Bifurcations of Liouville tori of coupled sextic anharmonic oscillators

Abstract

In the current paper, the problem of sextic anharmonic oscillators is investigated. There are three integrable cases of this problem. Emphasis is placed on two integrable cases, and a full description of each one is provided. The separated functions of the first and second integrability cases are transformed from a higher degree to the third and fourth degrees. Respectively, the periodic solution is obtained using Jacobi’s elliptic functions. The topology of phase space and Liouville tori’s bifurcations are discussed. The phase portrait is studied to determine singular points and classify their types in addition to the graphic representation for each of them. Finally, the numerical illustrations are introduced using the Poincaré surface section to emphasize the problem’s integrability.

Keywords

Anharmonic oscillators Hamilton–Jacobi’s equations periodic solution topology of the level sets phase portrait Poincaré surface section

Introduction

It is known that the $N$ degrees-of-freedom (DOF) Hamiltonian system is integrable if there are $N$ single-valued, analytic, universal integrals of motion in which they are functionally autonomous and in involution.¹ There is no general method of achieving the integrability of motion associated with Hamiltonian systems. The Painlevé (P) analysis is one of the most powerful methods for getting the adequate conditions for integrating the equations of motion (EOM). The dynamical system is governed by a system of ordinary differential equations (Odes) that possesses the P-property if the general solutions have no movable singularities other than the pole, where the solution is meromorphic or related to meromorphism. Several researchers have applied this analysis to determine the Hamiltonian system’s integrability, for example, Refs. 2–4 The conjecture Ablowitz–Ramani–Segur (ARS) asserts a nonlinear partial differential equation (PDE), which can be solved using the inverse scattering technique (IST). This can be done if the exact reduction of each nonlinear ordinary differential equation (Ode) has a type of Painlevé. P-analysis helps to find the necessary solution of the problem only. The goal after that is to know whether the problem can be transformed into quadrature, that is, separating variables and obtaining the second invariant integral. There is a group of issues in which no separation has been completed despite the determination of the integrability condition. The researchers hold to this limit for these types of issues.

Periodic solutions for the rigid body (RB) problem utilizing the small parameter method of Poincaré (SPMP), averaging method (AM), and other approaches had shed the attention of researchers during the recent decades.^5–23 The SPMP and its modifications are utilized in Refs. 5–11 to reduce the EOM for the EOM of the regulating system and their first-integrals into a suitable system of two second-order equations and only one integral. When the system’s frequency has integer values or their inverses, it is possible to construct approximate solutions for the latter system; however, these solutions comprise different singular points.^5–7 When the gyro moment (GM) around the third major axis is used, these singularities are dealt with in Ref. 8, as well as in Ref. 9, in accordance with the instance of Kovalevskaya. This method was most recently applied in Ref. 10 to investigate the movement of the RB in accordance with Bobylev–Steklov requirements when the body was impacted by the GM and various attracting fields. Furthermore, the authors of Ref. 11 proposed that the body’s center of mass is somewhat shifted from the axis of dynamic symmetry and that it is affected by the electromagnetic field and the general magnetic field. It is important to note that the obtained solutions have no singular points at all and can be adjusted to any value of the body’s frequency. On the flip side, one can look at Refs. 12–17 to investigate how the strategy of this approach might be applied to address various SB’s issues.

The AT, which has seen a lot of application over the last three decades, was employed in Refs. 18–23 to find the solutions for a symmetric RB spinning around a fixed point. This motion is studied in three specific circumstances: a uniform gravity field,^18,19 an external attracting center,^20,21 and in the presence of GM.^22,23 When the perturbing moments are assumed to be acting along the axes of inertia of the primary body, these situations are investigated. Based on some of the initial conditions, a small parameter is introduced to the EOM. The averaging systems of the various EOM are recognized and solved to achieve the desired solutions. In Ref. 24, the authors proposed a Hamiltonian formulation of a conservative oscillator to determine the frequency property. The theory of fractal variational is used in Ref. 25 to establish the fractal Toda oscillator, and the non-perturbative method is employed to obtain the appropriate analytical solution.

The problem of an anharmonic oscillator plays a significant role in different fields of applied mechanics. For example, in quantum mechanics, Dutt and Lakshmanan²⁶ studied the classical nonlinear quantum oscillator in chemical physics. However, Lakshmanan and Prabhakaran²⁷ found the energy levels of the sextic anharmonic oscillator. At the same time, many articles regarding galaxies dynamics have been examined in astrophysics for quartic, quantic, and sextic anharmonic oscillators. In Ref. 28, the authors introduced AM of first order in order to obtain the existence of four periodic solutions of the generalization of (FRW) potential. In Ref. 29, the Hamiltonian function of fifth degree with two real parameters has been studied to obtain the conditions on the two parameters and to determine the periodic solutions with the help of AM, while in Ref. 30, the authors discussed the Hamiltonian function of fifth degree with three real parameters using the same method of second order and proved the existence of periodic solutions. In Ref. 31, the authors studied the periodic orbits for the sextic anharmonic oscillator function using two different methods.

The main objective of this paper is to get a full description of the sextic coupled anharmonic oscillator,³² which is a model of elliptic galaxies motion, defined as

V_{6} = A x^{2} + B y^{2} + α x^{6} + β y^{6} + δ_{1} x^{4} y^{2} + δ_{2} x^{2} y^{4},

(1)

where

A, B, α, β, δ_{1}

, and

δ_{2} \in R

by studying the bifurcation and the phase portrait for two separated cases of a coupled anharmonic oscillator. This work is structured as follows; the integrability of potential (1) and the second invariant integral are introduced. The separation for the first case is discussed, and the periodic solution in terms of elliptic functions. The topological structure of the invariant level sets of the separated functions is discussed using Fomenko’s classification theorem.³³ The phase portrait of separation functions is described. The separation for the second case is studied, and the real phase space topology and bifurcation sets are obtained. The periodic solution for singular level sets of bifurcation is examined. The separated function is used to study the system’s phase portrait and determine the types of singular points. Finally, the regular motion in two integrable cases for different energy values is presented using Poincaré surface section.

