Approximate First Integrals of a Galaxy Model

Nonlinear Dynamics 28: 195–211, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Approximate First Integrals of a Galaxy Model

G. ÜNAL and G. GORALIDepartment of Engineering Sciences, Faculty of Sciences, Istanbul Technical University, Maslak,80626 Istanbul, Turkey

(Received: 16 February 2001; accepted: 7 September 2001)

Abstract. First-order approximate first integrals (conserved quantities) of a Hamiltonian dynamical system withtwo degrees of freedom which arises in the modeling of central part of a deformed galaxy [1] have been obtainedbased on the approximate Noether symmetries for resonances ω1 = ω2, ω1 = 2ω2 and 2ω1 = 3ω2. Further-more, KAM curves have been obtained analytically and they have been compared with the numerical ones on thePoincaré surface of section.

Keywords: Hamiltonian dynamical systems, approximate Noether symmetries, resonances, Noether’s theorem.

1. Introduction

Analytical investigations on the approximate first integrals was motivated by the regular be-haviour observed in the numerical studies of the nearly integrable Hamiltonian systems [2,3]. Various perturbative methods have been developed to construct approximate first integrals,e.g., direct method of Contopoulos [2] and Birkhoff–Gustavson normal form method. A com-prehensive study of these methods and others can be found in [4]. Yet, none of these methodsresort to the celebrated Noether’s theorem which provides a link between the exact Noethersymmetries of the dynamical systems and the exact first integrals. Extensions of Noether’stheorem to approximate symmetries are given for Hamiltonian systems in [5] and for Lag-rangian systems in [6]. Approximate symmetry group analysis of differential equations hasbeen developed by Baikov et al. [6]. Based on their definition, the author has given a methodwhich incorporates resonances in [5]. Here, we consider a Hamiltonian of the form

H = 1

2(p2

1 + p22 + Ax2

1 + Bx22 )− εx2

1x22 , (1)

where pi and xi are canonical momentum and position, respectively. This Hamiltonian arisesin the modeling of central part of a deformed galaxy [1]. Numerical results given in [7] showsthat this system exhibits chaotic behaviour. When we set A = ω2

1 and B = ω22, it becomes the

Hamiltonian of two coupled harmonic oscillators, where ε is a coupling parameter. Altoughthe parameter ε is allowed to have large values in [1, 7], here we will consider it as a smallpositive parameter (ε � 1).

Our main objective here is to obtain the approximate first integrals of the Hamiltoniansystem

x1

x2

x3

x4

=

0 0 1 00 0 0 1

−ω21 0 0 0

0 −ω22 1 0

x1

x2

x3

x4

+ ε

00

2x1x22

2x21x2

(2)

196 G. Ünal and G. Gorali

for resonances ω1 = ω2, ω1 = 2ω2 and 2ω1 = 3ω2. Here, x3 = p1 and x4 = p2. Approximatefirst integrals can be employed to obtain approximate solutions. Here, we will make use ofthem in obtaining KAM curves (intersections of the KAM tori with a surface of section) ofHamiltonian system given in (2). Furthermore, we will compare analytical results with thenumerical ones.

2. Approximate Noether Symmetries

Lie considered a differential equation as a surface in a certain space which is called a frame(this term has been coined by Ibragimov in [8]). He looked for transformations which leave theframe form invariant in the space spanned by the transformed coordinates. This is to say that ifone takes a picture of the new frame, one would observe the frame which has the same shapewith the picture of the frame taken in the original space. This led him to show that not onlythese transformations form a continuous group (called Lie symmetry group) but also theytransform solutions of a differential equation to new solutions. Many differential equationsarising in mathematical physics do not admit nontrivial exact symmetries. Is it possible torelax the exact invariance condition? An affirmative answer has been given by Baikov et al. in[6]. These transformations are called approximate symmetries.

Following Ibragimov [9] and Baikov et al. [6], first-order approximate invariance conditionof the Hamiltonian system (2) reads as

[X,F] = O(ε2), (3)

where [, ] is the Lie bracket and,

X = X0 + εX1, Xb =4∑l=1

ηlb(x)∂

∂xl,

F = F0 + εF1, Fb =4∑l=1

f lb(x)∂

∂xl(b = 0, 1).

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

4∑j=1

(ηj

0

∂f k0

∂xj− f

j

0

∂ηk0

∂xj

)= 0,

4∑j=1

(ηj

1

∂f k0

∂xj− f

j

0

∂ηk1

∂xj

)=

4∑j=1

(fj

1

∂ηk0

∂xj− η

j

0

∂f k1

∂xj

). (4)

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 method of characteristics. It can be verified that it forms an infinite dimensional Liealgebra.

Approximate First Integrals of a Galaxy Model 197

In order to remain in touch with the normal form theory [5], we now follow a differentavenue. Introducing the transformation

x = Sz, (5)

where

S =

iω1

iω1

0 0

0 0 iω2

iω2

−1 −1 0 0

0 0 −1 −1

to (2) yields

z1

z2

z3

z4

=

iω1 0 0 00 −iω1 0 00 0 iω2 00 0 0 −iω2

z1

z2

z3

z4

+ ε

i (z1−z2)(z3−z4)2

ω1ω22

i (z1−z2)(z3−z4)2

ω1ω22

i (z1−z2)2(z3−z4)

ω1ω22

i (z1−z2)2(z3−z4)

ω1ω22

. (6)

Determining system of equations for (6) becomes

4∑j=1

λjzj∂ηk0

∂zj− λkη

k0 = 0, (7)

4∑j=1

λjzj∂ηk1

∂zj− λkη

k1 =

4∑j=1

(fj

1

∂ηk0

∂zj− η

j

0

∂f k1

∂zj

), (8)

where λ1 = iω1 , λ2 = −iω1 , λ3 = iω1 , λ4 = −iω1. Solutions of (4) can be transformed tosolutions of (8) via [5]

ηk0 =4∑l=1

Sklηl0(S−1x, t) and ηk1 =

4∑l=1

Sklηl1(S−1x, t). (9)

The set of partial differential equations (7) is called homological equation in the context ofnormal form theory [10]. Its solutions in the space of homogenous polynomials read as [5,10]:

ηj

0 =∑

s1λ1+···+s4λ4−λj=0

Cj

0s1s2s3s4zs11 z

s22 z

s33 z

s44 , (10)

where C0s1...s4 are the group parameters, and the sum in (10) will be taken over all the resonantmonomials which satisfy the resonance conditions s1λ1 + · · · + s4λ4 − λj = 0.

