EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6341
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

An Exploration of the Topological Structure and
Bifurcation of Liouville Tori in Models of Galactic

Motion

T. S. Amer1,∗, F. M. El-Sabaa2, M. Fakharany1,3, H. M. Gad2

1 Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
2 Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt
3 Mathematics and Statistics Department, College of Science, Taibah University, Yanbu,
Saudi Arabia

Abstract. This paper examines two integrable cases: a generalized Hénon-Heiles (HH) system
and a quartic potential. For each case, the Liouville tori’s bifurcation (LTB) is analyzed. Periodic
solutions (PS) are derived using Jacobi elliptic functions, and the corresponding phase portraits
are presented with a classification of the singular points. Furthermore, the PS for both cases is
constructed based on the Lyapunov theorem. The possible applications of this study are primarily
confined to celestial mechanics and astrodynamics. Specifically, the generalization of the HH system
and quartic potentials often appears in modeling gravitational interactions between celestial bodies,
including the study of the stability and motion of planets, asteroids, and satellites. Additionally,
the classification of singular points and phase portraits provides valuable insights for identifying
chaotic or regular behaviors in various systems, such as weather models or economic systems.
Furthermore, understanding bifurcations and PS contributes to the design of control mechanisms
for nonlinear systems, including robotics and automated processes.

2020 Mathematics Subject Classifications: 37J35, 70H06, 37G10, 70F15

Key Words and Phrases: Hamilton-Jacobi Equations, Lyapunov’s method, Level sets’topology,
Periodic solution, Phase portrait

1. Introduction

In the current paper, we will discuss the two models of galactic motion to get a com-
plete description of these motions in view of studying the bifurcation and the topology of
invariant level sets (LS) of the separated functions namely, the third case of the generalized
HH and the non separated case of the quartic potential function (PF).

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6341

Email addresses: tarek.saleh@science.tanta.edu.eg (T. S. Amer),
fawzyfahmy@edu.asu.edu.eg (F. M. El-Sabaa), mohamed.elfakharany@science.tanta.edu.eg,
fakharany@aucegypt.edu (M. Fakharany), hadeermahmoud@edu.asu.edu.eg (H. M. Gad)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 2 of 22

The axisymmetric PF of the movement of a star around the galactic center in the
system of HH takes the following form

V =
1

2
r2 − 1

3
r3 cos 3θ.

This function is specified by 3-D cylindrical coordinates, where one of them is cyclic, and
the problem is reduced to the plane motion. The Hamiltonian function associated with
the potential V is

H =
1

2

(
P 2
X + P 2

Y +X2 + Y 2
)
+XY 2 − 1

3
X3.

This system can be integrated numerically for a fixed value of total energy, where
there exists an invariant torus around the PS of the system, while the motion exhibits
large stochasticity as the constant of energy increases. Hénon and Heiles [1] claim that
there is no change with the addition of potential higher-order terms, while Contopoulos
and Polymilis [2] demonstrated that the HH potential is considered as Toda’s lattice third-
order term that can be integrable in a 2-D system [3]. The generalized HH can be taken
the form

H =
1

2

(
P 2
X + P 2

Y + c1X
2 + c2Y

2
)
+ aXY 2 − b

3
X3. (1)

A large number of articles have been studied with the generalized HH. In [4], the second-
order averaging method with a small parameter is utilized to investigate the existence of
PS families under specific parameter conditions, characterizing the stability of periodic
orbits. Melnikov’s method, as applied in [5], confirms the non-integrability of neighboring
HH systems and determines the width of the principal chaotic layer responsible for this
behavior. In [6], second-order averaging theory is applied to study periodic orbits in
generalized HH systems. Furthermore, in quantum mechanics, the HH model serves as a
representation of atomic oscillators in triatomic molecules. In [7], the nonlinear HH system
is analyzed to study the wave packet behavior of bound states and the connection between
quantum and classical dynamics. Meanwhile, [8] investigates the non-uniform motion of
charged particles in a crystal under the HH potential. The other model of study in this
article is the quartic PF, which can be taken in the form

V = αx4 + δx2y2 + βy4. (2)

By adding two coupled quadratic harmonic oscillators ωx2 + σy2 to the function V , then
we have a model which was studied in several fields in physics: in quantum mechanics and
quantum field theory, see [9], in chemical physics [10] and in scalar field theory as presented
in [11]. In galactic motion, the model has been studied by many authors. Pucacco et al.[12]
studied the galactic potential (GP) V (x2, y2) which is the modeling of elliptical galaxies.
The Armbruster- Guckenheimer-Kim (AGK) potential [13] is another model that consists
of a 2-D harmonic potential with quartic terms. Friedmann-Roberston-Walker (FRW)
potential was introduced in [14], a model of GP, which modulates containing a positive or



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 3 of 22

negative gravitational coupled scalar field in the loop quantum cosmology scenario.

By using Painlevé property [15, 16], the generalized HH system and quartic potential
can be integrated for the special values of constants as follows:

In generalized HH:
(i) b/a = −1, c1 = c2, (ii) b/a = −6, c1, c2 arbitrary, (iii) b/a = −16, c1 = 16c2.
For quartic potential:
(i) α = β, δ = 6β, (ii) α = β, δ = 2β, (iii) β = 16α, δ = 12α, (iv) β = 8α, δ = 6α,
where there exists an invariant torus around the PS of the system. It is well known that
the first two cases of generalized HH and the first three cases of quartic potential have
been separated, and the second invariant integrals was given in quadratic in momenta see
for instance [15–20]. There are some problems in which the conditions for integrability
have been defined, but separation has not been achieved. The last cases of generalized HH
and quartic potential are belonging to this group, where the invariant integral for both
cases is quartic in momenta

IGHH = P 4
Y + 2Y 2(c2 + 2aX)P 2

Y + c22Y
4 − 4

3a
((
PXPY + a

6Y
3
)
Y 3 + (c2 + aX)XY 4

)
,

Iquad = p4x + 4x2(x2 + 6y2)p2x + 4x4p2x − 16x3ypxpy + 4x4(x4 + 4x2y2 + 4y4).