The integrability and the second invariant

The EOM associated with the potential function $V_{6}$ take the form

\begin{array}{l} \ddot{x} = - (2 A x + 6 α x^{5} + 4 δ_{1} x^{3} y^{2} + 2 δ_{2} x y^{4}), \\ \ddot{y} = - (2 B y + 6 β y^{5} + 2 δ_{1} x^{4} y + 4 δ_{2} x^{2} y^{3}) . \end{array}

(2)

By using the ARS conjecture, in which the above equations possess the P-property,³² the necessary integrability conditions are

\begin{array}{l} (i) A = B, α = β, δ_{1} = δ_{2} = 15 α, \\ (i i) A = B, α = β, δ_{1} = δ_{2} = 3 α, \\ (i i i) A = 4 B, α = 64 β, δ_{1} = 80 β, δ_{2} = 24 β \end{array}

To claim the integrability, it is required to get an additional integral (invariant integral) which is independent of the energy integral.

In the first case (i) and by using the transformations

u = x + y, v = x - y .

(3)

The Hamiltonian function becomes

H = p_{u}^{2} + p_{v}^{2} + \frac{A}{2} (u^{2} + v^{2}) + \frac{α}{2} (u^{6} + v^{6}) .

(4)

Therefore, the problem becomes separable, and the integrals of motion are

h - 2 p_{u}^{2} - A u^{2} - α u^{6} = - C,

(5)

h - 2 p_{v}^{2} - A v^{2} - α v^{6} = C,

(6)

where

h

is the energy integral and

C

is the second invariant integral that can be given as follows:

Adding the two equations (5) and (6), to get

2 C = 2 (p_{u}^{2} - p_{v}^{2}) + A (u^{2} - v^{2}) + α (u^{6} - v^{6}),

(7)

Hence, from Hamiltonian equations

p_{u} = \frac{\dot{u}}{2}, p_{v} = \frac{\dot{v}}{2},

we get

2 C = \frac{1}{2} ({\dot{u}}^{2} - {\dot{v}}^{2}) + A (u^{2} - v^{2}) + α (u^{6} - v^{6}) .

(8)

It should be noted that from (3), we have

u^{2} - v^{2} = 4 x y, u^{6} - v^{6} = 12 x^{5} y + 40 x^{3} y^{3} + 12 x y^{5}, {\dot{u}}^{2} - {\dot{v}}^{2} = 4 \dot{x} \dot{y} .

(9)

Thus, from equation (9), the second invariant integral becomes

C = \dot{x} \dot{y} + 2 A x y + α (6 x^{5} y + 20 x^{3} y^{3} + 6 x y^{5}) .

(10)

In the second case (ii), and by using the polar coordinates

x = r \cos θ, y = r \sin θ

(11)

The Hamiltonian function of equation (1) takes the form

H = \frac{1}{2} (p_{r}^{2} + \frac{C^{2}}{r^{2}}) + A r^{2} + α r^{6} .

(12)

As is well known, the variable $θ$ is a cyclic coordinate, so the invariant integral $C$ is given by

p_{θ} = r^{2} \dot{θ} = C,

With the help of equation (11), the invariant integral becomes

C = x \dot{y} - y \dot{x}

(13)

Finally, for the last case of Lakshmanan and Sahadevan,³² which is denoted previously by (iii), the corresponding potential function is

V = B (4 x^{2} + y^{2}) + β {(4 x^{2} + y^{2})}^{3} + β (32 y^{2} x^{4} + 12 x^{2} y^{4})

(14)

The parabolic coordinates that make the system of the case (iii) separable are

x = \frac{1}{2} (τ^{2} - σ^{2}), y = τ σ,

(15)

Therefore, the Hamiltonian function becomes

2 H (τ^{2} + σ^{2}) = p_{σ}^{2} + p_{τ}^{2} + 2 B (τ^{6} + σ^{6}) + 2 β (τ^{14} + σ^{14}) .

(16)

It is easy to obtain the integrals of motion in the following way

p_{τ}^{2} - 2 h τ^{2} + 2 B τ^{6} + 2 β τ^{14} = C,

(17)

p_{σ}^{2} - 2 h σ^{2} + 2 B σ^{6} + 2 β σ^{14} = - C .

(18)

The second integral $C$ can be obtained by subtracting the two equations (17) and (18), and using the Hamiltonian equations

\begin{array}{l} p_{τ} = \dot{τ} (τ^{2} + σ^{2}), \\ p_{σ} = \dot{σ} (τ^{2} + σ^{2}) . \end{array}

(19)

Furthermore, with the help of equation (15), the second invariant integral can be written in the form

C = 2 \dot{y} (\dot{x} y - \dot{y} x) + 4 B x y^{2} + 4 β [16 x^{2} (x^{2} + y^{2}) + 3 y^{4}] y^{2} x .

(20)

The first case of integrability

This section aims to discuss the integrability of case (i) in the previous section. The separation of the problem is explained, and the periodic solution is studied using Jacobi’s elliptic function. As well as the topological analysis of the problem, the singular points of the separated functions are specified through the study of the phase portrait.

Integration of the problem

In this subsection, we are going to find the periodic solution of the Hamiltonian system (4) using Jacobi’s elliptic function. Returning to equations (5) and (6), which can be reformulated as

\int \frac{d u}{\sqrt{F (u)}} = \sqrt{2} \int d t, \int \frac{d v}{\sqrt{F (v)}} = \sqrt{2} \int d t,

(21)

where

F (z) = - α z^{6} - A z^{2} + h \pm C, z = z (u, v)

(22)

An inspection of the previous equation shows that it is of the sixth degree, in which its solution can be found by reducing its degree.³⁴ In order to achieve this aim, we consider

f = h \pm C, z^{2} = {(x + \frac{A}{3 f})}^{- 1}

(23)

Therefore, equation (22) becomes

F (x) = f {(x + \frac{A}{3 f})}^{- 3} (x^{3} - p x - q),

(24)

where

p = \frac{A^{2}}{3 f^{2}}, q = \frac{α}{f} + \frac{2 A^{3}}{27 f^{3}}, d z = \frac{- 1}{2} {(x + \frac{A}{3 f})}^{\frac{- 3}{2}} d x .