Third-order resonant monomials can be obtained by solving the algebraic equations

s1λ1 + s2λ2 + s3λ3 + s4λ4 − λj = 0, s1 + s2 + s3 + s4 = 3. (11)


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

X0 = (a1z1z2z3 + a2z1z3z4 + a3z21z2 + a4z

21z4 + a5z2z

23 + a6z

23z4)

∂

∂z1

+ (b1z1z2z4 + b2z2z3z4 + b3z1z22 + b4z1z

24 + b5z

22z3 + b6z3z

24)∂

∂z2

+ (c1z1z2z3 + c2z1z3z4 + c3z21z2 + c4z

21z4 + c5z2z

23 + c6z

23z4)

∂

∂z3

+ (d1z1z2z4 + d2z2z3z4 + d3z1z22 + d4z1z

24 + d5z

22z3 + d6z3z

24)∂

∂z4, (12)

where a1, . . . , d6 are complex coefficients. Notice that X0 is written as a linear combination oftwenty four independent symmetries. In order for a symmetry vector field to be a Noether sym-metry, it must leave the Hamiltonian invariant and it must be divergence free (conservative)[9]. These conditions for X0 read as

4∑j=1

ηj

0

∂H0

∂zj= 0 and

4∑j=1

∂ηj

0

∂zj= 0, (13)

where H0 = 2(z1z2 + z3z4). Imposing the Noether symmetry conditions on X0 in (12) we find

a1 + b5 + d3 = 0, a2 + b2 + c1 + d1 = 0, a3 + b3 = 0,

a4 + b1 + c3 = 0, a5 + d5 = 0, a6 + c5 + d2 = 0,

b4 + c4 = 0, b6 + c2 + d4 = 0, c6 + d6 = 0,

a1 + 2b5 + 2c5 + d2 = 0, a2 + b2 + 2c6 + 2d6 = 0,

2a3 + 2b3 + c1 + d1 = 0, 2a4 + b1 + c2 + 2d4 = 0. (14)

Furthermore, solutions of (14) must yield real ηj0 when they are transformed according to (9),i.e.,

η10 = i

ω1(η1

0 − η20), η2

0 = i

ω1(η3

0 − η40), η3

0 = −η10 − η2

0, η40 = −η3

0 − η40

must be functions in real space. This leads to

η10 = η2∗

0 , η30 = η4∗

0 .

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

a1 = b∗1, a2 = b∗

2, a3 = b∗3, a4 = b∗

5, a5 = b∗4, a6 = b∗

6,

c1 = d∗1 , c2 = d∗

2 , c3 = d∗3 , c4 = d∗

5 , c5 = d∗4 , c6 = d∗

6 , (15)

where ( )∗ stands for the complex conjugate. Solving the set of algebraic equations in (14)under the conditions given in (15) leads to following Noether symmetries.

X10 = iz1z3z4

∂

∂z1− iz2z3z4

∂

∂z2, X

20 = iz1z2z3

∂

∂z3− iz1z2z4

∂

∂z4,


X30 = iz2

1z2∂

∂z1− iz1z

22∂

∂z2, X

40 = iz2

3z4∂

∂z3− iz3z

24∂

∂z4,

X50 = iz2z

23∂

∂z1− iz1z

24∂

∂z2+ iz2

1z4∂

∂z3− iz2

2z3∂

∂z4,

X60 = z2z

23∂

∂z1+ z1z

24∂

∂z2− z2

1z4∂

∂z3− z2

2z3∂

∂z4,

X70 = (z1z2z3 − z2

1z4)∂

∂z1+ (z1z2z4 − z2

2z3)∂

∂z2

+ (−z1z3z4 + z2z23)∂

∂z3+ (−z2z3z4 + z1z

24)∂

∂z4,

X80 = i(z1z2z3 + z2

1z4)∂

∂z1− i(z1z2z4 + z2

2z3)∂

∂z2

+ i(z1z3z4 + z2z23)∂

∂z3− i(z2z3z4 + z1z

24)∂

∂z4,

X90 = −z2

1z4∂

∂z1− z2

2z3∂

∂z2+ (−2z1z3z4 + z2

1z2 + 2z2z23)∂

∂z3

+ (−2z2z3z4 + z1z22 + 2z1z

24)∂

∂z4,

X100 = iz2

1z4∂

∂z1− iz2

2z3∂

∂z2+ i(2z1z3z4 − z2

1z2 + 2z2z23)∂

∂z3

+ i(−2z2z3z4 + z1z22 − 2z1z

24)∂

∂z4,

X110 = z2

3z4∂

∂z1+ z3z

24∂

∂z2+ (−2z1z3z4 + z2z

23)∂

∂z3+ (−2z2z3z4 + z1z

24)∂

∂z4,

X120 = iz2

3z4∂

∂z1− iz3z

24∂

∂z2+ i(2z1z3z4 + z2z

23)∂

∂z3− i(2z2z3z4 + z1z

24)∂

∂z4. (16)

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

X0 =12∑k=1

qkXk

0.

After substituting X0 into the right-hand side of (8), we notice that the following resonantmonomials appear.

