On the motion of some dynamical systems

Abstract

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