If there is a second independent invariant integral, then the 2-DOF system is integrable.
The second invariant can be given by different methods: Whittaker [21] considered the
invariant integral which is quadratic in momenta, and he studied some special cases to get
the integral. Dorizzi et al. [22] applied the Whittaker-Daraboux equation, and by rotation
of the coordinates, they introduced a partial differential equation in the form

2xVxy + 3Vy + y(Vyy − vxx) = 0, (3)

with the Hamiltonian function H = 1
2(p

2
x+p2y)+V (x, y), and they proved that there exists

a second invariant of motion which is quadratic in momenta. The solutions of the equation
(3) are given in several classes of the homogenous PF Vs(x, y):

V0 = 1, V1 = 2x, V2 = 4x2 + y2,
V3 = 8x3 + 4y2x, V4 = 16x4 + 12x2y2 + y4.
etc..

Nakagawa and Yoshida [23] considered a homogeneous PF

Vs =
1

r

((
r + x

2

)s+1

+ (−1)s
(
r − x

2

)s+1
)
, r2 = x2 + y2,

is separable in parabolic coordinates τ = r+x
2 , σ = r−x

2 , and the invariant integral Is is
given by

Is = py(ypx − xpy) +
1

2
y2Vs−1,



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 4 of 22

where the explicit functions

V2 =
1

4
(4x2 + y2), V3 =

1

8
(8x3 + 4xy2), V4 =

1

16
(16x4 + 12x2y2 + y4).

It is clear that both mentioned methods can not give the separability of the last two cases
of generalized HH and quartic potential. Verhoeven et al. [24] used the general solution
for the Kaup-Kupershmidt (KK) to get a separation of the variables for the third case of
generalized HH, while Ravoson et al. [25] used the method described by Vanhaecke [26]
to get the separability of the last case of the quartic potential.
The primary objective of this research article is to comprehensively elucidate the problem
under investigation through an examination of the bifurcation and the topology of invariant
LS of the separated functions employing Fomenko’s theory [27]. In [28], the Fomenko
classification is applied to the two fixed centers problem, while in [29], the bifurcation
behavior of the AGK GP is investigated. Waalkens et al. [30] investigated the topology of
energy surface bifurcations during three-dimensional motion. The bifurcation of a common
level set was studied in [31] for the first integral of Kovalevskaya problem, while the LTB
of the problem of the rigid body (RB) rotation around a fixed point in different cases
is obtained in [32]. In [33], the author investigated the RB motion while it is affected
by restoring, perturbing, and gyrostatic moments. In [34], it is expected that the gyro’s
direction is near the axis of dynamic symmetry, its angular velocity is sufficiently large, and
its perturbing moments are minimal in relation to its restoring ones. These circumstances
allow for the introduction of a tiny parameter. In the first and second approximations,
averaged systems of the equations of motion are found. In [35], the authors used Taylor
and the average methods to solve the problem while the body contained a cavity filled
with fluid and impacted by gyrostatic and constant torques.
The current paper is constructed as outlined below: In section 2, the separation of the
third case of generalized HH problem is introduced, and the topology of invariant LS of the
separated functions is discussed in subsection 2.1 using Fomenko’s classification theorem
[27]. Subsection 2.2 investigates the PS for singular LS of bifurcation. In subsection
2.3, the phase portrait of the separation functions is introduced, and the singular points
and their types are obtained. In section 3, the separation for the last case of quartic
potential (2) is examined, and the topology of real phase space and the sets of bifurcation
are investigated in subsection 3.1. The PS is examined in subsection 3.2. The separated
function is used in subsection 3.3 to examine the system’s phase portrait and discover the
various forms of singular points. In section 4, we obtain the PS about the equilibrium
points.

This study finds applications in celestial mechanics and astrodynamics, where gen-
eralized HH-systems and quartic potentials are used to model gravitational interactions
among celestial bodies. These models help analyze the stability and motion of planets,
asteroids, and satellites. Moreover, the classification of singular points and phase portraits
aids in identifying chaotic or regular behaviors in complex systems like weather models
and economic frameworks. The insights into bifurcations and PS further enable the design
of control mechanisms for nonlinear systems such as robotics and automation.



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 5 of 22

2. Separation of the third case of generalized HH

Hamiltonian function for generalized HH has the form

H =
1

2

(
P 2
X + P 2

Y + c1X
2 + c2Y

2
)
+ aXY 2 − b

3
X3.

It is shown in [36] that the three cases of generalized HH can be related to the Sawada-
Kotera (SK), KdV5, and equations of KK, respectively. Verhoeven et al. [24] used the KK
equations to get the separate functions of the Hamiltonian for the third case of generalized
HH, which is based on the following values of parameters

a =
1

4
, c1 = 16c2, c = c1c2,

with the transformation u = X + 2c2, and v = Y .
After performing the mathematical operations, they obtained the Hamiltonian

H =
1

2
(p2u + p2v) +

1

4
uv2 +

4

3
u3 − cu. (4)

Finally, the following transformation is used to get the separable Hamiltonian function

q1 = −6
p2v−k2,0

v2
− u,

q2 = −6
p2v+k2,0

v2
− u

p1 =
1

2v3

(
12p3v + 6uv2pv − v3pu − 12pvk2,0

)
,

p2 =
1

2v3

(
12p3v + 6uv2pv − v3pu + 12pvk2,0

)
.

