

33q4 � 28z2z33z4q8 � 56z2z33z4q10 � 56z2z33z4q12

and r22 ¼ ðr21Þ, r24 ¼ ðr23Þ

.

To kill the resonant monomials appearing in above equations we must set q6 ¼ 0, q8 ¼ 0,q10 ¼ 0, q12 ¼ 0 and q11 ¼ �q9. Furthermore, q7 is an arbitrary real number and, q1; . . . ; q5 mustsatisfy

q1 � q3 þ q5 ¼ 0; q1 þ 2q2 � 3q3 þ 3q5 ¼ 0; q1 � q4 þ q5 ¼ 0;

q2 � q3 þ q5 ¼ 0; 2q1 þ q2 � 3q4 þ 3q5 ¼ 0; q2 � q4 þ q5 ¼ 0:

From which we obtain

q1 ¼ q4 � q5; q2 ¼ q1; q3 ¼ q4; q11 ¼ �q9:

We have solved Eq. (8a) with the right-hand side which involves no resonant monomials. Ap-proximate symmetries involving the detuning parameter s becomes

Xs1 ¼ is z3

o

oz3

�� z4

o

oz4

�; X

s2 ¼ is2 z3

o

oz3

�� z4

o

oz4

�:


First- and second-order symmetry vector fields including terms with s take the form

X1 ¼1

6w31

ðiq9z41 þ 9iq9z21z22 þ 18iq9z21z3z4 þ 15iq9z21z

24 � 2iq9z1z32 þ 24iq9z1z2z3z4 � 6iq9z1z2z24

� 3iq9z22z23 � iq9z43 � 6iq9z33z4 � 21iq9z23z

24 þ 4iq9z3z34 þ iz41q7 � 9iz31q7z2 � 9iz21q7z

22

� 3iz21q7z23 þ 12iz21q7z3z4 þ 15iz21q7z

24 þ iz1q7z32 þ 15iz1q7z2z23 þ 18iz1q7z2z3z4 � 9iz1q7z2z24

� 6iq7z22z23 þ 6iq7z22z3z4 þ 3q4z31z3 � 3q4z31z4 � 12q4z21z2z3 þ 12q4z21z2z4 þ 9q4z1z22z3

� 9q4z1z22z4 � q4z1z33 � 3q4z1z23z4 þ 3q4z1z3z24 þ q4z1z34 þ 6q4z2z23z4 � 6q4z2z3z24 � q5z31z3� 9q5z31z4 þ 18q5z21z2z3 � 18q5z21z2z4 þ 27q5z1z22z3 þ 3q5z1z22z4 þ 3q5z1z33 � 9q5z1z23z4

þ 9q5z1z3z24 � 3q5z1z34 � 4q5z32z3 � 18q5z2z33 � 36q5z2z23z4 þ 6q5z2z3z24Þo

oz1þ ð�g11Þ

o

oz2

þ 1

6w31

ð�2iq9z31z3 � 18iq9z21z2z3 � 18iq9z21z2z4 � 36iq9z1z22z3 þ 6iq9z1z22z4 � 2iq9z1z33

� 12iq9z1z3z24 þ 2iq9z1z34 þ 4iq9z32z3 þ 6iq9z2z33 þ 6iq9z2z23z4 þ iz31q7z3 � 3iz31q7z4� 6iz21q7z2z3 þ 6iz21q7z2z4 � 15iz1q7z22z3 � 3iz1q7z22z4 � 3iz1q7z33 þ 21iz1q7z23z4 þ 3iz1q7z3z24þ 3iz1q7z34 þ 4iq7z32z3 þ 6iq7z2z33 þ 24iq7z2z23z4 � 6iq7z2z3z24 � 3q4z31z2 þ 6q4z21z

22 þ 3q4z21z

23

� 6q4z21z3z4 � 3q4z1z32 � 3q4z1z2z23 þ 6q4z1z2z3z4 þ 3q4z1z2z24 þ 3q4z22z23 � 6q4z22z3z4 � q4z43

þ 6q4z33z4 � 9q4z23z24 þ 4q4z3z34 þ q5z41 � 9q5z31z2 � 9q5z21z

22 � 3q5z21z

23 þ 36q5z21z3z4

þ 27q5z21z24 þ q5z1z32 � 9q5z1z2z23 þ 18q5z1z2z3z4 � 9q5z1z2z24 � 18q5z22z

23 þ 6q5z22z3z4Þ

o

oz3

þ ð�g13Þ o

oz4þ X

s1;

X2 ¼1

12w61

ð6iq4z31z23 � 12iq4z31z3z4 � 18iq4z21z2z23 � 18iq4z21z2z

24 � 36iq4z1z22z3z4 þ 18iq4z1z22z

24

� 2iq4z1z43 þ 8iq4z1z33z4 þ 8iq4z1z3z34 � 2iq4z1z44 � 6iq4z32z23 þ 12iq4z32z3z4 � 6iq4z32z