(25)

Hence, integrals (21) become

\frac{d x}{\sqrt{R (x)}} = - 2 \sqrt{2 f} d t,

(26)

where

R (x) = x^{3} - p x - q = (x - a) (x - b) (x - \bar{b}),

and

a

represents the real root of

R (x)

, while

b

and

\bar{b}

represent two complex roots.

Let us consider the following relations

\begin{array}{l} (x - b) (x - \bar{b}) = {(x - b_{1})}^{2} + a_{1}^{2}, \\ b_{1} = \frac{b + \bar{b}}{2}, a_{1}^{2} = \frac{- {(b - \bar{b})}^{2}}{4}, \\ {(b_{1} - a)}^{2} + a_{1}^{2} = B^{2} \end{array}

(27)

Then, $R (x)$ can be rewritten as

R (x) = (x - a) [{(x - b_{1})}^{2} + a_{1}^{2}] .

(28)

Taking into account the above equation, we can consider that

x = a + B \frac{1 + \cos φ}{1 - \cos φ}; \infty > x \geq a

(29)

Referring to equations (28) and (29), equation (26) becomes

\int_{0}^{φ} \frac{d φ}{\sqrt{1 - k^{2} \sin^{2} φ}} = 2 \sqrt{2 f B} (t - t_{0}) .

(30)

Therefore, the solution of (30) can be expressed in the following form

x = a + B \frac{1 + c n [2 \sqrt{2 f B} (t - t_{0}), k]}{1 - c n [2 \sqrt{2 f B} (t - t_{0}), k]},

(31)

where

\cos φ = c n (2 \sqrt{2 f B} (t - t_{0})) .

Making use of equations (23) and (31), to get

z = η s n [\sqrt{2 f B} (t - t_{0}), k] {\frac{1 + d n [2 \sqrt{2 f B} (t - t_{0}), k]}{1 + η_{1} c n [2 \sqrt{2 f B} (t - t_{0}), k]}}^{\frac{1}{2}},

(32)

where

η = {(a + B + \frac{A}{3 f})}^{\frac{- 1}{2}}, η_{1} = \frac{B - a - A / 3 f}{B + a + A / 3 f}, k^{2} = \frac{B - a + b_{1}}{2 B},

(33)

with the period $T$

T = \frac{1}{2 \sqrt{2 f B}} s n^{- 1} (1, k) = \frac{1}{2 \sqrt{2 f B}} K (k), K (k) = \int_{0}^{\frac{π}{2}} \frac{d φ}{\sqrt{1 - k^{2} \sin^{2} φ}},

where

K (k)

is the first kind of the complete elliptic integral.

Topological analysis

In the current section, we examine the topological analysis of the real invariant manifold utilizing the theory of Fomenko.³³ The Fomenko classification had been investigated in several areas in mechanics: in Celestial Mechanics, El-Sabaa et al.³⁵ studied the bifurcation of the problem of two fixed centers. The bifurcation set was constructed by Vozmischeva^36–38 in which the domain of possible motion on the configurational space was classified, while Waalkens et al.³⁹ studied the bifurcation in the topology of energy surfaces for the motion in three dimensions. In El-Sabaa et al.,⁴⁰ the bifurcation of AGK galactic potential was analyzed. In Refs. 41 and 42, the authors studied the bifurcation of invariant manifolds in the GHH system and discussed the bifurcation of the common level set of the first integral of Kovalevskaya’s case for the problem of a RB. In addition, the bifurcation of Liouville tori for the different cases of the RB rotation around one fixed point of its configuration is studied.⁴³

The common level set of the first integral is investigated with a view to the phase space analysis. Fomenko classification is used to describe all generic bifurcation of Liouville tori. Therefore, the definitions below are introduced.⁴⁴

1. The function $F (x) = (f_{1} (x), . . ., f_{n} (x))$ is called momentum mapping if there exists a simplistic manifold system $M^{2 n}$ such that $F : M^{2 n} \to R^{n},$ and $f_{1}, f_{2}, . . ., f_{n}$ are independent integrals.

2. When the rank $d F (x) < n,$ and $F (x) \in R^{n}$ represent a critical value, then $x \in M$ is said to be a critical point of the momentum mapping.

3. The bifurcation diagram is the set $B = F (k) \subset R^{n}$ that is the union of several pieces $B_{k}$ .

4. The set of all critical points of the momentum mapping $Z$ is a subset of $M$ .

In our problem, the level sets have the topology

M_{S} = {(u, v, p_{u}, p_{v}) \in R^{4} : H = h, F = C \subset R^{4}},

(34)

where

M_{S}

is a finite union of two-dimension tori for non-critical values of both

F

and

H

The energy-momentum diagram can be estimated according to the set of critical points

(u, v, p_{u}, p_{v}) \to (H, F),

If $B$ is the discriminant locus of the polynomials $F (u)$ and $F (v)$ , in which their coefficients are given as functions of $h$ and $C$ , then

B = B_{1} \cup B_{2} = [(h, C) \in R^{2} / d i s c (F (u)) = 0] \cup [(h, C) \in R^{2} / d i s c (F (v)) = 0] .

(35)

The topological sort of $M_{S}$ can be changed by the point $(h, C)$ passes through $B$ . The set $R^{2} / B$ constitutes from 24 connected components, as displayed in Figure 1. So, in every connected component of $R^{2} / B$ , the level set $M_{S}$ has the same topological sort. For the values of $H$ and $F$ in which are non-critical, the level set $M_{S}$ is a bounded union of low-dimensional tori.¹ Their number depends on the number and location of the allowed ovals on the Riemann surface associated with the elliptical curves $Γ_{1}$ and $Γ_{2}$ where

Γ_{1} : ω_{1} = \sqrt{F (u)}, Γ_{2} : ω_{2} = \sqrt{F (v)} .

Figure 1.

Shows the bifurcation portraits $B = B_{1} \cup B_{2}$ .

Table 1 displays the roots of both

F (u)

and

F (v)

, while Table 2 describes the topological sort of

M_{S}

is a torus, or a set of non-elements.⁴⁵

Table 1.