Therefore the Hamiltonian system (4) takes the form

H = p21 + p22 +
1

12
(q31 + q32)−

c

4
(q1 + q2). (5)

Using Hamilton Jacobi equations

h+

(
∂W

∂q1

)2

+

(
∂W

∂q2

)2

+
1

12
(q31 + q32)−

c

4
(q1 + q2) = 0, (6)

where W is a complete integral and it has the form

W = W1(q1) +W2(q2). (7)

Thus, equation (6) can be introduced as

2
(
dW1
dq1

)2
+

q31
6 − c

2q1 + h = α,

2
(
dW2
dq2

)2
+

q32
6 − c

2q2 + h = −α,
(8)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 6 of 22

and W can be obtained in the following form

W =

∫ √
ϕ(z)dz, (9)

where

ϕ(z) = ±2α− 2h− z3

3
+ cz, z = z(q1, q2). (10)

Thus, the differential equation can be written as

dz√
ϕ(z)

= t− t0. (11)

2.1. Topological analysis

The Fomenko categorization is used for identifying all generic LTB in this section.
Thus, the definitions below are considered [37].

(i) If there is a simple manifold system M2n with F : M2n → R2n, and f1, f2, . . . , fn
as independent integrals, the function F (x) = (f1, f2, . . . , fn) is referred to as mo-
mentum mapping.

(ii) If the rank satisfies dF (x) < n, and F (x) ∈ Rn, then x ∈ M is recognized as a
critical point of the momentum mapping.

(iii) The bifurcation graph is the set B = F (k) ⊂ Rn formed by the union of many parts
Bk.

(iv) The set of all critical points associated with the momentum mapping Z is contained
within the set M .

In the present problem the LS has the topology

LS = {(q1, q2, q̇1, q̇2) ∈ R4 : H = h, F = α ⊂ R4},

where LS is formed by a finite union of 2-D tori corresponding to non-critical values of F
and H. The set of critical points allows for the determination of the energy-momentum
diagram.

(q1, q2, q̇1, q̇2) → (H,F ),

where D represents the discriminant region of the functions ϕ(q1) and ϕ(q2), where their
coefficients are given in terms of h and α

D = D1 ∪D2 = [(h, α) ∈ R2/disc(ϕ(q1)) = 0] ∪ [(h, α) ∈ R2/disc(ϕ(q2)) = 0]. (12)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 7 of 22

The point (h, α) that passes through D can modify the topological kind of LS . The
set R2/D comprised of 16 connected parts as depicted in Figure1. As a result, in every
connected part of R2/D, the level set LS contains the preceding one type. When H and
F are noncritical, the level set LS forms a finite union of lower-dimensional tori [38].
The number of acceptable ovals on the Riemann surface associated to elliptical curves Γj

determines their numbers, where

Γj : ωj =
√
ϕ(qj ), j = 1, 2.

Table1 describes the roots of both ϕ(q1) and ϕ(q2), whereas Table2 illustrates that LS is
either a torus or a set of non-elements [39]].

Figure 1: Diagram of bifurcation D = D1 ∪D2.

Figure 2: The LTB.



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 8 of 22

Table 1: Topological class of LS and the real roots of ϕ(q1) and ϕ(q2) for (h, α) ∈ R2/D.

Domain Roots of ϕ(q1) Roots of ϕ(q2)

1 0 0
2 0 0
3 a > 0 0
4 a > 0 0
5 a > 0 0
6 a > 0 b > 0
7 a > 0 b > 0
8 0 b > 0
9 0 b > 0
10 0 b > 0
11 0 0
12 0 0
13 0 0
14 0 0
15 0 0
16 0 0

Figure 3: The bifurcations of invariant Liouville tori are linked to the bifurcations of the polynomial roots ϕ(q1)
and ϕ(q2).

2.2. Periodic solution

According to the preceding section, we have PS on the specified curve denoted by c1
where 3(h + α) = −1. This observation is supported by the data presented in table 4,
where the q1 parameter taking value in interval [0, a] and q2 = 2. Solving equation (11),
the function ϕ(z) is a third-degree polynomial that has a real root a and two roots d and
d̄ which are complex

ϕ(z) = −z3

3
+ cz + n = (a− z)(z − d)(z − d̄), (13)

and, n = 2(α− h).
Using the following relations

(z − d)(z − d̄) = (x− d1)
2 + a21, (14)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 9 of 22

Table 2: Allowable ovals on diagram D.

Domain q1 − plane∆1 q2 − plane∆2 Topological type

1 ∅ ∅ ∅
2 [0, a] ∅ ∅
3 [0, a] ∅ ∅
4 [0, a] ∅ ∅
5 [0, a] ∅ ∅
6 [0, a] [0, b] T
7 [0, a] [0, b] T
8 ∅ [0, b] ∅
9 ∅ [0, b] ∅
10 ∅ [0, b] ∅
11 ∅ ∅ ∅
12 ∅ ∅ ∅
13 ∅ ∅ ∅
14 ∅ ∅ ∅
15 ∅ ∅ ∅
16 ∅ ∅ ∅

Table 3: The generic bifurcations of LS between domains i and ℓ, where i = 6, 7 and ℓ = (1 : 5, 8 : 16).

6 → 7 6 → ℓ 7 → ℓ

T → T T → ∅ T → ∅

where

d1 =
d+ d̄

2
, a21 =

−(d− d̄)2

4
, (d1 − a)2 + a21 = A2.

Then, ϕ(z) can be rewritten as

ϕ(z) = (a− z)
(
(z − d1)

2 + a21
)
. (15)

Taking into account that we can consider that

z = a−A
1 + cosφ

1− cosφ
; −∞ < z ≤ a. (16)

Referring to equations (15) and (16), then equation (11) becomes∫ φ

0

dφ√
1− k21 sin

2 φ
=

√
A(t− t0). (17)

Table 4: The topological type of LS on diagram D corresponds to the two no-coplanar circles, denoted as
(S ∧ S), which intersect at a single shared point.