24

þ 2iq4z2z43 þ 12iq4z2z23z24 � 8iq4z2z3z34 þ 2iq4z2z44 þ 3iq5z51 � 21iq5z41z2 � 12iq5z31z

23

þ 96iq5z31z3z4 � 42iq5z21z32 � 12iq5z21z2z

23 � 12iq5z21z2z

24 þ 15iq5z1z42 þ 288iq5z1z22z3z4

� 36iq5z1z22z24 þ 9iq5z1z43 þ 12iq5z1z33z4 þ 12iq5z1z3z34 þ 9iq5z1z44 � iq5z52 � 108iq5z32z

23

� 24iq5z32z3z4 þ 4iq5z32z24 � 75iq5z2z43 � 138iq5z2z23z

24 þ 36iq5z2z3z34 � 3iq5z2z44 þ 7q9z41z3

þ 17q9z41z4 þ 20q9z31z2z3 þ 6q9z21z22z4 � 6q9z21z

33 þ 6q9z21z

23z4 þ 42q9z21z

34 þ 52q9z1z32z3

� 4q9z1z32z4 � 12q9z1z2z33 � 84q9z1z2z3z24 � 12q9z1z2z34 � q9z42z3 þ q9z42z4 þ 78q9z22z23z4

� 18q9z22z3z24 þ 2q9z22z

34 þ 3q9z53 þ 21q9z43z4 þ 42q9z23z

34 � 9q9z3z44 þ q9z54 � 2z41z3q7

þ 26z41q7z4 � 4z31z2z3q7 þ 24z21z22q7z4 þ 6z21z

33q7 � 18z21z

23q7z4 þ 42z21q7z

34 � 44z1z32z3q7

þ 20z1z32q7z4 � 48z1z2z33q7 � 24z1z2q7z34 þ 2z42z3q7 � 2z42q7z4 þ 18z22z23q7z4 � 42z22z3q7z

24

þ 6z22q7z34Þ

o

oz1þ ð�g11Þ

o

oz2þ 1

12w61

ð3iq4z41z3 � 3iq4z41z4 � 12iq4z31z2z3 � 18iq4z21z22z4


� 4iq4z21z33 þ 12iq4z21z

23z4 þ 4iq4z21z

34 � 12iq4z1z32z3 þ 12iq4z1z32z4 þ 8iq4z1z2z33

þ 24iq4z1z2z3z24 � 8iq4z1z2z34 þ 3iq4z42z3 � 3iq4z42z4 þ 12iq4z22z23z4 � 12iq4z22z3z

24

þ 4iq4z22z34 þ iq4z53 � 5iq4z43z4 � 10iq4z23z

34 þ 5iq4z3z44 � iq4z54 � 6iq5z41z3 � 54iq5z41z4

þ 96iq5z31z2z3 � 12iq5z21z22z4 þ 18iq5z21z

33 � 138iq5z21z

23z4 � 150iq5z21z

34 þ 96iq5z1z32z3

� 24iq5z1z32z4 þ 12iq5z1z2z33 þ 36iq5z1z2z3z24 þ 36iq5z1z2z34 � 6iq5z42z3 þ 2iq5z42z4

� 138iq5z22z23z4 þ 54iq5z22z3z

24 � 6iq5z22z

34 � q9z51 � 13q9z41z2 � 14q9z31z

23 � 20q9z31z3z4

� 34q9z21z32 � 54q9z21z2z

23 � 102q9z21z2z

24 þ 11q9z1z42 þ 36q9z1z22z3z4 þ 6q9z1z22z

24

þ 3q9z1z43 þ 12q9z1z33z4 � 36q9z1z3z34 þ 3q9z1z44 � q9z52 � 50q9z32z23 þ 4q9z32z3z4

� 2q9z32z24 � 9q9z2z43 þ 18q9z2z23z

24 þ 12q9z2z3z34 � q9z2z44 � z51q7 þ 11z41z2q7

þ 4z31z23q7 þ 4z31z3q7z4 � 10z21z

32q7 � 48z21z2z

23q7 þ 12z21z2q7z

24 þ 11z1z42q7

þ 60z1z22z3q7z4 � 24z1z22q7z24 � 3z1z43q7 þ 48z1z33q7z4 þ 24z1z3q7z34 þ 9z1q7z44

� z52q7 � 32z32z23q7 � 20z32z3q7z4 þ 4z32q7z

24 � 15z2z43q7 � 54z2z23q7z

24 þ 24z2z3q7z34

� 3z2q7z44Þo

oz3þ ð�g13Þ

o

oz4þ X

s2:

