Existence Of Quasi-periodic Solutions Of Two Coupled Mathieu-duffing Equations

This paper mainly study the existence of quasi-periodic solutions of two cou-ple Mathieu-Duffing equations,and one of the equations with external forces,namely the harmonic driving forces.In recent years,the two equations have very important applications in mechanical engineering,engineering and the automobile suspension system.We will apply KAM theory to analyse the existence of quasi-periodic solutions of two couple Mathieu-Duffing equations.Firstly,we introduction the background of the equations,and give a KAM theorem,to analyse the existence of quasi-periodic solutions of the above equation in the four-dimensional phase space.This paper is to study the system through a series of reversible transforma-tions into a normal form to which can apply KAM theory,and then the normal form is reduced by the Newton iteration method.There will be small divisor problems in the iterative procedure,the parameters must meet the Diophantine condition and we have to estimate the measure.Finally,the reduced system has a quasi-periodic solution near the equilibrium solution for most of parameters with-in the considered parameter set.The variable transformations are reversible,so the original system also has a quasi-periodic solution for most of the parameters.