Topological type of $M_{S}$ and real roots of the polynomials $F (u)$ and $F (v)$ for $(C, h) \in R^{2} / B$ .

Domain	Roots of $F (u)$	Roots of $F (v)$
1	$u_{1} < 0 < u_{2}$	$v_{1} < 0 < v_{2}$
2	$u_{1} < 0 < u_{2}$	$0$
3	$u_{1} < 0 < u_{2}$	$0$
4	$u_{1} < 0 < u_{2}$	$0$
5	$u_{1} < 0 < u_{2}$	$0$
6	$0$	$0$
7	$0$	$0$
8	$0$	$0$
9	$0$	$0$
10	$0$	$0$
11	$0$	$0$
12	$0$	$0$
13	$0$	$0$
14	$0$	$0$
15	$0$	$0$
16	$0$	$0$
17	$0$	$0$
18	$0$	$0$
19	$0$	$0$
20	$0$	$v_{1} < 0 < v_{2}$
21	$0$	$v_{1} < 0 < v_{2}$
22	$0$	$v_{1} < 0 < v_{2}$
23	$0$	$v_{1} < 0 < v_{2}$
24	$u_{1} < 0 < u_{2}$	$v_{1} < 0 < v_{2}$

Table 2.

Presents allowable ovals and topological sort of $M_{S}$ for $(C, h) \in R^{2} / B$ .

Domain	$u - p l a n e Δ_{1}$	$v - p l a n e Δ_{2}$	Topological type
1	$[0, u_{2}]$	$[v_{1}, 0]$	$T$
2	$[0, u_{2}]$	$\emptyset$	$\emptyset$
3	$[0, u_{2}]$	$\emptyset$	$\emptyset$
4	$[0, u_{2}]$	$\emptyset$	$\emptyset$
5	$[0, u_{2}]$	$\emptyset$	$\emptyset$
6	$\emptyset$	$\emptyset$	$\emptyset$
7	$\emptyset$	$\emptyset$	$\emptyset$
8	$\emptyset$	$\emptyset$	$\emptyset$
9	$\emptyset$	$\emptyset$	$\emptyset$
10	$\emptyset$	$\emptyset$	$\emptyset$
11	$\emptyset$	$\emptyset$	$\emptyset$
12	$\emptyset$	$\emptyset$	$\emptyset$
13	$\emptyset$	$\emptyset$	$\emptyset$
14	$\emptyset$	$\emptyset$	$\emptyset$
15	$\emptyset$	$\emptyset$	$\emptyset$
16	$\emptyset$	$\emptyset$	$\emptyset$
17	$\emptyset$	$\emptyset$	$\emptyset$
18	$\emptyset$	$\emptyset$	$\emptyset$
19	$\emptyset$	$\emptyset$	$\emptyset$
20	$\emptyset$	$[v_{1}, 0]$	$\emptyset$
21	$\emptyset$	$[v_{1}, 0]$	$\emptyset$
22	$\emptyset$	$[v_{1}, 0]$	$\emptyset$
23	$\emptyset$	$[v_{1}, 0]$	$\emptyset$
24	$[0, u_{2}]$	$[v_{1}, 0]$	$T$

It is worth noticing that the intersection of the discriminants of the separated functions, which contain the energy and separated constants, produces twenty-four domains, as mentioned before and displayed in Figure 1. Therefore, we select the roots of the equation in each domain. For the imaginary roots, we write zero as in Table 1, while we write the real ones to determine the kind of topology in each domain as mentioned in Table 2. For the columns of $u$ and $v$ , we consider the roots greater and less than zero, respectively, in which we denote for the case of no real roots by $\emptyset$ . Moreover, each interval in Table 2 represents an oval, and when we have two ovals in one row, then we obtain a kind of topological torus. Otherwise, the kind of topology is $\emptyset$ .

The system’s generic bifurcations are displayed in Table 3, while the bifurcation Liouville tori and the correspondence between bifurcations of the polynomials’ roots are plotted in Figures 2 and 3.

Table 3.

Reveals the generic bifurcations of $M_{S}$ passing from domain $i$ to domain $j$ where $i = 1,24$ and $j = (2 : 23)$ .

$1 \to 24$	$1 \to j$	$24 \to j$
$T \to T$	$T \to \emptyset$	$T \to \emptyset$

Figure 2.

Displays bifurcation of Liouville tori, where $(S \land S)$ is a union of two circles having exactly one common point.

Figure 3.

Illustrates the correspondence between bifurcations of invariant Liouville tori and bifurcations of $F (u)$ and $F (v)$ roots.

Separated function and its phase portrait

This section aims to explain the topological translation of the path by applying the phase portrait as examined in Ref. 46. For this purpose, let us consider

F = 2 p^{2} + A q^{2} + α q^{6} - k,

(36)

where

k = h \pm C

The singular points of the function $F$ can be obtained from the solutions of the following equations

\frac{\partial F}{\partial p} = 4 p = 0, \frac{\partial F}{\partial q} = 2 A q + 6 α q^{5} = 0 .

(37)

Then, we have three distinct points $(0,0)$ and $(0, \pm \sqrt[4]{\frac{- A}{3 α}})$ .

1. Take the point $(x, y)$ closest to the point (0,0), then the function $F$ has the form

F = 2 x^{2} + A y^{2} - k,

(38)

in which the term “degree greater than two” is neglected. According to (38), the point

(0,0)

is a hyperbolic point where

M < 0

, as shown in Figure 4.

2. Based on the same method, we can find that the points $(0, \pm \sqrt[4]{\frac{- A}{3 α}})$ are elliptic points, as sketched in Figure 5, in which $A < 0$ and $M > 0$ .

M = [\begin{array}{l} \frac{\partial^{2} F}{\partial x^{2}} \frac{\partial^{2} F}{\partial x \partial y} \\ \frac{\partial^{2} F}{\partial x \partial y} \frac{\partial^{2} F}{\partial y^{2}} \end{array}]

Figure 4.

Shows the hyperbolic point $(0,0)$ .

Figure 5.

Reveals two elliptic points $(0, \pm \sqrt[4]{\frac{- A}{3 α}})$ and one hyperbolic point $(0,0)$ .