3. Approximate first integrals

According to an approximate version of Noether theorem, to each approximate Noethersymmetry ðX ¼ X0 þ eX1 þ e2X2Þ there corresponds an approximate first integralðI ¼ I0 þ eI1 þ e2I2Þ [5]. Hence, first-order approximate first integral of (2) can be obtained from

XcX ¼ dI þOðe3Þ; ð17Þ
where, c is the interior product, X ¼ dx1 ^ dx3 þ dx2 ^ dx4 (^ is the wedge product), d is the ex-terior derivative. Evaluating (17) in ascending order of � we find
oIjox1

¼ �g3j ;oIjox2

¼ �g4j ;oIjox3

¼ g1j ;oIjox4

¼ g2j ðj ¼ 0; 1Þ:

Notice that these are not equations to solve to determine the approximate first integral. We in-tegrate these equations for the symmetries found in the previous section (after transforming intooriginal variables) to obtain

I0 ¼1

48w1

ð3x41q4w41 þ 12x31x4q9w

31 þ 6x21x

22q4w

41 � 12x21x2x3q9w

31 þ 6x21x

23q4w

21 þ 6x21x

24q4w

21

� 12x21x24q5w

21 þ 12x1x22x4q9w

31 þ 24x1x2x3x4q5w2

1 þ 12x1x23x4q9w1 � 4x1x34q9w1 � 8x1x34w1q7

þ 3x42q4w41 þ 4x32x3q9w

31 þ 8x32x3w

31q7 þ 6x22x

23q4w

21 � 12x22x

23q5w

21 þ 6x22x

24q4w

21

� 12x2x33q9w1 þ 12x2x3x24q9w1 þ 12x2x3x24w1q7 þ 3x43q4 þ 6x23x24q4 þ 3x44q4Þ; ð18Þ