Curve ∆1 ∆2 LS = ∆1 ×∆2

c1 [0, a] [0, b] S
c2 [0, a] [0, b] S × (S ∧ S)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 10 of 22

Therefore, equation (11) has the following solution

z = a−A
1 + cn[

√
A(t− t0), k1]

1− cn[
√
A(t− t0), k1]

, (18)

where cosφ = cn(
√
A(t− t0)), with the period T

T =
1√
A
sn−1(1, k1) =

1√
A
K(k1), K(k1) =

∫ π/2

0

dφ√
1− k21 sin

2 φ
,

where K(k1) denotes the complete elliptic integral of the first kind.

2.3. The phase portrait after separation

The phase portrait is a successful tool to get the type of singularity for separated
functions. The singularity of the rotation of a RB revolving around a fixed point in the
Kovalevskaya case is studied in [40], while in [41] the singular points in the problem of
two fixed centers are classified. It is defined that the type of singular point is elliptic,
hyperbolic, or parabolic according to the Jacobian. We consider the function

F = p2 +
q3

6
− c

2
q + h− α. (19)

The singular points have been obtained from the following equations,

∂F

∂p
= 2p = 0,

∂F

∂q
=

q2

2
− c

2
= 0. (20)

Then, from equation (20) there exists two singular points (0,±
√
c).

In order to determine the type of these points, put p = x and q = y ±
√
c in equation

(19) and ignoring terms of degree greater than two, thus the point (0,
√
c) is elliptic point

where the Jacobian J = FxxFyy − FxyFyx > 0, and the point (0,−
√
c) is hyperbolic point

where the Jacobian J = FxxFyy − FxyFyx < 0 as shown in Figure 4.

3. Separation of the last case of quartic potential

The Hamiltonian function associated with the quartic potential in the last case is given
by

H =
1

2

(
p2x + p2y

)
+ α(x4 + 6x2y2 + 8y4), (21)

for simplicity, we can consider that α = 1. In [28], the authors used the transformation

u =
p2x + c

x2
+ 2x2 + 4y2, v =

p2x − c

x2
+ 2x2 + 4y2, (22)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 11 of 22

Figure 4: The elliptic point (0,
√
c) and the hyperbolic point (0,−

√
c).

where

x2 =
2c

u− v
, y2 =

u+ v

8
− c

u− v
− (v̇ − u̇)2

16(u− v)2
.

Hence the separation equations are written in the form

u̇2 = 2u3 − 8u(2h+ C), v̇2 = 2v3 − 8v(2h− C), (23)

where h and C represent the energy integral and second invariant integral respectively∫
du√
P1(u)

=

∫
dt,

∫
dv√
P2(v)

=

∫
dt, (24)

where
P1(u) = 2u3 − 8u(2h+ C), P2(v) = 2v3 − 8v(2h− C). (25)

3.1. Bifurcation of the problem

Following the method described in subsection 2.1, the LS of this problem possesses the
topology

AS = {(u, v, u̇, v̇) ∈ R4 : H = h, F = C ⊂ R4}. (26)

For non-critical values of both F and H, AS consists of a limited union of 2-D tori.
The calculation of the energy momentum diagram involves the utilization of the set of

critical points (u, v, u̇, v̇) → (H,F ).
If G is the discriminant locus of the polynomials P1(u) and P2(v), with coefficients

depending on h and C, then

G = G1 ∪G2 = [(h,C) ∈ R2/disc(P1(u)) = 0] ∪ [(h,C) ∈ R2/disc(P2(v)) = 0]. (27)

The set R2/G is comprised of of eight connected components, as shown in Figure 5.
Table 5 indicates the real roots of P1(u) and P2(v) aiming to obtain the ovals (a related
component for the set of fixed points on the curve Γj , (j = 1, 2)); where Γ1 : ω1 =

√
P1(u)

and Γ2 : ω2 =
√

P2(v), while Table 6 shows the topological kind of AS is a torus, or it



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 12 of 22

Table 5: Topological type of AS and real roots of the polynomials P1(u) and P2(v) for (h,C) ∈ R2/G.

Domain Roots of P1(u) Roots of P2(v)

1 u1 < 0 < u3 v1 < 0 < v3
2 u1 < 0 < u3 v2 = 0
3 u1 < 0 < u3 v2 = 0
4 u2 = 0 v2 = 0
5 u2 = 0 v2 = 0
6 u2 = 0 v1 < 0 < v3
7 u2 = 0 v1 < 0 < v3
8 u1 < 0 < u3 v1 < 0 < v3

might be an empty set. The topological kind is a tours as in Dm (m = 1, 8) or empty as
in Dm∗ (m∗ = 2, 3, 4, 5, 6, 7). To analyze all generic LTB of the issue as demonstrated in
Table 7, it is sufficient to examine the bifurcations of P1(u) and P2(v), the correspondence
between invariant LTB and bifurcations of roots of them, see figures 6 and 7.

Figure 5: Diagram of bifurcation G = G1 ∪G2.

3.2. Periodic solution

From the previous section, the PS lies on the curves c1 where C = 2h as shown in
Table 8, and the u3 parameter taking value in interval [0, u3] and v1 = 0. According to
the initial equation of (25), the function P1(u) can be characterized as a cubic polynomial
whose real roots u1, 0 and u3. Here

P1(u) = u(u− u1)(u− u3); u3 ≤ u, (28)

u =
u3

1− sin2 φ
, (29)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 13 of 22

Table 6: The allowable ovals and the topological classification of AS for (h,C) ∈ R2/G.

Domain u− plane ∆1 v − plane ∆2 Topological type

1 [0, u3] [v1, 0] T
2 [0, u3] ∅ ∅
3 [0, u3] ∅ ∅
4 ∅ ∅ ∅
5 ∅ ∅ ∅
6 ∅ [v1, 0] ∅
7 ∅ [v1, 0] ∅
8 [0, u3] [v1, 0] T

Table 7: The generic bifurcations of AS between domains i and j, where i = 1, 8 and j = (2 : 7).