The second case of integrability

This section aims to analyze the second case (ii) of integrability. The separation of the problem is explained, and the topological analysis of the problem is discussed. Moreover, the periodic solution is obtained through the use of Jacobi’s elliptic function. The phase portrait is introduced, and the singular points of the separated functions are determined.

Separation of the problem

As mentioned above, the Hamiltonian function (12) for case (ii) can be written as follows

r^{2} p_{r}^{2} = - 2 α r^{8} - 2 A r^{4} + 2 h r^{2} - C^{2},

(39)

p_{r} = \frac{1}{r} \sqrt{Q (r)},

(40)

where

Q (r) = - 2 α r^{8} - 2 A r^{4} + 2 h r^{2} - C^{2} .

(41)

Consequently, from equations (40) and (41), we can obtain

\int \frac{r d r}{\sqrt{Q (r)}} = \int d t .

(42)

Bifurcation of the problem

In order to study the bifurcation diagram of this case, we are going to reduce the degree of the equation (41) through substitution $r^{2} = x,$ and then the function $Q (x)$ becomes

Q (x) = - 2 α x^{4} - 2 A x^{2} + 2 h x - C^{2} .

(43)

By following up the study using the same mentioned previous method in subsection (3.2). Therefore, let $Ω$ be the discriminate of the polynomial $Q (x)$

\begin{array}{l} Ω = {(h, C) \in R^{2} : 64 (8 C^{2} - 32 C^{4} + 32 C^{6} - 4 h^{2} + 72 C^{2} h^{2} - 27 h^{4})^{2} = 0, C = p_{θ} \geq 0}, θ \in [0,2 π] \end{array}

(44)

The set $R^{2} / Ω$ constitutes eight connected components as presented in Figure 6. Table 4 reveals the real roots of $Q (x)$ in order to find the ovals (a connected component of the fixed points set on the curve) $Γ_{j} (j = 1,2)$ where $Γ_{1} : ω_{1} = \sqrt{Q (x)}$ and $Γ_{2} : ω_{2} = \sqrt{Q (θ)}$ , while Table 5 shows the topological type of $A_{S}$ , $A_{S} = {(x, y, \dot{x}, \dot{y}) \in R^{4} : H = h, F = C} \subset R^{4}$ .

Figure 6.

Represents the bifurcation diagram $Ω$ .

Table 4.

Topological category of $A_{S}$ and real roots of the polynomial $Q (x)$ for $(C, h) \in R^{2} / Ω$ .

Domain	Roots of $Q (x)$
1	$0 < x_{3} < x_{4}$
2	$0$
3	$0$
4	$x_{1} < x_{2} < 0$
5	$x_{1} < x_{2} < 0$
6	$0$
7	$0$
8	$0 < x_{3} < x_{4}$

Table 5.

Represents admissible ovals and topological type of $A_{S}$ for $(C, h) \in R^{2} / Ω$ .

Domain	$x - p l a n e Δ_{1}$	$θ - p l a n e Δ_{2}$	Topological type
1	$[x_{3}, x_{4}]$	$[0,2 π]$	$T$
2	$\emptyset$	$[0,2 π]$	$\emptyset$
3	$\emptyset$	$[0,2 π]$	$\emptyset$
4	$[x_{1}, x_{2}]$	$[0,2 π]$	$T$
5	$[x_{1}, x_{2}]$	$[0,2 π]$	$T$
6	$\emptyset$	$[0,2 π]$	$\emptyset$
7	$\emptyset$	$[0,2 π]$	$\emptyset$
8	$[x_{3}, x_{4}]$	$[0,2 π]$	$T$

In other words, Table 4 represents the case of one only separated function $(x)$ , therefore we have obtained eight domains as seen in Figure 6 when the discriminant of this function is drawn due to that the variable $θ$ is cyclic which is defined over the interval $[0,2 π]$ . The column $x$ contains real roots, in which they are categorized in terms of greater and less than zero. In Table 5, each interval expresses an oval, and the kind of topology is determined.

The topological type is a tour as, in

D_{j} (j = 1,4,5,8)

or empty, as in

D_{j} (j = 2,3,6,7)

. To study all generic bifurcations of Liouville tori of the problem as seen in Table 6, it is sufficient to look at the polynomial bifurcations

Q (x)

, the correspondence between bifurcations of invariant Liouville tori, and bifurcations of

Q (x)

roots, see Figures 7 and 8.

Table 6.

Shows the generic bifurcations of $A_{S}$ passing from domain $m = 1,4,5,8$ to domain $n = 2,3,6,7$ and from domain $i = 1,4,5,8$ to domain $j = 1,4,5,8$ .

$m \to n$	$i \to j; i \neq j$
$T \to \emptyset$	$T \to T$

Figure 7.

Shows bifurcation of Liouville tori.

Figure 8.

Shows the correspondence between bifurcations of invariant Liouville tori and bifurcations of roots of polynomial $Q (x)$ .

Periodic solution

Based on the above section, we have families of periodic solutions on the curves

c_{2}, c_{3}, c_{4},

and

c_{6}

(see Table 7), in which the periodic solution on the curve

c_{2}

can be investigated. Considering the parameter

θ

belongs to the admissible interval

[0,2 π]

which is a constant and the values of

x

parameter belong to the interval

[x_{3}, x_{4}]

. Therefore, we can solve the following integral equation

\int \frac{d x}{\sqrt{Q (x)}} = 2 \int d t,

(45)

where

Q (x) = - 2 α x^{4} - 2 A x^{2} + 2 h x - C^{2} .

Table 7.

Represents the topological type of $A_{S}$ on the diagram $Ω$ , where $(S \land S)$ is the two no-coplanar circles having one common point.