r11 = (3iq7 + q8)z

31z2z4 + (−q1 + q3 + 2q5)z

31z

24 + 2iq6z

31z

24

+ (3iq7 − q8 − 3iq9 + q10)z21z

22z3 + (−3iq7 − q8 − 2iq9 − 2q10 + 4iq11)z

21z3z

24

+ (q1 + 2q2 − 3q3 − 6q5 + 6iq6 − 3iq7)z1z22z

23

+ (q8 + 2iq9 + 2q10 + 8iq11)z1z2z23z4 + (−3iq9 + 3q10 − 3iq11)z

22z

33

− 2iq6z1z23z

24 + 3q12z

22z

33 + (−2q1 + 2q4 + 4q5 − 4iq6)z2z

33z4


+ 4iq6z21z2z3z4 + (−3iq11 + q12)z

33z

24,

r21 = (−3iq7 − q8 + 3iq9 + q10)z

21z

22z4 + (q1 + 2q2 − 3q3 − 6q5 − 6iq6)z

21z2z

24

+ (3iq9 + 3q10 + 3iq11 + 3q12)z21z

34 + (−3iq7 + q8)z1z

32z3 − 4iz1z

22z3z4q6

+ (3iq7 + q8 − 2iq9 + 2q10 − 8iq11)z1z2z3z24

+ (−2q1 + 2q4 + 4q5 + 4iq6)z1z3z34

+ (−q1 + q3 + 2q5 − 2iq6)z32z

23 + (3iq7 − q8 + 2iq9 − 2q10 − 4iq11)z

22z

23z4

+ 2iq6z2z23z

24 + (3iq11 + q12)z

23z

34,

r31 = (−3iq9 − q10)z

31z

22 + (−2q2 + 2q3 + 4q5 + 4iq6)z

31z2z4 + 2iq6z

21z

22z3

+ (−3iq9 − 3q10 − 3iq11 − 3q12)z31z

24

+ (3iq7 + q8 + 14iq9 + 2q10 + 8iq11)z21z2z3z4

+ (2q1 + q2 − 3q4 − 6q5 − 6iq6)z21z3z

24

+ (3iq7 − q8 + 10iq9 − 2q10 + 4iq11)z1z22z

23

+ (−3iq7 − q8 − 6iq9 − 2q10 − 9iq11 − 3q12)z1z23z

24

+ (−q2 + q4 + 2q5 − 2iq6)z22z

33

+ (−3iq7 + q8 − 6iq9 + 2q10 − 6iq11 + 2q12)z2z33z4 − 4iq6z1z2z

23z4,

r41 = (3iq9 − q10)z

21z

32 − 2iq6z

21z

22z4

+ (−3iq7 − q8 − 10iq9 − 2q10 − 4iq11)z21z2z

24 + (−q2 + q4 + 2q5 + 2iq6)z

21z

34

+ (−3iq7 + q8 − 14iq9 + 2q10 − 8iq11)z1z22z3z4

+ (−2q2 + 2q3 + 4q5 − 4iq6)z1z32z3

+ (3iq7 + q8 + 6iq9 + 2q10 + 6iq11 + 2q12)z1z3z34

+ (3iq9 − 3q10 + 3iq11 − 3q12)z32z

23 + (2q1 + q2 − 3q4 − 6q5 + 6iq6)z

22z

23z4

+ (3iq7 − q8 + 6iq9 − 2q10 + 9iq11 − 3q12)z2z23z

24 + 4iq6z1z2z3z

24. (17)

In order to be able to find ηk1 in the space of homogeneous polynomials, we now have tokill these resonant monomials by setting their coefficients equal to zero. This leads to q6 =0, . . . , q12 = 0, and the linear algebraic equations

−q1 + q3 + 2q5 = 0, q1 + 2q2 − 3q3 − 6q5 = 0,

−2q1 + 2q4 + 4q5 = 0, −2q2 + 2q3 + 4q5 = 0,

2q1 + q2 − 3q4 − 6q5 = 0, −q2 + q4 + 2q5 = 0.

Solutions can be found as

q1 = 2q5 + q4, q2 = q1, q3 = q4. (18)


This, indeed, indicates that the symmetry-breaking occurring [5]. Remaining monomials inUk

1 are not resonant. Hence, infinitesimals of the first-order approximate symmetry can bewritten as [5]

ηk1 =∑

sk1,...,sk4

Ksk1...sk4

sk1λ1 + · · · + sk4λ4 − λkzsk11 . . . z

sk44 , (19)

whereKsk1...sk4 are the coefficients of the corresponding non-resonant monomials appearing inUk

1 . This leads to

X1 = 1

4ω41

g

((−2iz3

1z23q5 − 2iz3

1z23q4 + 4iz3

1z3z4q5 + 4iz31z3z4q4 + 12iz2

1z2z23q5

+ 8iz21z2z

23q4 + 12iz2

1z2z24q5 + 8iz2

1z2z24q4 + 12iz1z

22z3z4q5 + 12iz1z

22z3z4q4

− 6iz1z22z

24q5 − 6iz1z

22z

24q4 − iz1z

43q5 + 4iz1z

33z4q5 + 4iz1z

33z4q4 + 4iz1z3z

34q5

+ 4iz1z3z34q4 − iz1z

44q5 + 8iz3

2z23q5 − 4iz3

2z3z4q5 + 4iz2z43q5

+ 12iz2z23z

24q5 + 8iz2z

23z

24q4 − 4iz2z3z

34q5 − 4iz2z3z

34q4)

∂

∂z1

+ (4iz31z3z4q5 − 8iz3

1z24q5 + 6iz2

1z2z23q5 + 6iz2

1z2z23q4 − 12iz2

1z2z3z4q5

− 12iz21z2z3z4q4 − 12iz1z

22z

23q5 − 8iz1z

22z

23q4 − 12iz1z

22z

24q5

− 8iz1z22z

24q4 + 4iz1z

33z4q5 + 4iz1z

33z4q4 − 12iz1z

23z

24q5 − 8iz1z

23z

24q4

− 4iz1z44q5 − 4iz3

2z3z4q5 − 4iz32z3z4q4 + 2iz3

2z24q5 + 2iz3

2z24q4

+ iz2z43q5 − 4iz2z

33z4q5 − 4iz2z

33z4q4 − 4iz2z3z

34q5 − 4iz2z3z

34q4

+ iz2z44q5)

∂

∂z2+ (−iz4

1z3q5 + 4iz41z4q5 + 4iz3

1z2z3q5

+ 4iz31z2z3q4 + 12iz2

1z22z4q5 + 8iz2

1z22z4q4 − 2iz2

1z33q5 − 2iz2

1z33q4

+ 12iz21z

23z4q5 + 8iz2

1z23z4q4 + 8iz2

1z34q5 + 4iz1z

32z3q5 + 4iz1z

32z3q4

− 4iz1z32z4q5 − 4iz1z

32z4q4 + 4iz1z2z

33q5 + 4iz1z2z

33q4 + 12iz1z2z3z

24q5

+ 12iz1z2z3z24q4 − 4iz1z2z

34q5 − iz4

2z3q5 + 12iz22z

23z4q5 + 8iz2

2z23z4q4

− 6iz22z3z

24q5 − 6iz2

2z3z24q4)

∂

∂z3+ (iz4

1z4q5 + 4iz31z2z3q5 + 4iz3

1z2z3q4

− 4iz31z2z4q5 − 4iz3

1z2z4q4 − 12iz21z

22z3q5 − 8iz2

1z22z3q4 + 6iz2

1z23z4q5

+ 6iz21z

23z4q4 − 12iz2

1z3z24q5 − 8iz2

1z3z24q4 − 4iz1z

32z4q5 − 4iz1z

32z4q4

+ 4iz1z2z33q5 − 12iz1z2z

23z4q5 − 12iz1z2z

23z4q4 − 4iz1z2z

34q5

− 4iz1z2z34q4 − 4iz4

2z3q5 + iz42z4q5 − 8iz2

2z33q5 − 12iz2

2z3z24q5

− 8iz22z3z

24q4 + 2iz2

2z34q5 + 2iz2

2z34q4)

∂

∂z4g

). (20)


Transforming (19) and (20) according to (9) we have obtained corresponding approximateNoether symmetry in the real space. This has been given in (A.1).

Following the procedure performed for the resonance condition ω1 = ω2 given above, wehave obtained the following first-order approximate Noether symmetry vector fields for theresonance condition ω1 = 2ω2.