T → T T → ∅ T → ∅

and

du =
2u3 sinφ cosφ

(1− sin2 φ)2
dφ. (30)

Substituting equations (28)-(30) into the initial equation of (25) to obtain the solution

u(t) =
u3

1− sn2
[
n
2 (t− t0), k

] , (31)

with period T

T =
2

n
sn−1(1, k) =

2

n
K(k),

where

K(k) =

∫ π/2

0

dφ√
1− k2 sin2 φ

, n =
√
u3 − u1, k2 =

u1
u1 − u3

.

Here, K(k) represents a complete elliptic integral of the first kind.

3.3. The phase portrait of the separated function

Let the function
F = p2 − 2q3 + 8q(2h+ C). (32)

The singular points of F can be determined by evaluating the solutions of the subsequent
equations

∂F

∂p
= 2p = 0,

∂F

∂q
= −6q2 + 8(2h+ C) = 0. (33)

Table 8: Topological kind of AS on G.

Curve ∆1 ∆2 LS = ∆1 ×∆2

c1 [0, u3] [v1 = 0] S
c2 [0, u3] [v1, 0] S × (S ∧ S)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 14 of 22

Figure 6: The LTB.

Figure 7: The relation between invariant LTB and bifurcations of roots of P1(u) and P2(v).

From equation (33) we have

p = 0, or q = ±
√

8(2h+ C)

6
.

Then, the two singular points

(
0,

√
8(2h+C)

6

)
,

(
0,−

√
8(2h+C)

6

)
.

To identify the types of points, we employ the methodology outlined in subsection 2.3,
yielding the subsequent findings

(i) The point

(
0,

√
8(2h+C)

6

)
is hyperbolic, where J = GxxGyy −GxyGyx < 0,

(ii) The point

(
0,−

√
8(2h+C)

6

)
is elliptic, where J = GxxGyy −GxyGyx > 0, as plotted

in Figure 8.

4. Periodic solutions by applying Lyapunov’s method

The PS close to the equilibrium positions of the systems (1) and (21) are introduced by
using Lyapunov’s method [42]. Many scientific articles study this method. The problem



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 15 of 22

Figure 8: The elliptic point

(
0,−

√
8(2h+C)

6

)
and the hyperbolic point

(
0,
√

8(2h+C)
6

)
.

of rotating RB around a fixed point is studied in [43–56]. In [44], a new approach to solve
the RB problem with constant body torques. In [45], the authors decoupled the Euler
equations and solved them with gyrostatic impact. In [46], the problem is solved with
time-varying constant body torques. In [51, 52], the author obtained the PS with the
energy dissipation that influences the motion.

For the case of generalized HH:
The governing equations for the Hamiltonian (1) governing by the PF V are given by

Ẍ + Vx = 0, Ÿ + Vy = 0, (34)

with the integral related with energy

1

2

(
Ẋ2 + Ẏ 2

)
+ V = h. (35)

The equilibrium points (Xp, Yp) can be obtained by solving the equations Vx = Vy = 0,
such that for each point the equation (35) gives the initial value of the energy constant
h. Inserting X = Xp + ξ, Y = Ypη in equation (34) to investigate the disturbance motion
around the equilibrium points. After some adjustments, the following differential equations
may be obtained

ξ̈ + Lξ +Mη = 0, η̈ +Nη +Mξ = 0, (36)

where VXX(Xp, Yp) = L, VY Y (Xp, Yp) = N, VXY (Xp, Yp) = M . Introducing the time
transformation

u = νt, (37)

then, the equations (36) become

ν2
d2ξ

du2
+ Lξ +Mη = 0, ν2

d2η

du2
+Nη +Mξ = 0. (38)

We consider the solution of the system (38) as

ξ =

∞∑
j=1

εjX(j), η =

∞∑
j=1

εjY (j), h = h0 +

∞∑
j=2

εjh(j), (39)



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 16 of 22

where h(j) represent constants and X(j), Y (j) express about T -periodic functions of t with
the frequency ν that may be formulated as

T =
2π

ν
=

2π

ν0

1 +

∞∑
j=2

εjTj

 . (40)

Substituting (39) and (40) in (38) and taking the first approximation, then we get

ν20
d2X(1)

du2
+ LX(1) +MY (1) = 0, ν20

d2Y (1)

du2
+NY (1) +MX(1) = 0. (41)

In order to accomplish our objective, we express the supposed solution for equations (41)
in the Fourier series

X(s) = a01s +

s∑
r=1

(a
(r)
1s cos ru+ b

(r)
1s sin ru),

Y (s) = a02s +
s∑

r=1

(a
(r)
2s cos ru+ b

(r)
2s sin ru),

(42)

by taking first approximation of expressions (42) and substituting into (41), then we get
the following solutions

X = Xp + ε[b̃(ν20 −N) cosu+ ãM sinu],

Y = Yp + ε[b̃M cosu+ ã(ν20 − L) sinu],

h = h0 +
ε2

4

(
(ã2 + b̃2)ν60 + (ã2(N − 2L) + b̃2(α− 2β))ν40+

(ã2(L2 − 2LN + 3M2) + b̃2(N2 − 2LN + 3M2))ν20 + (ã2L+ b̃2N)(LN −M2)
)
,

(43)

where ã and b̃ are unrestricted variables. The system’s frequency is determined by

ν0 =

√
(L+N)±

√
(L+N)2 − 4(LN −M2)

2
. (44)

Taking into account that, the frequency’s value ν0 relies on the values of L, M and N .
So, let’s discuss this scenario.

