Nonlinear Dynamics Study Of Two Coupled Van Der Pol Systems

Nonlinear equations has been a mathematician and physicist focuses on problems of van der Pol and coupling system is a basic model of the nonlinear field.In recent years,the van der Pol equation of coupling analytical approximate solution of dynamic character and the related problem has become an important field at home and abroad,related literature and results emerge in endlessly.The purpose of this paper is to study two kinds of coupling van der Pol system dynamic character of simple.The first kind of equations for the three degree of freedom coupling van der Pol equation,we mainly use the homotopy analysis method research such van der Pol oscillator loop system and the approximate expressions of the periodic solutions.For the second type of equation,which is the two-degree-of-freedom delay-coupled van der Pol equation,we mainly take coupling intensity alpha and time-delay tau as bifurcation parameters to study its 5:7resonant doubled Hopf bifurcation.This thesis mainly divided into three chapters.The first chapter simply introduced the research background and research status of this article.The second chapter introduces the homotopy analysis method of the theory,then the homotopy analysis method was applied to three degrees of freedom coupling van der Pol oscillator ring and find out the analytical approximate solutions.In particular,we divide the system into four categories: first,all the stator synchronous movement;Second,the two oscillators move simultaneously,while the third vibrator moves in an independent manner(except that it vibrates in the same period as the second oscillator).Third,the motion of adjacent oscillators on the ring that are one third of a cycle apart from each other;Fourth,two oscillator phase 1/2 cycle,while the third vibrator 2 times on their frequency vibration.The use of four different types of van der Pol oscillator ring to show the effectiveness of the homotopy analysis method and widely applied,and the method are compared with those of the numerical integral method,the results showed that the analytical solution and numerical solution of high.In the third chapter,we first according to the distribution of eigenvalues for two degrees of freedom coupled with time-delay van der Pol equation of double Hopf bifurcation parameter of the critical value.After that,The 5:7 bihopf bifurcation diagram of the system was analyzed by using the canonical equation obtained by the multi-scale method,the 6 regions divided in the parameter plane were obtained,and the dynamic behaviors in different regions were analyzed.Finally,the effectiveness of the method was verified by numerical simulation in different regions.