Application Of The Perturbation-incremental Method In Solving Plane Differential System

In the theory and application of plane differential systems,the qualitative and quantitative research of limit cycles and homoclinic(heteroclinic)orbits has always been the focus of scholars.In this thesis,the perturbation-incremental method is used to quantitatively analyze several plane differential systems,and the approximate analytical solutions of homoclinic orbits and limit cycles of systems are obtained.This paper has been divided into six chapters as follows:The first chapter summarizes the research backgrounds,research significances,current situation and development trend of this paper.In second chapter simply introduce the limit cycle,homoclinic(heteroclinic)orbit,Fourier series and harmonic balance method.In the third chapter,we simply introduce the idea of perturbation-incremental method which combines the perturbation method with the incremental method,and describe the specific calculation process,that is,from the zero-order approximate solution obtained by the perturbation method to the approximate analytical solutions obtained by the parameter incremental method with arbitrary parameters.In the fourth chapter,the homoclinic orbits of a class of isochronous systems are studied by using the perturbation-incremental method.The zero-order perturbation solution and the approximate analytical solutions of different parameter states are obtained.The two-dimensional streamline and phase diagrams of different parameter states are also given.In the fifth chapter,two kinds of plane differential systems with isochronous centers are studied by using the perturbation-incremental method.A pair of symmetric homoclinic orbits of systems are obtained,and the approximate analytical expressions in the zero-order perturbation state and an iteration process are given.Finally,the two-dimensional streamline and phase diagrams in the corresponding states are given.In the sixth chapter,the application of perturbation-incremental method in solving the limit cycle of Duffing-Van der Pol equation is introduced.The zero-order perturbation solution of the system,the approximate analytical expression and phase diagrams of the limit cycle with corresponding parameters are obtained.Finally,the effectiveness of the method is verified by comparing with the results obtained by numerical integration method.