(i) If LN −M2 < 0, this means that the equilibrium points (Xp, Yp) are saddle points.
In the current case, one family of periodic orbits close to the equilibrium points with
the frequency (44) in a positive sign is obtained.

(ii) If LN−M2 > 0, which means that there is an extremum value for the PF at (Xp, Yp).
The inequality L+N > 2

√
LN −M2 with the second condition produce the criteria

for the stability of the equilibrium (Xp, Yp), where the roots of the frequency are
pure imaginary, consequently we get two families of PS around the minimum stable
points.



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 17 of 22

For the case of quartic potential:
The PS can be obtained by the same steps as the previous steps. The governing equations
for the Hamiltonian (21) may be represented as follows

ẍ+ Vx = 0, ÿ + Vy = 0, (45)

and these equations satisfy the energy integral

1

2

(
ẋ2 + ẏ2

)
+ V = h. (46)

Moreover, the equilibrium points (xp, yp) can be obtained by solving the equations Vx =
Vy = 0, in which from equation (46) there is an initial value of the energy constant h for
each equilibrium point.

Inserting x = xp + ξ, y = yp + η into equation (45), we get after some calculations the
following linear equations

ξ̈ + γξ + µη = 0, η̈ + κη + µξ = 0, (47)

where Vxx(xp, yp) = γ, Vyy(xp, yp) = κ, Vxy(xp, yp) = µ.
Introducing the time transformation, (u = νt) and following the same steps in equa-

tions (38)-(40) and taking the first approximation, then we get

ν20
d2x(1)

du2
+ γx(1) + µy(1) = 0, ν20

d2y(1)

du2
+ κy(1) + µx(1) = 0. (48)

As well known the solution of (48) can be obtained using the Fourier series

x(s) = a01s +
s∑

r=1

(a
(r)
1s cos ru+ b

(r)
1s sin ru),

y(s) = a02s +
s∑

r=1

(a
(r)
2s cos ru+ b

(r)
2s sin ru).

(49)

Setting the first approximation of expressions (49) into (48), then we get the following
solutions

x = xp + ε[B̃(ν20 − κ) cosu+ õ sinu],

y = yp + ε[B̃µ cosu+ Ã(ν20 − γ) sinu],

h = h0 +
ε2

4

(
(Ã2 + B̃2)ν60 + (Ã2(κ− 2γ) + B̃2(γ − 2κ))ν40+

(Ã2(γ2 − 2γκ+ 3µ2) + B̃2(κ2 − 2γκ+ 3µ2))ν20 + (Ã2γ + B̃2κ)(γκ− µ2)
)
,

(50)

where à and B̃ are undefined chosen parameters. The frequency is given by

ν0 =

√
(γ + κ)±

√
(γ + κ)2 − 4(γκ− µ2)

2
. (51)

Thus, there are two types of equilibrium points:



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 18 of 22

(i) If γκ− µ2 < 0, this means that the points (xp, yp) of equilibrium are saddle points,
which give a family of PS with the positive sign of frequency (51).

(ii) If γκ − µ2 > 0, the equilibrium points (xp, yp) are either minimum or maximum

for the PF (2). This condition with the inequality γ + κ > 2
√

γκ− µ2 gives the
equilibrium stability condition (xp, yp) (i.e., the roots of the frequency are pure
imaginary), consequently there are two families of PS around the minimum stable
points.

5. Conclusion

The present study investigates the comprehensive characterization of two galactic mo-
tion models. The topological type of the LS is introduced, which is a torus or an empty
set. Moreover, the PS of the two models is obtained. The phase portrait of separated func-
tions is analyzed and through it, we get the singular points that are elliptic and hyperbolic.
The elliptic points are found to exhibit stability, while the remaining points demonstrate
instability. The Lyapunov theorem [42] is applied to study the PS. Moreover, we have
obtained two families of PS around the stable equilibrium point, whereas each PF has
one family that is near to the saddle points. The outcomes of this research are applicable
in celestial mechanics and astrodynamics, particularly for modeling gravitational interac-
tions between celestial bodies like planets and asteroids. By classifying singular points
and examining phase portraits, the study also provides tools for understanding chaotic
behaviors in weather forecasting and economic modeling. Additionally, the analysis of
bifurcations and PS supports advancements in robotics and automation systems.

Authors Statements

T. S. Amer: Methodology, Data curation, Visualization, Conceptualization, Writing,
Reviewing, Editing.
F. M. El-Saba: Methodology, Validation, Visualization, Conceptualization.
M. Fakharany: Methodology, Conceptualization, Resources, Reviewing.
H. M. Gad: Methodology, Conceptualization, Investigation, Data curation, Writing - orig-
inal draft.

Conflict of Interest

The authors declare that they have no conflict of interest.



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 19 of 22

Funding Acknowledgement

This research received no specific grant from any funding agency in the public, com-
mercial, or not-for-profit sectors.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed
during the current study.

References

[1] Michel Hénon and Carl Heiles. The applicability of the third integral of motion: some
numerical experiments. Astronomical Journal, 69:73–79, 1964.

[2] G. Contopoulos and C. Polymilis. Approximations of the 3-particle toda lattice.
Physica D: Nonlinear Phenomena, 24(1-3):328–342, 1987. Publisher: Elsevier BV.

[3] Morikazu Toda. Vibration of a Chain with Nonlinear Interaction. Journal of the
Physical Society of Japan, 22(2):431–436, 1967. Publisher: Physical Society of Japan.

[4] Dante Carrasco and Claudio Vidal. Periodic Solutions, Stability and Non-Integrability
in a Generalized Hénon-Heiles Hamiltonian System. Journal of Nonlinear Mathemat-
ical Physics, 20(2):199, 2021. Publisher: Springer Science and Business Media LLC.

