The High Order Melnikov Function For The Time-Periodic Equations With Heteroclinic Orbit And Its Chaotic Dynamics

In the study of ordinary differential equations,a question of concern is whether the equations appear chaotic dynamics.We usually use the simple zero point of the first order Melnikov function to determine the transversal intersections of the stable manifold and the unstable manifold of the saddle point,so that we can determine the chaotic dynamics of the system.It is an important question that how to check the chaotic dynamics of the system if the first order Melnikov function is always zero.To solve this problem,the high order Melnikov functions are used.At present,the high order Melnikov function for time-periodic equation with a homoclinic orbit has been solved.This dissertation studies the theoretical derivation and calculation of the high order Melnikov function for time-periodic equation with a heteroclinic orbit.We give two theorems to determine the chaotic dynamics based on the second order Melnikov function.The content and structure of this paper are as follows:The first chapter is the background.In this part we briefly introduce Smale horseshoe,homoclinic tangles,heteroclinic tangles and the high order Melnikov function for time-periodic equation with a homoclinic orbit.The second chapter is the derivation of the high order Melnikov function for timeperiodic equation with a heteroclinic orbit.First,we introduce the concept of splitting distance.Then we use the new variables to transform the equation into a normalized equation.After that,we solve the normalized equation.We derive the high order Melnikov function from the integral form.Finally,We give two theorems to determine the chaotic dynamics by the second order Melnikov function.We have obtained the chaotic dynamics of the Duffing equation by the theorems.The third chapter is the prospect.We give the idea of the high order Melnikov function when the first order Melnikov function and the second order Melnikov function are always zero.In particular,we give the calculation process of the third order Melnikov function.