X10 = iz1z3z4

∂

∂z1− iz2z3z4

∂

∂z2, X

20 = iz1z2z3

∂

∂z3− iz1z2z4

∂

∂z4,

X30 = iz2

3z4∂

∂z3− iz3z

24∂

∂z4, (21)

X1 = 1

3ω41

g

((−2iz3

1z23q1 + 6iz3

1z24q1 + 12iz2

1z2z23q1 + 24iz2

1z2z23q2

+ 12iz21z2z

24q1 + 24iz2

1z2z24q2 + 6iz1z

22z

23q1 + 24iz1z

22z

23q2

− 2iz1z22z

24q1 − 8iz1z

22z

24q2 + 24iz1z

33z4q3 + 24iz1z3z

34q3

− 24iz2z33z4q1 + 24iz2z

33z4q3 + 24iz2z

23z

24q1 − 8iz2z3z

34q1

− 8iz2z3z34q3)

∂

∂z1+ (2iz2

1z2z23q1 + 8iz2

1z2z23q2

− 6iz21z2z

24q1 − 24iz2

1z2z24q2 − 12iz1z

22z

23q1 − 24iz1z

22z

23q2 − 12iz1z

22z

24q1

− 24iz1z22z

24q2 + 8iz1z

33z4q1 + 8iz1z

33z4q3 − 24iz1z

23z

24q1

+ 24iz1z3z34q1 − 24iz1z3z

34q3 − 6iz3

2z23q1 + 2iz3

2z24q1 − 24iz2z

33z4q3

− 24iz2z3z34q3)

∂

∂z2+ (12iz3

1z2z4q2 + 24iz21z

22z4q2

− 4iz21z

33q2 − 2iz2

1z33q3 + 6iz2

1z23z4q1 + 12iz2

1z23z4q2 − 12iz2

1z3z24q1

− 12iz21z3z

24q2 + 18iz2

1z3z24q3 − 4iz1z

32z4q2 + 12iz1z2z

33q3 + 36iz1z2z3z

24q3

− 12iz22z

33q2 + 6iz2

2z33q3 + 6iz2

2z23z4q1 + 12iz2

2z23z4q2 − 4iz2

2z3z24q1

− 4iz22z3z

24q2 − 6iz2

2z3z24q3)

∂

∂z3+ (4iz3

1z2z3q2 − 24iz21z

22z3q2

+ 4iz21z

23z4q1 + 4iz2

1z23z4q2 + 6iz2

1z23z4q3 − 6iz2

1z3z24q1

− 12iz21z3z

24q2 + 12iz2

1z34q2 − 6iz2

1z34q3 − 12iz1z

32z3q2

− 36iz1z2z23z4q3 − 12iz1z2z

34q3 + 12iz2

2z23z4q1 + 12iz2

2z23z4q2

− 18iz22z

23z4q3 − 6iz2

2z3z24q1 − 12iz2

2z3z24q2 + 4iz2

2z34q2

+ 2iz22z

34q3)

∂

∂z4g

). (22)

Corresponding approximate Noether symmetry vector field in the real space has been givenin (A.2).


Similar considerations lead to the following approximate Noether symmetries for the res-onance condition 2ω1 = 3ω2.

X10 = iz1z3z4

∂

∂z1− iz2z3z4

∂

∂z2, X

20 = iz2

1z2∂

∂z1− iz1z

22∂

∂z2,

X30 = iz1z2z3

∂

∂z3− iz1z2z4

∂

∂z4, X

40 = iz2

3z4∂

∂z3− iz3z

24∂

∂z4, (23)

X1 = 1

40ω41

g

((−18iz3

1z23q1 − 27iz3

1z23q2 + 90iz3

1z3z4q2 + 90iz31z

24q1

− 135iz31z

24q2 + 90iz2

1z2z23q1 + 135iz2

1z2z23q3 + 90iz2

1z2z24q1

+ 135iz21z2z

24q3 + 90iz1z

22z

23q1 − 405iz1z

22z

23q2 + 270iz1z

22z

23q3

+ 270iz1z22z3z4q2 − 18iz1z

22z

24q1 − 81iz1z

22z

24q2

− 54iz1z22z

24q3 + 135iz1z

33z4q4 + 135iz1z3z

34q4 − 270iz2z

33z4q1

+ 270iz2z33z4q4 + 180iz2z

23z

24q1 − 54iz2z3z

34q1 − 54iz2z3z

34q4)

∂

∂z1

+ (18iz21z2z

23q1 + 81iz2

1z2z23q2 + 54iz2

1z2z23q3 − 270iz2

1z2z3z4q2

− 90iz21z2z

24q1 + 405iz2

1z2z24q2 − 270iz2

1z2z24q3 − 90iz1z

22z

23q1 − 135iz1z

22z

23q3

− 90iz1z22z

24q1 − 135iz1z

22z

24q3 + 54iz1z

33z4q1 + 54iz1z

33z4q4 − 180iz1z

23z

24q1

+ 270iz1z3z34q1 − 270iz1z3z

34q4 − 90iz3

2z23q1 + 135iz3

2z23q2

− 90iz32z3z4q2 + 18iz3

2z24q1 + 27iz3

2z24q2 − 135iz2z

33z4q4

− 135iz2z3z34q4)

∂

∂z2+ (60iz3

1z2z3q2 − 180iz31z2z4q2

+ 180iz31z2z4q3 + 180iz2

1z22z4q3 − 27iz2

1z33q3 − 18iz2

1z33q4

+ 60iz21z

23z4q1 + 90iz2

1z23z4q3 − 180iz2

1z3z24q1 − 135iz2

1z3z24q3

+ 270iz21z3z

24q4 + 60iz1z

32z3q2 − 36iz1z

32z4q2 − 36iz1z

32z4q3

+ 90iz1z2z33q4 + 270iz1z2z3z

24q4 − 135iz2

2z33q3 + 90iz2

2z33q4

+ 60iz22z

23z4q1 + 90iz2

2z23z4q3 − 36iz2

2z3z24q1 − 27iz2

2z3z24q3

− 54iz22z3z

24q4)

∂

∂z3+ (36iz3

1z2z3q2 + 36iz31z2z3q3

− 60iz31z2z4q2 − 180iz2

1z22z3q3 + 36iz2

1z23z4q1 + 27iz2

1z23z4q3

+ 54iz21z

23z4q4 − 60iz2

1z3z24q1 − 90iz2

1z3z24q3 + 135iz2

1z34q3

− 90iz21z

34q4 + 180iz1z

32z3q2 − 180iz1z

32z3q3 − 60iz1z

32z4q2

− 270iz1z2z23z4q4 − 90iz1z2z

34q4 + 180iz2

2z23z4q1 + 135iz2

2z23z4q3

− 270iz22z

23z4q4 − 60iz2

2z3z24q1 − 90iz2

2z3z24q3 + 27iz2

2z34q3


+ 18iz22z

34q4)

∂

∂z4g

). (24)

Corresponding approximate Noether symmetry vector field in the real space has been givenin (A.2).

