الفهرس 
Bifurcations of Liouville tori of a coupled sextic anharmonic oscillatorsOnly 14 pages are availabe for public view
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Abstract
Summary
The main objectives of this thesis is to
1- Introduce the topological analysis of the problems under study,
2- Investigate the periodic solutions in terms of Jacobi’s elliptic functions,
3- Study the phase portrait and determine the singular points,
4- Employ Poincaré surface section (PSS) to prove that the motion is regular in the integrable cases of the problem of sextic anharmonic oscillators,
5- Use the averaging theory to discuss the existence of periodic solutions,
6- Get the family of periodic solutions and study their stability using Lyapunov’s theorem,
7- Apply Kovalevskaya exponents (KE) method to discuss the non-integrability in a cosmological scalar field,
8- Discuss the governing system of equations of motion (EOM) utilizing Lagrange’s equations and using the multiple scales technique (MST) to obtain the approximate solutions up to the third approximation,
9- Study the modulation equations (ME) providing the solvability conditions that correspond to the arising resonance cases for stable solutions, and apply the criterion of Routh-Hurwitz to check the stability of the fixed points at the steady-state solutions.
This thesis consists of six chapters:
In chapter (1), 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. Therefore, 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 the singular points and to classify their types in addition to the graphical representation for each of them. Finally, the numerical illustrations are introduced using the PSS to emphasize the problem’s integrability.
Chapter (2)
The goal of this chapter is to investigate the periodic solutions for the Hamiltonian function that governs the sextic galactic potential function in accordance with two different methods. The first method is applied using the averaging theory of first-order. The sufficient conditions on the parameters for the stability are given and analyzed. The numerical examples of families of periodic orbits are introduced. Meanwhile, the second method is presented using Lyapunov’s theorem for the holomorphic integral, where the periodic solutions depend on the type of the equilibrium points
Chapter (3)
This chapter is concerned with the study of two integrable cases. The first case is a generalized Hénon-Heiles (GHH) system, and the second one is a quartic potential. The bifurcation of Liouville tori is introduced for each case. Moreover, the Jacobi elliptic functions are used to obtain the periodic solution. The phase portrait is presented and the singular points are classified. Additionally, we construct the periodic solutions for both cases based on the Lyapunov theorem.
Chapter (4) aims to study KE to show the non-integrability of the Hamiltonian system that describes the scalar field universe. In addition to studying Lyapunov’s theory of holomorphic integral, in which periodic solutions depend on equilibrium points’ type.
In chapter (5), we are going to study a general model of a harmonically excited damped spring pendulum in which its pivot point moves in a Lissajous curve. Moreover, a rigid body is considered to connect with the end point of the pendulum. Lagrange’s equations are used to obtain the governing system of motion that comprises of three nonlinear differential equations from second-order. The ME and the analytical solutions up to the third-order of approximation are obtained utilizing MST. The steady-state solutions are achieved and the graphs of time histories and resonance curves are plotted. The numerical results are obtained using the algorithms of Runge–Kutta fourth-order and compared with the analytical ones. This comparison shows high agreement between them and reveals the good accuracy of the used perturbation technique. Mathematica software is used to calculate the equations and generate the presented graphs.
Chapter (6)
This chapter concerns a novel problem of the planar motion of dynamical system consisting of two degrees of freedom (DOF) double rigid-body-pendulum (RBP), whose pivot point has been constrained to move in a Lissajous curve. The nonlinear differential EOM are derived utilizing the second kind of Lagrange’s equations. Higher estimations of the analytical solutions are achieved using the MST. These solutions are compared with the numerical ones, which are obtained using the fourth-order Runge-Kutta method, to show the high accuracy between them. In light of the categorized resonance cases; the solvability conditions and the ME are obtained. Criterion of Routh-Hurwitz is applied to check the stability of the fixed points at the steady-state solutions. The nonlinear stability analysis of the ME is examined and discussed through the stability and instability ranges of the plotted curves of the frequency responses. The times histories of the solutions are graphed as well as the curves of resonance to explore the significance of various values of the physical parameters on the systems’ behaviour. The stability and instability areas have been examined for the cases of external and combination resonances. One of the most important applications of this work in our daily life is to utilize the vibrational motion for the robots devices, pumps compressors, rotor dynamics, and transportation devices.