[5] Philip Holmes. Proof of non-integrability for the Hénon-Heiles Hamiltonian near
an exceptional integrable case. Physica D: Nonlinear Phenomena, 5(2-3):335–347,
September 1982. Publisher: Elsevier BV.

[6] Jaume Llibre and Lidia Jiménez-Lara. Periodic orbits and non-integrability of
Hénon–Heiles systems. Journal of Physics A: Mathematical and Theoretical,
44(20):205103, 2011. Publisher: IOP Publishing.

[7] Mordechai Bixon and Joshua Jortner. Quantum dynamics of the Henon–Heiles sys-
tem. The Journal of Chemical Physics, 77(8):4175–4187, 1982. Publisher: AIP Pub-
lishing.

[8] A.I. Akhiezer, V.I. Truten’, and N.F. Shul’ga. Dynamic chaos in the motion of charged
particles through a crystal. Physics Reports, 203(5), May 1991. Publisher: Elsevier
BV.

[9] R. Rajaraman and Erick J. Weinberg. Internal symmetry and the semiclassical
method in quantum field theory. Physical Review D, 11(10):2950–2966, 1975. Pub-
lisher: American Physical Society (APS).

[10] R. Dutt and M. Lakshmanan. Application of coherent state representation to classical
x6 and coupled anharmonic oscillators. Journal of Mathematical Physics, 17(4):482–
484, 1976. Publisher: AIP Publishing.

[11] R. Friedberg, T. D. Lee, and A. Sirlin. Class of scalar-field soliton solutions in three
space dimensions. Physical Review D, 13(10):2739–2761, 1976. Publisher: American
Physical Society (APS).



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 20 of 22

[12] Giuseppe Pucacco, Dino Boccaletti, and Cinzia Belmonte. Quantitative predictions
with detuned normal forms. Celestial Mechanics and Dynamical Astronomy, 102(1-
3):163–176, 2008. Publisher: Springer Science and Business Media LLC.

[13] Dieter Armbruster, John Guckenheimer, and Seunghwan Kim. Chaotic dynamics in
systems with square symmetry. Physics Letters A, 140(7-8):416–420, 1989. Publisher:
Elsevier BV.

[14] E Calzetta and C El Hasi. Chaotic Friedmann-Robertson-Walker cosmology. Classical
and Quantum Gravity, 10(9):1825–1841, 1993. Publisher: IOP Publishing.

[15] B. Grammaticos, B. Dorizzi, and R. Padjen. Painleve property and integrals of motion
for the Henon-Heiles system. Physics Letters A, 89(3):111–113, 1982. Publisher:
Elsevier BV.

[16] M. Lakshmanan and R. Sahadevan. Painlevé analysis, Lie symmetries, and integrabil-
ity of coupled nonlinear oscillators of polynomial type. Physics Reports, 224(1-2):1–93,
1993. Publisher: Elsevier BV.

[17] Tassos Bountis, Harvey Segur, and Franco Vivaldi. Integrable Hamiltonian systems
and the Painlevé property. Physical Review A, 25(3):1257–1264, 1982. Publisher:
American Physical Society (APS).

[18] S. Kasperczuk. Integrability of the Yang-Mills Hamiltonian system. Celestial Me-
chanics & Dynamical Astronomy, 58(4):387–391, 1994. Publisher: Springer Science
and Business Media LLC.

[19] A. Ramani, B. Dorizzi, and B. Grammaticos. Painlevé Conjecture Revisited. Physical
Review Letters, 49(21):1539–1541, 1982. Publisher: American Physical Society (APS).

[20] Stefan Wojciechowski. Separability of an integrable case of the Henon-Heiles system.
Physics Letters A, 100(6):277–278, 1984. Publisher: Elsevier BV.

[21] Edmund T. Whittaker. A treatise on the analytical dynamics of particles and rigid
bodies: with an introduction to the problem of three bodies. Cambridge Mathemat-
ical Library. Cambridge Univ. Press, Cambridge, 4. ed., transferred to digital print
edition, 1999.

[22] B. Dorizzi, B. Grammaticos, and A. Ramani. A new class of integrable systems.
Journal of Mathematical Physics, 24(9):2282–2288, 1983. Publisher: AIP Publishing.

[23] Katsuya Nakagawa and Haruo Yoshida. A list of all integrable two-dimensional ho-
mogeneous polynomial potentials with a polynomial integral of order at most four in
the momenta. Journal of Physics A: Mathematical and General, 34(41):8611–8630,
2001. Publisher: IOP Publishing.

[24] C. Verhoeven, M. Musette, and R. Conte. Integration of a generalized Hénon–Heiles
Hamiltonian. Journal of Mathematical Physics, 43(4):1906–1915, 2002. Publisher:
AIP Publishing.

[25] V. Ravoson, A. Ramani, and B. Grammaticos. Generalized separability for a Hamil-
tonian with nonseparable quartic potential. Physics Letters A, 191(1-2):91–95, 1994.
Publisher: Elsevier BV.

[26] Pol Vanhaecke. Explicit techniques for studying two-dimensional integrable systems.
PhD thesis, 1991.

[27] A. T. Fomenko. Integrability and Nonintegrability in Geometry and Mechanics.



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 21 of 22

Springer Netherlands, Dordrecht, 1988.
[28] F. M. El-Sabaa, M. Hosny, and S. K. Zakria. Bifurcations of Liouville tori of a

two fixed center problem. Astrophysics and Space Science, 363(4), 2018. Publisher:
Springer Science and Business Media LLC.

[29] F. M. El-Sabaa, M. Hosny, and S. K. Zakria. Bifurcations of Armbruster Gucken-
heimer Kim galactic potential. Astrophysics and Space Science, 364(2):1–9, 2019.
Publisher: Springer Science and Business Media LLC.

[30] Holger Waalkens, Holger R. Dullin, and Peter H. Richter. The problem of two fixed
centers: bifurcations, actions, monodromy. Physica D: Nonlinear Phenomena, 196(3-
4):265–310, 2004. Publisher: Elsevier BV.