3. Approximate First Integrals

According to an approximate version of Noether theorem, to each approximate Noether sym-metry (X = X0 + εX1) there corresponds an approximate first integral (I = I0 + εI1) [5].Hence, first-order approximate first integral of (2) can be obtained from

X �$ = dI +O(ε2), (25)

where � is the interior product, $ = dx1 ∧ dx3 + dx2 ∧ dx4 (∧ is the wedge product), d is theexterior derivative. Evaluating (25) in ascending order of ε we find

∂Ij

∂x1= −η3

j ,∂Ij

∂x2= −η4

j ,∂Ij

∂x3= η1

j ,∂Ij

∂x4= η2

j (j = 0, 1). (26)

Integration of (26) for the infinitesimals given in (A.1) yields

I = 1

16ω1(x4

1q4ω41 + 2x2

1x22q4ω

41 + 6x2

1x22ω

41q5 + 2x2

1x23q4ω

21

+ 2x21x

24q4ω

21 + 2x2

1x24ω

21q5 + 8x1x2x3x4ω

21q5 + x4

2q4ω41

+ 2x22x

23q4ω

21 + 2x2

2x23ω

21q5 + 2x2

2x24q4ω

21 + x4

3q4 + 2x23x

24q4

+ 6x23x

24q5 + x4

4q4)+ ε1

64ω51

(−15x41x

22q5ω

61 − 10x4

1x22ω

61q4

+ 5x41x

24q5ω

41 + 2x4

1x24ω

41q4 − 8x3

1x2x3x4q5ω41 + 8x3

1x2x3x4ω41q4

− 15x21x

42q5ω

61 − 10x2

1x42ω

61q4 + 6x2

1x22x

23q5ω

41 − 8x2

1x22x

23ω

41q4

+ 6x21x

22x

24q5ω

41 − 8x2

1x22x

24ω

41q4 + 6x2

1x23x

24q5ω

21 + 8x2

1x23x

24ω

21q4

− 3x21x

44q5ω

21 + 2x2

1x44ω

21q4 − 8x1x

32x3x4q5ω

41 + 8x1x

32x3x4ω

41q4

+ 24x1x2x33x4q5ω

21 + 8x1x2x

33x4ω

21q4 + 24x1x2x3x

34q5ω

21 + 8x1x2x3x

34ω

21q4

+ 5x42x

23q5ω

41 + 2x4

2x23ω

41q4 − 3x2

2x43q5ω

21 + 2x2

2x43ω

21q4 + 6x2

2x23x

24q5ω

21

+ 8x22x

23x

24ω

21q4 + 9x4

3x24q5 + 6x4

3x24q4 + 9x2

3x44q5 + 6x2

3x44q4). (27)

Notice that the first-order approximate symmetry analysis does not allow us to determinethe values of q4 and q5. Therefore, approximate first integral given in (27) is not unique at thisorder of approximation.

Approximate first integral I in (27) and the energy integral H in (1) define hypersurfacesin the phase space of the dynamical system given in (2). Their intersection is a surface onwhich the projections of orbits survive. The section of this surface by the plane x2 = 0 givesthe invariant curves (invariant under the flow operator generated by (2)) . In order to obtain


invariant curves analytically, we first set x2 = 0 both in (27) and (1). We next calculate x4

from the energy h = H(x1, x3, x4) as

x4 =√

2h− ax21 − x2

3 , (28)

where a = ω21 and then we substitute it into I to obtain

P = 1

8a1/2(−a2x4

1q5 + 2ahx21q5 − 4ax2

1x23q5 + 2h2q4 + 6hx2

3q5 − 3x43q5)

+ ε1

32a5/2(−4a3x6

1q5 − 2a2hx41q4 + 11a2hx4

1q5 − 4a2x41x

23q5

+ 4ah2x21q4 − 6ah2x2

1q5 − 8ahx21x

23q4 − 6ahx2

1x23q5 + 12h2x2

3q4

+ 18h2x23q5 − 6hx4

3q4 − 9hx43q5). (29)

Similarly we have obtained the following first-order approximate first integral and its sectionon x2 = 0 for the resonance case ω1 = 2ω2.

I = 1

128ω1(8x2

1x22ω

41q2 + 32x2

1x24ω

21q2 + x4

2ω41q3 + 8x2

2x23ω

21q2 + 8x2

2x24ω

21q3

+ 32x23x

24q2 + 16x4

4q3)+ ε1

1152ω51

(−96x41x

22q2ω

61 + 384x4

1x24q2ω

41

+ 768x31x2x3x4q2ω

41 − 42x2

1x42q2ω

61 − 21x2

1x42ω

61q1 − 24x2

1x42ω

61q3

− 288x21x

22x

23q2ω

41 − 336x2

1x22x

24q2ω

41 + 24x2

1x22x

24ω

41q1 + 1152x2

1x23x

24q2ω

21

+ 96x21x

44q2ω

21 + 48x2

1x44ω

21q1 + 384x2

1x44ω

21q3 − 96x1x

32x3x4ω

41q1

+ 192x1x32x3x4ω

41q3 + 768x1x2x

33x4q2ω

21 − 512x1x2x3x

34q2ω

21

− 128x1x2x3x34ω

21q1 + 768x1x2x3x

34ω

21q3 + 42x4

2x23q2ω

41 + 21x4

2x23ω

41q1

− 48x42x

23ω

41q3 − 192x2

2x43q2ω

21 + 336x2

2x23x

24q2ω

21 − 24x2

2x23x

24ω

21q1

+ 768x43x

24q2 − 96x2

3x44q2 − 48x2

3x44q1 + 768x2

3x44q3), (30)

P = 1

8a1/2(−2a2x4

1q2 + a2x41q3 + 4ahx2

1q2 − 4ahx21q3 − 4ax2

1x23q2 + 2ax2

1x23q3

+ 4h2q3 + 4hx23q2 − 4hx2

3q3 − 2x43q2 + x4

3q3)+ ε1

24a5/2(−6a3x6

1q2 + 8a3x61q3

+ a3x61q1 + 8a2hx4

1q2 − 32a2hx41q3 − 4a2hx4

1q1 − 30a2x41x

23q2 + 32a2x4

1x23q3

+ a2x41x

23q1 + 8ah2x2

1q2 + 32ah2x21q3 + 4ah2x2

1q1 + 48ahx21x

23q2

− 96ahx21x

23q3 − 42ax2

1x43q2 + 40ax2

1x43q3 − ax2

1x43q1 − 8h2x2

3q2 + 64h2x23q3

− 4h2x23q1 + 40hx4

3q2 − 64hx43q3 + 4hx4

3q1 − 18x63q2 + 16x6

3q3 − x63q1). (31)

For the resonance case 2ω1 = 3ω2, approximate first integral and its section on x2 = 0 hasthe form

I = 1

864ω1(54x4

1q2ω41 + 72x2

1x22ω

41q3 + 108x2

1x23q2ω

21 + 162x2

1x24ω

21q3 + 16x4

2ω41q4


+ 72x22x

23ω

21q3 + 72x2

2x24ω

21q4 + 54x4

3q2 + 162x23x

24q3 + 81x4

4q4)