Curve	$Δ_{1}$	$Δ_{2}$	$A_{S} = Δ_{1} \times Δ_{2}$
$c_{1}$	$[x_{3}, x_{4}]$	$[0,2 π]$	$S \times (S \land S)$
$c_{2}$	$[x_{3}, x_{4}]$	$[0,2 π]$	$S$
$c_{3}$	$[x_{3}, x_{4}]$	$[0,2 π]$	$S$
$c_{4}$	$[x_{1}, x_{2}]$	$[0,2 π]$	$S$
$c_{5}$	$[x_{1}, x_{2}]$	$[0,2 π]$	$S \times (S \land S)$
$c_{6}$	$[x_{1}, x_{2}]$	$[0,2 π]$	$S$

According to the $Q (x)$ formula, we can discuss the following cases

1. If $α > 0$ , the real motion is bounded where $(x_{4} \leq x \leq x_{3}) \cup (x_{2} \leq x \leq x_{1})$ such that

Q (x) = (x_{1} - x) (x - x_{2}) (x - x_{3}) (x - x_{4}) .

(46)

Let us use the following substitution³⁴

x = \frac{x_{2} (x_{1} - x_{3}) - x_{3} (x_{1} - x_{2}) \sin^{2} φ}{(x_{1} - x_{3}) - (x_{1} - x_{2}) \sin^{2} φ},

(47)

then the solution of equation (45) can be represented in the form of an elliptic function as

x (t) = \frac{x_{2} (x_{1} - x_{3}) - x_{3} (x_{1} - x_{2}) s n^{2} [a (t - t_{0}), k_{1}]}{(x_{1} - x_{3}) - (x_{1} - x_{2}) s n^{2} [a (t - t_{0}), k_{1}]}

(48)

Referring to our substitution in the previous section $(r^{2} = x)$ , then the solution of equation (42) becomes

r (t) = {(\frac{x_{2} (x_{1} - x_{3}) - x_{3} (x_{1} - x_{2}) s n^{2} [a (t - t_{0}), k_{1}]}{(x_{1} - x_{3}) - (x_{1} - x_{2}) s n^{2} [a (t - t_{0}), k_{1}]})}^{1 / 2},

(49)

with period

T

T = \frac{1}{a} s n^{- 1} (1, k_{1}) = \frac{1}{a} K (k_{1}),

(50)

where

\begin{array}{l} K (k_{1}) = \int_{0}^{\frac{π}{2}} \frac{d φ}{\sqrt{1 - k_{1}^{2} \sin^{2} φ}}, \\ k_{1}^{2} = \frac{(x_{1} - x_{2}) (x_{3} - x_{4})}{(x_{1} - x_{3}) (x_{2} - x_{4})}, \\ a = \sqrt{(x_{1} - x_{3}) (x_{2} - x_{4}) .} \end{array}

(51)

2. If $α < 0$ , the real motion is bounded where $(x_{3} \leq x \leq x_{2})$ such that $Q (x) = (x_{1} - x) (x_{2} - x) (x - x_{3}) (x - x_{4})$

Using the same procedure described in case (i), we can get the following solution

x (t) = \frac{x_{3} (x_{2} - x_{4}) - x_{4} (x_{2} - x_{3}) s n^{2} [b (t - t_{0}), k_{2}]}{(x_{2} - x_{4}) - (x_{2} - x_{3}) s n^{2} [b (t - t_{0}), k_{2}]},

(52)

r (t) = {(\frac{x_{3} (x_{2} - x_{4}) - x_{4} (x_{2} - x_{3}) s n^{2} [b (t - t_{0}), k_{2}]}{(x_{2} - x_{4}) - (x_{2} - x_{3}) s n^{2} [b (t - t_{0}), k_{2}]})}^{1 / 2},

(53)

T = \frac{1}{b} s n^{- 1} (1, k_{2}) = \frac{1}{b} K (k_{2}); K (k_{2}) = \int_{0}^{\frac{π}{2}} \frac{d φ}{\sqrt{1 - k_{2}^{2} \sin^{2} φ}},

(54)

k_{2}^{2} = \frac{(x_{1} - x_{4}) (x_{2} - x_{3})}{(x_{2} - x_{4}) (x_{1} - x_{3})}, b = \sqrt{(x_{2} - x_{4}) (x_{1} - x_{3})} .

(55)

Here,

K (k_{1})

and

K (k_{2})

are known as the first kind of the complete elliptic integral.

Phase portrait of the problem

Consider the function

G = p^{2} q^{2} + 2 A q^{4} + 2 α q^{8} - 2 h q^{2} + C^{2} .

(56)

The singular points of the previous function $G$ can be obtained from the solutions of the following equations

\begin{array}{l} \frac{\partial G}{\partial p} = 2 p q^{2} = 0, \\ \frac{\partial G}{\partial q} = 2 q p^{2} + 8 A q^{3} + 16 α q^{7} - 4 h q = 0 \end{array}

(57)

From first equation of (57), we have $p = 0 or q = 0$ .

Similarly, from the second equation of (57), one obtains

q = 0, or p^{2} = 2 h - 4 A q^{2} - 8 α q^{6} .

(58)

Therefore, at $p = 0$ , one gets

h - 2 A q^{2} - 4 α q^{6} = 0 .

(59)

Hence, we have two real roots $\pm q^{*}$ , where

q^{*} = \sqrt{\frac{- 2 {(3)}^{2 / 3} + {(3)}^{1 / 3} {(9 h + \sqrt{24 + 81 h^{2}})}^{2 / 3}}{6 {(9 h + \sqrt{24 + 81 h^{2}})}^{1 / 3}}}

At $q = 0$ , the second equation in (58) produces $p = \pm \sqrt{2 h}$ .

According to the above, we have five singular points $(0,0), (\pm \sqrt{2 h}, 0)$ and $(0, \pm q^{*})$ .

To know the types of points, we follow the same method mentioned in subsection (3.3) and the following results are obtained

1. Points $(0,0)$ and $(\pm \sqrt{2 h}, 0)$ are parabolic points, where $M = G_{x x} G_{y y} - G_{x y} G_{y x} = 0$ .

2. Points $(0, \pm q^{*})$ are elliptic points, where $M = G_{x x} G_{y y} - G_{x y} G_{y x} > 0$ , as plotted in Figures 9 and 10.

Figure 9.

Shows the parabolic points $(0,0)$ and $(\pm \sqrt{2 h}, 0)$ .

Figure 10.

Represents two elliptic points $(0, \pm q^{*})$ and three parabolic points $(0,0)$ and $(\pm \sqrt{2 h}, 0)$ .

