Study On Periodic Solutions For Several Coupled Nonlinear Dynamics Systems

In the past few decades,with the development of science and technology and the con-tinuous deepening of theoretical research,nonlinear problems have aroused widespread concern.The research of nonlinear dynamics mainly focused on three aspects of bifurca-tion,chaos and soliton.It is hard to get the exact solutions for the nonlinear equations,Hence it becomes critical to seek the approximate analysis solution.At present,the methods of studying the nonlinear system include perturbation method,harmonic bal-ance method,multiple scales method,Lindstedt-Poincare methods,homotopy analysis method and so on.This paper mainly studies the dynamic response behavior of nonlinear systems.The first chapter introduces the research background and the present research status of the nonlinear systems.In the second chapter,the primary resonances of the van der Pol system with parameter excitation is studied by using the multi-scale method and the ho-motopy analysis method.Firstly,we study the nonlinear dynamic response of the system when the ratio of internal resonance are different,and obtain the four-dimensional average equation of the rectangular coordinate form by the multiple scales method,thereby the periodic motion of the system is found;Secondly,using the homotopy analysis method,we obtain the four periodic solutions,where there are two sets of in-phase periodic solutions and two sets of out-of-phase periodic solutions.Finally,we get the frequency response curves using these two methods,and found that these differences were negligible.The third chapter uses the multi-frequency homotopy analysis method to study two-degree-of-freedom coupling Duffing system.On the one hand,we construct the step of solving two-degree-of-freedom nonlinear dynamical system by multi-frequency homotopy analysis method,and get the single-period solution and double-period solution of this system.On the other hand,comparing the periodic solutions obtained by multi-frequency homotopy analysis method and 4th-order Runge-kutta method,we find that the approximate so-lution agrees well with the numerical solution.The fourth chapter analyzes the chaotic behavior of discrete nonlinear dynamical systems.