+ ε1

5760ω51

(−1008x41x

22q2ω

61 − 108x4

1x22ω

61q3 + 648x4

1x24q2ω

41

+ 243x41x

24ω

41q3 − 2592x3

1x2x3x4q2ω41 + 3888x3

1x2x3x4ω41q3 − 336x2

1x42ω

61q3

− 224x21x

42ω

61q1 − 48x2

1x42ω

61q4 − 1080x2

1x22x

23ω

41q3 − 1512x2

1x22x

24ω

41q3

+ 288x21x

22x

24ω

41q1 + 2430x2

1x23x

24ω

21q3 + 486x2

1x44ω

21q3 + 324x2

1x44ω

21q1

+ 243x21x

44ω

21q4 − 1152x1x

32x3x4ω

41q1 + 1728x1x

32x3x4ω

41q4 − 2592x1x2x

33x4q2ω

21

+ 3888x1x2x33x4ω

21q3 − 2592x1x2x3x

34ω

21q3 − 864x1x2x3x

34ω

21q1

+ 3888x1x2x3x34ω

21q4 + 336x4

2x23ω

41q3 + 224x4

2x23ω

41q1 − 432x4

2x23ω

41q4

+ 1008x22x

43q2ω

21 − 972x2

2x43ω

21q3 + 1512x2

2x23x

24ω

21q3 − 288x2

2x23x

24ω

21q1

− 648x43x

24q2 + 2187x4

3x24q3 − 486x2

3x44q3 − 324x2

3x44q1 + 2187x2

3x44q4), (32)

P = 1

32a1/2(2a2x4

1q2 − 6a2x41q3 + 3a2x4

1q4 + 12ahx21q3 − 12ahx2

1q4 + 4ax21x

23q2

− 12ax21x

23q3 + 6ax2

1x23q4 + 12h2q4 + 12hx2

3q3 − 12hx23q4 + 2x4

3q2

− 6x43q3 + 3x4

3q4)+ ε9

640a5/2(−8a3x6

1q2 + 3a3x61q3 + 3a3x6

1q4 + 4a3x61q1

+ 16a2hx41q2 − 18a2hx4

1q3 − 12a2hx41q4 − 16a2hx4

1q1 − 8a2x41x

23q2

− 27a2x41x

23q3 + 33a2x4

1x23q4 + 4a2x4

1x23q1 + 24ah2x2

1q3 + 12ah2x21q4

+ 16ah2x21q1 + 60ahx2

1x23q3 − 120ahx2

1x23q4 + 8ax2

1x43q2 − 63ax2

1x43q3

+ 57ax21x

43q4 − 4ax2

1x43q1 − 24h2x2

3q3 + 108h2x23q4 − 16h2x2

3q1 − 16hx43q2

+ 78hx43q3 − 108hx4

3q4 + 16hx43q1 + 8x6

3q2 − 33x63q3 + 27x6

3q4 − 4x63q1). (33)

Level curves (or contours) of the functions (P ) given in (29, 31, 33) are invariant curves.Contour plots of the functions (P ) are given on the x1–x3 plane in Figures 1a, 1c, 1e (whereabscissa is x1 and ordinate is x3) for various values of the parameters given in the figurecaption. Poincaré surfaces of section have also been obtained numerically by using a fourth-order Runge–Kutta integrator. They are plotted on the x1x3 plane in Figures 1b, 1d, 1f (whereabscissa is x1 and ordinate is x3) for various values of the parameters given in the figure cap-tion. The points on the surface of section are just the iteration points of the symplectic Poincarémap. It is a discrete dynamical system, and its dimension one less than the dimension of thecorresponding continuous dynamical system. Notice that the analytically obtained invariantcurves agree well with the numerical results.

Fixed points of the Poincaré map correspond to the periodic orbits of the continuousdynamical system (2). Elliptic (hyperbolic) fixed points correspond to the local minima (max-ima) of the functions given in (29, 31, 33). There are five fixed points for ω1 = ω2, threeof them are elliptic and the remaining two are hyperbolic (see Figures 1a and 1b). Ellipticfixed points are surrounded by the islands which correspond to KAM curves (intersection of


Figure 1. Analytical versus numerical results for a = 0.1, and (a), (b) ω1 = ω2, h = 0.00765, ε = 0.1, q4 = 1,q5 = 1, (c), (d) ω1 = 2ω2, h = 0.00765, ε = 0.05, q1 = 1, q2 = 1, q3 = 1, (e), (f) 2ω1 = 3ω2, h = 0.01,ε = 0.089, q1 = 2, q2 = 2, q3 = 0.2, q4 = 0.2.


the KAM tori with the surface x2 = 0). Hyperbolic fixed points are connected to each otherwith separatrix (it is, indeed, a heteroclinic orbit). When ω1 = 2ω2, we have only one ellipticfixed point surrounded by KAM curves (see Figures 1c and 1d). In case of 2ω1 = 3ω2, thereare two elliptic fixed point located inside of the separatrix ( eight figure in Figures 1 e andf). Separatrix is, inded, a homoclinic orbit involving a hyperbolic fixed point. Existence ofthe invariant tori with irrational rotation numbers (nonresonant tori) under small perturbationsis guaranteed by the celebrated KAM theorem [10]. Last but not least, drastic changes inthe form of approximate symmetries given in (A.1–A.3) with resonance relations betwen ω1

and ω2, hence, in the shape of invariant curves given in Figure 1 is just a manifestation ofbifurcation occurring in the Hamiltonian system given in (2).

A. Appendix: Approximate Noether symmetries

First-order approximate Noether symmetry vector field for ω1 = ω2 is given by

X = 1

4ω1(x2

1x3q4ω21 + 2x1x2x4q5ω

21 + x2

2x3q5ω21 + x2

2x3q4ω21 + x3

3q4

+ 3x3x24q5 + x3x

24q4)

∂

∂x1+ 1

4ω1(x2

1x4q5ω21 + x2

1x4q4ω21

+ 2x1x2x3q5ω21 + x2

2x4q4ω21 + 3x2

3x4q5 + x23x4q4 + x3

4q4)∂

∂x2

+ 1

4(−x3

1q4ω31 − 3x1x

22q5ω

31 − x1x

22q4ω

31 − x1x

23q4ω1 − x1x

24q5ω1

− x1x24q4ω1 − 2x2x3x4q5ω1)

∂

∂x3+ 1

4(−3x2

1x2q5ω31 − x2

1x2q4ω31

− 2x1x3x4q5ω1 − x32q4ω

31 − x2x

23q5ω1 − x2x

23q4ω1 − x2x

24q4ω1)

