Using Homotopy Theory And The Approximate Solution Of Nonlinear Dynamical Systems

With the development of society and science technology, the investigation on nonlinear dynamics systems is more and more important. At the moment, most of the study is concentrating on weakly nonlinear dynamics systems. However, strongly ones which don't contain small parameters are rarely paid attention, even without universal approach. For these equations, homotopy theory is applied to nonlinear dynamics systems. The homotopy analysis technique based on the parameter expansion ( PE-HAM ) and homotopy perturbation method (HPM ) combined with method of linearization, parameter expansion and modified Lindsted-Poincare are applied to solving strongly nonlinear dynamics systems. Some examples are given to prove that the method of homotopy theory is very effective and convenient to study nonlinear dynamics systems. The contents of this paper are:In chapter 1, we introduce the development of nonlinear dynamics systems status, summarize up the current situation of strongly nonlinear dynamics systems and a sea -air oscillator model for the ENSO. And, simply introduce the basic idea and development history of homotopy theory;In chapter 2, the basic idea of the homotopy analysis technique based on the parameter expansion( PE-HAM ) is proposed to study conservative Duffing oscillator with 5th order nonlinearity and the strongly dissipated oscillation system with harmonic excitation respectively to obtain analytically approximate solution. The numerical simulation proves that the approximate solution obtained by this method is of high accuracy;In chapter 3, firstly, a systematic and regular method is summarized, which can be applied to solve approximate solutions and periods of strongly nonlinear oscillation problem by using homotopy perturbation method combining method of linearization. And an approximate period formula for a category of Duffing equation is obtained. At the same time, we prove that even its first order approximations are of high accuracy. Secondly, the homotopy perturbation method combining the parameter-expanding method is applied to solving the strongly nonlinear problems. Some examples are given to prove that the approximate solutions obtained by this method are of high accuracy;In chapter 4, the homotopic mapping is introduced to solve a sea-air oscillator model for the ENSO. Under certain conditions, from the perspective of mathematics and physics, combining parameter expansion method with homotopy perturbation method, we successfully obtain the approximate solutions of two types of sea-air oscillator model. Therefore, we can directly discuss certain quantitative characteristics related to physics and find its principles in order to know more about it;Finally, in chapter 5, concluding remarks are employed to close this paper.