[31] Mikhail P Kharlamov. Bifurcation of common levels of first integrals of the ko-
valevskaya problem. Journal of Applied Mathematics and Mechanics, 47(6):737–743,
1983.

[32] A Ouazzani-Th, J Kharbach, and M Ouazzani-Jamil. Phase space topology and
bifurcation of liouville torii in the goryatchev-tchaplygin top. Moroccan Journal of
Condensed Matter, 2, 1999.

[33] TS Amer. On the rotational motion of a gyrostat about a fixed point with mass
distribution. Nonlinear Dynamics, 54(3):189–198, 2008.

[34] TS Amer. The rotational motion of the electromagnetic symmetric rigid body. Appl.
Math. Inf. Sci, 10(4):1453–1464, 2016.

[35] AA Galal, TS Amer, AH Elneklawy, and HF El-Kafly. Studying the influence of
a gyrostatic moment on the motion of a charged rigid body containing a viscous
incompressible liquid. The European Physical Journal Plus, 138(10):1–13, 2023.

[36] Allan P Fordy. The hénon-heiles system revisited. Physica D: Nonlinear Phenomena,
52(2-3):204–210, 1991.

[37] Boris A. Dubrovin, Anatolij Timofeevič Fomenko, and Sergĕı Petrovich Novikov.
Modern Geometry—Methods and Applications: Part II: The Geometry and Topology
of Manifolds. Springer Science & Business Media, 1985.

[38] Vladimir Igorevich Arnol’d. Mathematical methods of classical mechanics, volume 60.
Springer Science & Business Media, 2013.

[39] Anatolij T Fomenko. Visual geometry and topology. Springer Science & Business
Media, 2012.

[40] FM El-Sabaa. Bifurcation of kovalevskaya polynomial. International Journal of The-
oretical Physics, 34(10), 1995.

[41] FM El-Sabaa, M Hosny, and SK Zakria. Bifurcations of liouville tori of generalized
two-fixed center problem. Italian Journal of Pure and Applied Mathematics, 43:331–
352, 2020.

[42] Aleksandr Mikhailovich Lyapunov. The general problem of the stability of motion.
International journal of control, 55(3):531–534, 1992.

[43] TS Amer, HF El-Kafly, AH Elneklawy, and AA Galal. Analyzing the dynamics of a
charged rotating rigid body under constant torques. Scientific Reports, 14(1):9839,
2024.

[44] Amer TS, Elneklawy AH, and El-Kafly HF. A novel approach to solving euler’s



T. S. Amer et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6341 22 of 22

nonlinear equations for a 3dof dynamical motion of a rigid body under gyrostatic and
constant torques. Journal of Low Frequency Noise, Vibration and Active Control,
44(1):111–129, 2025.

[45] TS Amer and IM Abady. Solutions of euler’s dynamic equations for the motion of a
rigid body. Journal of Aerospace Engineering, 30(4):04017021, 2017.

[46] TS Amer, AH Elneklawy, and HF El-Kafly. Analysis of euler’s equations for a symmet-
ric rigid body subject to time-dependent gyrostatic torque. Journal of Low Frequency
Noise, Vibration and Active Control, page 14613484241312465, 2025.

[47] TS Amer, AM Farag, and WS Amer. The dynamical motion of a rigid body for the
case of ellipsoid inertia close to ellipsoid of rotation. Mechanics Research Communi-
cations, 108:103583, 2020.

[48] TS Amer, H Elkafly, and AA Galal. The 3d motion of a charged solid body using
the asymptotic technique of kbm. Alexandria Engineering Journal, 60(6):5655–5673,
2021.

[49] TS Amer, HF El-Kafly, AH Elneklawy, and WS Amer. Modeling analysis on the
influence of the gyrostatic moment on the motion of a charged rigid body subjected
to constant axial torque. Journal of Low Frequency Noise, Vibration and Active
Control, 43(4):1593–1610, 2024.

[50] Asmaa Amer, TS Amer, and AA Galal. Simulation of a subjected rigid body motion
to an external force and moment. Journal of Vibration Engineering & Technologies,
12(3):2775–2790, 2024.

[51] TS Amer, HF El-Kafly, AH Elneklawy, and AA Galal. Stability analysis of a rotating
rigid body: The role of external and gyroscopic torques with energy dissipation. Jour-
nal of Low Frequency Noise, Vibration and Active Control, page 14613484251324586,
2025.

[52] TS Amer, AH Elneklawy, and HF El-Kafly. Dynamical motion of a spacecraft con-
taining a slug and influenced by a gyrostatic moment and constant torques. Journal of
Low Frequency Noise, Vibration and Active Control, page 14613484251322235, 2025.

[53] FM El-Sabaa, TS Amer, HM Gad, and MA Bek. Existence of periodic solutions and
their stability for a sextic galactic potential function. Astrophysics and Space Science,
366(8):74, 2021.

[54] TS Amer, Asma Alanazy, AH Elneklawy, WS Amer, and HF El-Kafly. A novel study
on the fourth first integral for the rotatory motion of an impacted charged rigid body
by external torques. Journal of Low Frequency Noise, Vibration and Active Control,
page 14613484251347080, 2025.

[55] TS Amer, HF El-Kafly, AH Elneklawy, and AA Galal. Analyzing the spatial motion
of a rigid body subjected to constant body-fixed torques and gyrostatic moment.
Scientific Reports, 14(1):5390, 2024.

[56] TS Amer, WS Amer, M Fakharany, AH Elneklawy, and HF El-Kafly. Modeling of
the euler-poisson equations for rigid bodies in the context of the gyrostatic influences:
an innovative methodology. European Journal of Pure and Applied Mathematics,
18(1):5712–5712, 2025.