I1 ¼1

60w41

ð15x41x2q4w51 � 15x41x3q9w

41 þ 60x31x2x4q9w

41 þ 30x31x3x4q5w

31 þ 10x21x

32q4w

51

þ 30x21x22x3q9w

41 þ 30x21x

22x3w

41q7 þ 15x21x2x

23q4w

31 � 30x21x2x

23q5w

31 þ 15x21x2x

24q4w

31

� 60x21x2x24q5w

31 � 5x21x

33q9w

21 þ 10x21x

33w

21q7 þ 45x21x3x

24q9w

21 þ 15x21x3x

24w

21q7

þ 20x1x32x4q9w41 þ 30x1x22x3x4q5w

31 þ 30x1x2x23x4q9w

21 � 10x1x2x34q9w

21 � 20x1x2x34w

21q7

þ 20x1x33x4q5w1 � 60x1x3x34q5w1 � 5x52q4w51 � 5x42x3q9w

41 � 15x42x3w

41q7 � 5x32x

23q4w

31

þ 30x32x23q5w

31 � 5x32x

24q4w

31 þ 25x22x

33q9w

21 þ 10x22x

33w

21q7 þ 15x22x3x

24q9w

21

� 15x22x3x24w

21q7 � 20x2x43q5w1 þ 60x2x23x

24q5w1 � 2x53q9 þ 4x53q7 þ 40x33x

24q9

þ 10x33x24q7 � 15x3x44q7Þ � s

x24x2

1

�þ x22

�; ð19Þ

I2 ¼1

2880w71

ð205x61q5w61 þ 870x51x4q9w

51 þ 150x51x4w

51q7 þ 495x41x

22q4w

61 � 1230x41x

22q5w

61

þ 570x41x2x3q9w51 þ 1290x41x2x3w

51q7 � 105x41x

23q5w

41 � 45x41x

24q4w

41 � 1830x41x

24q5w

41

þ 2820x31x22x4q9w

51 � 600x31x

22x4w

51q7 � 360x31x2x3x4q4w

41 þ 4080x31x2x3x4q5w

41

þ 300x31x23x4q9w

31 þ 300x31x

23x4w

31q7 � 1060x31x

34q9w

31 � 640x31x

34w

31q7 � 330x21x

42q4w

61

þ 1845x21x42q5w

61 � 180x21x

32x3q9w

51 � 480x21x

32x3w

51q7 � 270x21x

22x

23q4w

41

� 1620x21x22x

23q5w

41 þ 180x21x

22x

24q4w

41 � 2250x21x

22x

24q5w

41 þ 660x21x2x

33q9w

31

þ 660x21x2x33w

31q7 þ 3420x21x2x3x

24q9w

31 þ 900x21x2x3x

24w

31q7 � 345x21x

43q5w

21

� 270x21x23x

24q4w

21 þ 2700x21x

23x

24q5w

21 þ 30x21x

44q4w

21 þ 1665x21x

44q5w

21 þ 510x1x42x4q9w

51

þ 450x1x42x4w51q7 þ 240x1x32x3x4q4w

41 þ 3240x1x32x3x4q5w

41 � 900x1x22x

23x4q9w

31

� 1080x1x22x23x4w

31q7 � 660x1x22x

34q9w

31 � 180x1x22x

34w

31q7 � 360x1x2x33x4q4w

21

� 2640x1x2x33x4q5w21 þ 240x1x2x3x34q4w

21 � 5400x1x2x3x34q5w

21 � 570x1x43x4q9w1

þ 150x1x43x4w1q7 � 780x1x23x34q9w1 þ 480x1x23x

34w1q7 � 114x1x54q9w1 þ 378x1x54w1q7

þ 55x62q4w61 � 78x52x3q9w

51 þ 486x52x3w

51q7 þ 30x42x

23q4w

41 � 1935x42x

23q5w

41

� 75x42x24q4w

41 � 1180x32x

33q9w

31 � 640x32x

33w

31q7 þ 780x32x3x

24q9w

31 þ 720x32x3x

24w

31q7

� 45x22x43q4w

21 þ 2010x22x

43q5w

21 þ 180x22x

23x

24q4w

21 þ 630x22x

23x

24q5w

21 � 75x22x

44q4w

21

þ 570x2x53q9w1 � 150x2x53w1q7 � 300x2x33x24q9w1 � 1020x2x33x

24w1q7 þ 150x2x3x44q9w1

� 360x2x3x44w1q7 � 115x63q5 � 225x43x24q4 þ 690x43x

24q5 þ 150x23x

44q4 � 1035x23x

44q5

� 25x64q4Þ � s2x24x21

�þ x22

�: ð20Þ

Notice that the first-order approximate symmetry analysis does not allow us to determine thevalues of q4, q5, q7 and q9. Therefore, approximate first integral given in (17) is not unique at thisorder of approximation.Approximate first integral I in (17) and the energy integral H in (1) define hypersurfaces in the

phase space of the dynamical system given in (2). Their intersection is a surface on which theprojections of orbits survive. The section of this surface by the plane x1 ¼ 0 gives the invariantcurves (invariant under the flow operator generated by (2)). In order to obtain invariant curvesanalytically, we first set x1 ¼ 0 both in (17) and (1). We next calculate x3 from the energyh ¼ Hðx2; x3; x4Þ as

Fig. 1. Analytical versus numerical comparisons for the parameter values a ¼ 0:25, h ¼ 0:05, e ¼ 0:1 (a), (b) s ¼ 0:05,((c), (d)) s ¼ 0:0, ((e), (f)) s ¼ �0:05.



x3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2h� bx22 � x24 þ

2e3x22

r; ð21Þ

where b ¼ x21 þ es and then we substitute it into I to obtain the reduced approximate first integral

in terms of x2 and x4. Contour curves have been plotted in Fig. 1(a), (c) and (e) for q4 ¼ �20,q5 ¼ 1 and q7 ¼ q9 ¼ 0. These are, indeed, celebrated KAM curves which have been proven toexist under small perturbations [9]. As the detuning parameter s has been varied from s ¼ 0:05 to)0.05 continuously we have observed that system undergoes complicated bifurcations which havebeen discussed in [8] in a much general context. Numerically obtained Poincar�e surface of sectionsagree well with the numerical ones in Fig. 1(b), (d) and (f).

Acknowledgement

G. €Unal greatly acknowledges the support provided by IISAMM (International Institute forSymmetry Analysis and Mathematical Modelling).

References

[1] Contopoulos G. On the existence of a third integral of motion. Astron J 1962;68:1–14.

[2] Henon M, Heiles C. The applicability of the third integral of motion, some numerical experiments. Astron J

1964;69:73–9.

[3] Gustavson FG. On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron J

1966;71:670–80.

[4] Lichtenberg AJ, Lieberman MA. Regular and stochastic motion. New York: Springer-Verlag; 1992.

[5] €Unal G. Approximate generalized symmetries, normal forms and approximate first integrals. Phys Lett A

2000;269:13–30.

[6] Baikov VA, Gazizov RK, Ibragimov NH. Approximate symmetries. Math USSR Sbornik 1989;64:427–41.

[7] Ibragimov NH. Transformation groups applied to mathematical physics. Moscow: Nauka; 1985. English

translation published by D. Reidel, Dordrecht.

[8] Meyer KR, Hall GR. Introduction to Hamiltonian dynamical systems and the N -body problem. New York:

Springer-Verlag; 1992.

[9] Arnold VI. Mathematical methods of classical mechanics. New York: Springer-Verlag; 1978.

[10] €Unal G, Gorali G. Approximate first integrals of a galaxy model. Nonlinear Dynam 2002;28:195–211.

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Approximate first integrals of the Hénon–Heiles system revisited