Numerical illustrations

In this section, the Poincaré surface section (PSS) is investigated numerically for the given invariant curves of the considered problem. The solution of the problem reveals a trajectory in the phase space $(x, y, \dot{x}, \dot{y})$ , and on the energy surface $h = c o n s t$ in which it can be introduced in the 3-D invariant hypersurface $(x, y, \dot{y})$ . The sequential points with $\dot{x} > 0$ where a cross orbit of the plane $x = 0$ in phase space demonstrates a locus of 2-D area maintaining mapping on the $(y, \dot{y})$ plane, which is known as Poincaré mapping.

It is known that the energy integral has the form

\frac{1}{2} ({\dot{x}}^{2} + {\dot{y}}^{2}) + V (x, y) = h,

(60)

\dot{x} = \dot{y} = 0

, equation (60) becomes

V (x, y) = h,

(61)

in which for different values of

h

, equipotential lines are obtained

Consider $x = 0$ in equation (61), the section of the curve of zero velocity $V (0, y) = h$ is obtained. Using the energy equation (60) to obtain ${\dot{x}}^{2}$ which should be non-negative, then the condition $\frac{1}{2} {\dot{y}}^{2} + V (x, y) \leq h$ is obtained. Considering $x = 0$ , one can use $(y, \dot{y})$ as coordinates on the 2-D surface, which are restricted by the following relation

\frac{1}{2} {\dot{y}}^{2} + V (0, y) \leq h .

(62)

Now, we have the following figures:

• For the first case (i), Figure 11 represents the equipotential lines for distinct values of $h$ , while Figure 12 shows the regular orbits located on the smooth curves in the Poincaré surface section.

• For the second case (ii), Figure 13 describes the equipotential lines for distinct values of $h$ , and Figure 14 reveals the regular orbits lie on the smooth curves in the Poincaré surface section.

• Figure 15 displays the zero velocity curve for the two integrable cases.

• Figures 16 and 17 explain the solution of $x$ and $y$ via $t$ for the first case and second case at $h = 0.5$ .

Figure 11.

Shows equipotential lines of the first case for the different values of $h$ .

Figure 12.

Represents invariant curves for the case (i) where (a) $h = 0.5$ and (b) $h = 10$ .

Figure 13.

Displays the equipotential lines of the case (ii) for the different values of $h$ .

Figure 14.

Shows the invariant curves for the case (ii) where (a) $h = 0.5$ and (b) $h = 10$ .

Figure 15.

Describes curve of zero velocities of the two integrable cases.

Figure 16.

Portraits the solution of $x$ and $y$ via $t$ for the first case at $h = 0.5$ .

Figure 17.

Portraits the solution of $x$ and $y$ via $t$ for the second case at $h = 0.5$ .

Finally, Figure 18 explains the trajectory on the energy surface $h = 0.5$

Figure 18.

Displays the trajectory on the energy surface $h = 0.5$ : (a) represents the case (i), (b) represents the case (ii).

Therefore, the numerical illustrations are introduced using PSS to confirm the integrability of the two cases (i) and (ii), where PSS are plotted in the $(y, \dot{y})$ plane as shown in Figures 12 and 14, which display PSS for the two cases (i) and (ii) at various values of energy constants $h$ and show that the structure of PSS is completely regular and there are no areas of stochastic motion, and this confirms the integrability of both systems (i) and (ii). Moreover, the numerical solutions of the EOM for the two integrable cases are presented in Figures 16 and 17 when $h = 0.5$ , and the trajectory is plotted in the $x y$ plane at the same value of $h$ , as shown in Figure 18.

Conclusions

In the current paper, the complete picture of our problem has been studied: The topological type of the level set is a torus, or empty set, where we constructed the topological translation of the trajectory by applying the phase portrait. The singular points of the separated functions, which are elliptic and hyperbolic points, allow us to understand the trajectories and the behaviors of these points. The elliptic points are stable, and the others are unstable. It can be seen that the separated functions in the first case are transformed from the sixth degree to the third one, and the separated functions in the second case are transformed from the eighth degree to the fourth one, where the quadrature can be integrated through the elliptic integral. The first and second cases are calculated numerically using the Poincaré surface section to confirm our work, where the trajectories lie on the closed curves for different values of the energy constant, which are called invariant curves.It must be noted that the original system (2) can be solved using non- perturbative method.^47,48 Therefore we can obtain this solution in future.

Footnotes

Author contributions

FM El-Sabaa: Supervision, Methodology, Validation, Data curation, Visualization, Reviewing; TS Amer: Supervision, Methodology, Conceptualization, Visualization, Writing, Reviewing.; HM Gad: Writing - original draft, Investigation, Data curation, Editing; MA Bek: Supervision, Methodology, Validation, Reviewing.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Tarek S Amer

Hadeer M Gad

References

Aronlod

. Mathematical methods of classical mechanics. New York, NY: Graduate Texts in Mathematics Springer-Verlag, 1989.

Ramani

Grammaticos

Bountis

. The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys Rep 1989; 180: 159–245.

Rajasekar

Raj

. The Painlevé property, integrability and chaotic behaviour of a two-coupled Duffing oscillators. Pramana - J Phys 1996; 47: 183–198.

Elmandouh

. On the dynamics of armbruster guckenheimer kim galactic potential in a rotating reference frame. Astrophys Space Sci 2016; 361: 182–194.

Arkhangel’skii

. On the motion about a fixed point of a fast spinning heavy solid. J Appl Math Mech 1963; 27: 1314–1333.

El-Barki

Ismail

. Limiting case for the motion of a rigid body about a fixed point in the Newtonian force field. Z Angew Math Mech 1995; 7511: 821–829.

Ismail

. The motion of a fast spinning disc which comes out from the limiting case. Comput Methods Appl Mech Eng 1998; 161: 67–76.

Ismail

Amer

. The fast spinning motion of a rigid body in the presence of a gyrostatic momentum. Acta Mech 2002; 154: 31–46.

Amer

. The rotational motion of a symmetric rigid body similar to Kovalevskaya's case. Iran J Sci Technol Trans Sci 2018; 42(3): 1427–1438.

10.