∂

∂x4

+ ε1

32ω31

g

(1

ω21

(−4x31x2x4q5ω

41 + 4x3

1x2x4q4ω41 + 6x2

1x22x3q5ω

41

− 8x21x

22x3q4ω

41 + 6x2

1x3x24q5ω

21 + 8x2

1x3x24q4ω

21 − 4x1x

32x4q5ω

41

+ 4x1x32x4q4ω

41 + 36x1x2x

23x4q5ω

21 + 12x1x2x

23x4q4ω

21 + 12x1x2x

34q5ω

21

+ 4x1x2x34q4ω

21 + 5x4

2x3q5ω41 + 2x4

2x3q4ω41 − 6x2

2x33q5ω

21 + 4x2

2x33q4ω

21

+ 6x22x3x

24q5ω

21 + 8x2

2x3x24q4ω

21 + 18x3

3x24q5 + 12x3

3x24q4 + 9x3x

44q5

+ 6x3x44q4)

∂

∂x1+ 1

ω21

(5x41x4q5ω

41 + 2x4

1x4q4ω41 − 4x3

1x2x3q5ω41

+ 4x31x2x3q4ω

41 + 6x2

1x22x4q5ω

41 − 8x2

1x22x4q4ω

41 + 6x2

1x23x4q5ω

21

+ 8x21x

23x4q4ω

21 − 6x2

1x34q5ω

21 + 4x2

1x34q4ω

21 − 4x1x

32x3q5ω

41 + 4x1x

32x3q4ω

41

+ 12x1x2x33q5ω

21 + 4x1x2x

33q4ω

21 + 36x1x2x3x

24q5ω

21 + 12x1x2x3x

24q4ω

21

+ 6x22x

23x4q5ω

21 + 8x2

2x23x4q4ω

21 + 9x4

3x4q5 + 6x43x4q4 + 18x2

3x34q5


+ 12x23x

34q4)

∂

∂x2+ (30x3

1x22q5ω

41 + 20x3

1x22q4ω

41 − 10x3

1x24q5ω

21 − 4x3

1x24q4ω

21

+ 12x21x2x3x4q5ω

21 − 12x2

1x2x3x4q4ω21 + 15x1x

42q5ω

41 + 10x1x

42q4ω

41

− 6x1x22x

23q5ω

21 + 8x1x

22x

23q4ω

21 − 6x1x

22x

24q5ω

21 + 8x1x

22x

24q4ω

21

− 6x1x23x

24q5 − 8x1x

23x

24q4 + 3x1x

44q5 − 2x1x

44q4 + 4x3

2x3x4q5ω21 − 4x3

2x3x4q4ω21

− 12x2x33x4q5 − 4x2x

33x4q4 − 12x2x3x

34q5 − 4x2x3x

34q4)

∂

∂x3

+ (15x41x2q5ω

41 + 10x4

1x2q4ω41 + 4x3

1x3x4q5ω21 − 4x3

1x3x4q4ω21 + 30x2

1x32q5ω

41

+ 20x21x

32q4ω

41 − 6x2

1x2x23q5ω

21 + 8x2

1x2x23q4ω

21 − 6x2

1x2x24q5ω

21 + 8x2

1x2x24q4ω

21

+ 12x1x22x3x4q5ω

21 − 12x1x

22x3x4q4ω

21 − 12x1x

33x4q5 − 4x1x

33x4q4

− 12x1x3x34q5 − 4x1x3x

34q4 − 10x3

2x23q5ω

21 − 4x3

2x23q4ω

21 + 3x2x

43q5

− 2x2x43q4 − 6x2x

23x

24q5 − 8x2x

23x

24q4)

∂

∂x4g

). (A.1)

First-order approximate Noether symmetry vector field for ω1 = 2ω2 is given by

X = 1

16ω1(x2

2x3q1ω21 + 4x3x

24q1)

∂

∂x1+ 1

8ω1(4x2

1x4q2ω21 + x2

2x4q3ω21

+ 4x23x4q2 + 4x3

4q3)∂

∂x2+ 1

16(−x1x

22q1ω

31 − 4x1x

24q1ω1)

∂

∂x3

+ 1

32(−4x2

1x2q2ω31 − x3

2q3ω31 − 4x2x

23q2ω1 − 4x2x

24q3ω1)

∂

∂x4

+ εg

(1

96ω51

(64x31x2x4q2ω

41 − 8x2

1x22x3q1ω

41 − 32x2

1x22x3q2ω

41

+ 32x21x3x

24q1ω

21 + 128x2

1x3x24q2ω

21 − 8x1x

32x4q1ω

41 + 16x1x

32x4q3ω

41

+ 64x1x2x23x4q1ω

21 + 64x1x2x

23x4q2ω

21 − 32x1x2x

34q1ω

21 + 64x1x2x

34q3ω

21

+ 7x42x3q1ω

41 − 8x4

2x3q3ω41 − 16x2

2x33q1ω

21 − 32x2

2x33q2ω

21 + 24x2

2x3x24q1ω

21

+ 64x33x

24q1 + 128x3

3x24q2 − 16x3x

44q1 + 128x3x

44q3)

∂

∂x1

+ 1

24ω51

(16x41x4q2ω

41 + 16x3

1x2x3q2ω41 + x2

1x22x4q1ω

41

− 14x21x

22x4q2ω

41 + 48x2

1x23x4q2ω

21 + 4x2

1x34q1ω

21 + 8x2

1x34q2ω

21

+ 32x21x

34q3ω

21 − 2x1x

32x3q1ω

41 + 4x1x

32x3q3ω

41 + 16x1x2x

33q2ω

21

− 8x1x2x3x24q1ω

21 − 32x1x2x3x

24q2ω

21 + 48x1x2x3x

24q3ω

21 − x2

2x23x4q1ω

21

+ 14x22x

23x4q2ω

21 + 32x4

3x4q2 − 4x23x

34q1 − 8x2

3x34q2 + 64x2

3x34q3)

∂

∂x2


+ 1

96ω31

(8x31x

22q1ω

41 + 16x3

1x22q2ω

41 − 32x3

1x24q1ω

21 − 64x3

1x24q2ω

21

− 64x21x2x3x4q1ω

21 − 64x2

1x2x3x4q2ω21 + 7x1x

42q1ω

41 + 4x1x

42q3ω

41

+ 16x1x22x

23q1ω

21 + 16x1x

22x

23q2ω

21 + 24x1x

22x

24q1ω

21 − 64x1x

23x

24q1

− 64x1x23x

24q2 − 16x1x

44q1 − 64x1x

44q3 + 8x3

2x3x4q1ω21 − 16x3

2x3x4q3ω21

− 64x2x33x4q2 + 32x2x3x

34q1 − 64x2x3x

34q3)