Amer

El-Kafly

, et al. Modelling of the rotational motion of 6-DOF rigid body according to the Bobylev-Steklov conditions. Results Phys 2022; 35: 105391.

11.

Farag

Amer

. The periodic solutions of a symmetric charged gyrostat for a slightly relocated center of mass. Alex Eng J 2022; 61: 7155–7170.

12.

.Periodic solutions of quasilinear autonomous systems which have first integrals. J Appl Math Mech 1963; 272: 551–557.

13.

Elfimov

. Existence of periodic solutions of equations of motion of a solid body similar to the Lagrange gyroscope. J Appl Math Mech 1978; 42(2): 262–269.

14.

Amer

. Motion of a rigid body analogous to the case of euler and poinsot. Analysis 2004; 24: 305–315.

15.

Amer

. On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector. Nonlinear Dyn 2008; 54: 249–262.

16.

Amer

. On the motion of a flywheel in the presence of attracting center. Results Phys 2017; 7: 1214–1220.

17.

Amer

. Modelling and analyzing the rotatory motion of a symmetric gyrostat subjected to a Newtonian and magnetic fields. Results Phys 2021; 24: 104102.

18.

Akulenko

Leshchenko

Chernousko

. Perturbed motions of a rigid body that are close to regular precession. Izv Akad Nauk SSSR MTT 1986; 21(5): 3–10.

19.

Akulenko

Leshchenko

Kozochenko

. Evolution of rotations of a rigid body under the action of restoring and control moments. J Comput Syst Sci 2002; 41(5): 868–874.

20.

Amer

Abady

. On the motion of a gyro in the presence of a Newtonian force field and applied moments. Math Mech Solids 2018; 23(9): 1263–1273.

21.

Ismail

Amer

El Banna

, et al. Electromagnetic gyroscopic motion. J Appl Math 2012; 2012: 1–14.

22.

Amer

. On the rotational motion of a gyrostat about a fixed point with mass distribution. Nonlinear Dyn 2008; 54: 189–198.

23.

Amer

. The rotational motion of the electromagnetic symmetric rigid body. Appl Math Inf Sci 2016; 10(4): 1453–1464.

24.

J-H

Hou

W-F

Qie

, et al. Hamiltonian-based frequency-amplitude formulation for nonlinear oscillators. FU Mech Eng 2021; 19(2): 199–208.

25.

Feng

G-Q

Niu

J-Y

. An analytical solution of the fractal toda oscillator. Results Phys 2023; 44: 106208.

26.

Dutt

Lakshmanan

. Application of coherent state representation to classical and coupled anharmonic oscillators. J Math Phys 1976; 17: 482–484.

27.

Lakshmanan

Prabhakaran

. The energy levels of ananharmonic oscillator. Lettere Al Nuovo Cimento (1971-1985) 1973; 7: 689–693.

28.

Llibre

Makhlouf

. Periodic orbits of the generalized Friedmann–Robertson–Walker Hamiltonian systems. Astrophys Space Sci 2013; 344: 45–50.

29.

Ortega

. Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five. Astrophys Space Sci 2016; 361: 1–8.

30.

Uribe

Quispe

. Existence and stability of periodic orbits for a Hamiltonian system with homogeneous potential of degree five. Differ Equ Dyn Syst 2020: 1–23.

31.

El-Sabaa

Amer

Gad

, et al. Existence of periodic solutions and their stability for a sextic galactic potential function. Astrophys Space Sci 2021; 366(8): 74.

32.

Lakshmanan

Sahadevan

. Painlevé analysis, lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type. Phys Rep 1993; 224(1–2): 1–93.

33.

Fomenko

. Integrability and nonintegrability in geometry and mechanics. Berlin, Germany: Springer Science & Business Media, 2012.

34.

Doob

Grothendieck

Heinz

. Handbook of elliptic integrals for engineers and scientists. Berlin and Heidelberg: Springer, 1971.

35.

El-Sabaa

Hosny

Zakria

. Bifurcations of Liouville tori of a two fixed center problem. Astrophys Space Sci 2018; 363: 77.

36.

Vozmischeva

. The two center and Lagrange problem in the Lobachevsky space. In: International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 1–10 September 1999, pp. 283–298.

37.

Vozmischeva

. Classification of motions for generalization of the two centers problem on a sphere. Celest Mech Dyn Astron 2000; 77: 37–48.

38.

Vozmischeva

. The Lagrange and two-center problems in the Lobachevsky space. Celest Mech Dyn Astron 2002; 84: 65–85.

39.

Waalkens

Dullin

Richter

. The problem of two fixed centers: bifurcations, actions, monodromy. Physica D: Nonlinear Phenom 2004; 196: 265–310.

40.

El-Sabaa

Hosny

Zakria

. Bifurcations of Armbruster Guckenheimer Kim galactic potential. Astrophys Space Sci 2019; 364: 34–39.

41.

Gavrilov

. Bifurcations of invariant manifolds in the generalized Hénon-Heiles system. Physica D: Nonlinear Phenomena 1989; 34: 223–239.

42.

Kharlamov

. Bifurcation of common levels of first integrals of the Kovalevskaya problem. J Appl Math Mech 1983; 47: 737–743.

43.

Ouazzani-Th

Kharbach

Ouazzani-Jamil

. Phase space topology and bifurcation of Liouville torii in the Goryatchev-Tchaplygin top. MJCM 1999; 2: 91–99.

44.

Dubrovin

Fomenko

Novikov

. Modern geometry-methods and applications, Part II: the geometry and topology of manifolds. New York, NY: Springer, 1985.

45.

Fomenko

. Visual geometry and topology. Berlin, Germany: Springer, 1994.

46.

El-Sabaa

. Bifurcation of Kovalevskaya polynomial. Int J Theor Phys (Dordr) 1995; 34: 2071–2083.

47.

El-Dib

. Estimated the frequencies of a coupled damped nonlinear oscillator with the non-Perturbative method. Journal of Low Frequency Noise, Vibration and Active Control 2023; 42(1): 110–125.

48.

El-Dib

Elgazery

Alyousef

. A heuristic approach to the prediction of a periodic solution for a damping nonlinear oscillator with the nonperturbative technique. Front. Phys 2023; 11: 1122592.