∂

∂x3+ 1

96ω31

(16x41x2q2ω

41

− 64x31x3x4q2ω

21 + 7x2

1x32q1ω

41 + 14x2

1x32q2ω

41 + 8x2

1x32q3ω

41

+ 48x21x2x

23q2ω

21 + 28x2

1x2x24q1ω

21 − 8x2

1x2x24q2ω

21 + 8x1x

22x3x4q1ω

21

+ 32x1x22x3x4q2ω

21 − 48x1x

22x3x4q3ω

21 − 64x1x

33x4q2 + 32x1x3x

34q1

− 64x1x3x34q3 − 7x3

2x23q1ω

21 − 14x3

2x23q2ω

21 + 16x3

2x23q3ω

21 + 32x2x

43q2

− 28x2x23x

24q1 + 8x2x

23x

24q2)

∂

∂x4g

). (A.2)

First-order approximate Noether symmetry vector field for 2ω1 = 3ω2 is given by

X = 1

36ω1(9x2

1x3q2ω21 + 4x2

2x3q1ω21 + 9x3

3q2 + 9x3x24q1)

∂

∂x1

+ 1

24ω1(9x2

1x4q3ω21 + 4x2

2x4q4ω21 + 9x2

3x4q3 + 9x34q4)

∂

∂x2

+ 1

36(−9x3

1q2ω31 − 4x1x

22q1ω

31 − 9x1x

23q2ω1 − 9x1x

24q1ω1)

∂

∂x3

+ 1

54(−9x2

1x2q3ω31 − 4x3

2q4ω31 − 9x2x

23q3ω1 − 9x2x

24q4ω1)

∂

∂x4

+ εg

(1

2880ω51

(−1296x31x2x4q2ω

41 + 1944x3

1x2x4q3ω41 − 72x2

1x22x3q1ω

41

− 972x21x

22x3q3ω

41 + 162x2

1x3x24q1ω

21 + 2187x2

1x3x24q3ω

21 − 576x1x

32x4q1ω

41

+ 864x1x32x4q4ω

41 + 2592x1x2x

23x4q1ω

21 − 3888x1x2x

23x4q2ω

21

+ 1944x1x2x23x4q3ω

21 − 1296x1x2x

34q1ω

21 + 1944x1x2x

34q4ω

21 + 448x4

2x3q1ω41

− 432x42x3q4ω

41 − 648x2

2x33q1ω

21 + 2016x2

2x33q2ω

21 − 972x2

2x33q3ω

21

+ 720x22x3x

24q1ω

21 + 1458x3

3x24q1 − 1296x3

3x24q2 + 2187x3

3x24q3 − 648x3x

44q1

+ 2187x3x44q4)

∂

∂x1+ 1

320ω51

(72x41x4q2ω

41 + 27x4

1x4q3ω41 − 144x3

1x2x3q2ω41

+ 216x31x2x3q3ω

41 + 32x2

1x22x4q1ω

41 − 168x2

1x22x4q3ω

41 + 270x2

1x23x4q3ω

21

+ 72x21x

34q1ω

21 + 108x2

1x34q3ω

21 + 54x2

1x34q4ω

21 − 64x1x

32x3q1ω

41

+ 96x1x32x3q4ω

41 − 144x1x2x

33q2ω

21 + 216x1x2x

33q3ω

21 − 144x1x2x3x

24q1ω

21


− 432x1x2x3x24q3ω

21 + 648x1x2x3x

24q4ω

21 − 32x2

2x23x4q1ω

21 + 168x2

2x23x4q3ω

21

− 72x43x4q2 + 243x4

3x4q3 − 72x23x

34q1 − 108x2

3x34q3 + 486x2

3x34q4)

∂

∂x2

+ 1

2880ω31

(72x31x

22q1ω

41 + 2016x3

1x22q2ω

41 + 108x3

1x22q3ω

41

− 162x31x

24q1ω

21 − 1296x3

1x24q2ω

21 − 243x3

1x24q3ω

21 − 2592x2

1x2x3x4q1ω21

+ 3888x21x2x3x4q2ω

21 − 1944x2

1x2x3x4q3ω21 + 448x1x

42q1ω

41 + 48x1x

42q4ω

41

+ 648x1x22x

23q1ω

21 + 108x1x

22x

23q3ω

21 + 720x1x

22x

24q1ω

21 − 1458x1x

23x

24q1

− 243x1x23x

24q3 − 648x1x

44q1 − 243x1x

44q4 + 576x3

2x3x4q1ω21 − 864x3

2x3x4q4ω21

+ 1296x2x33x4q2 − 1944x2x

33x4q3 + 1296x2x3x

34q1 − 1944x2x3x

34q4)

∂

∂x3

+ 1

720ω31

(252x41x2q2ω

41 + 27x4

1x2q3ω41 + 324x3

1x3x4q2ω21 − 486x3

1x3x4q3ω21

+ 112x21x

32q1ω

41 + 168x2

1x32q3ω

41 + 24x2

1x32q4ω

41 + 270x2

1x2x23q3ω

21

+ 252x21x2x

24q1ω

21 − 108x2

1x2x24q3ω

21 + 144x1x

22x3x4q1ω

21 + 432x1x

22x3x4q3ω

21

− 648x1x22x3x4q4ω

21 + 324x1x

33x4q2 − 486x1x

33x4q3 + 324x1x3x

34q1

− 486x1x3x34q4 − 112x3

2x23q1ω

21 − 168x3

2x23q3ω

21 + 216x3

2x23q4ω

21

− 252x2x43q2 + 243x2x

43q3 − 252x2x

23x

24q1 + 108x2x

23x

24q3)

∂

∂x4g

). (A.3)

References

1. Siopis, C. V., Kandrup, H. E., Contopoulos, G., and Dvorak, R., ‘Universal properties of escape’, Annals ofthe New York Academy of Sciences 773, 1995, 221–230.

2. Contopoulos, G., ‘On the existence of a third integral of motion’, The Astronomical Journal 68, 1962, 1–14.3. Henon, M. and Heiles, C., ‘The applicability of the third integral of motion, some numerical experiments’,

The Astronomical Journal 69, 1964, 73–79.4. Lichtenberg, A. J. and Lieberman, M. A., Regular and Stochastic Motion, Springer-Verlag, New York, 1992.5. Ünal, G., ‘Approximate generalized symmetries, normal forms and approximate first integrals’, Physics

Letters A 269, 2000, 13–30.6. Baikov, V. A., Gazizov, R. K., and Ibragimov N. H., ‘Approximate symmetries’, Mathematics USSR Sbornik

64, 1989, 427–441.7. Contopoulos, G., Voglis, N., Efthyiopoulos, C., and Grousouzakou, ‘Invariant spectra of dynamical systems’,

Annals of the New York Academy of Sciences 773, 1995, 145–167.8. Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, New York,

1999.9. Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics, Nauka, Moscow, 1985 [English

translation published by D. Reidel, Dordrecht].10. Arnold, V. I., Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.

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