Study On Liénard Equation And Duffing Equation Under Nonresonance Condition

In this paper, a Lienard type equation is deduced from the basic equation of atmospheric motion in stratified atmosphere, and then we consider the existence of the periodic solutions for Lienard and Duffing type equations. For the class of nonlinear second order ordinary differential equations (ODEs), we firstly consider one of their special one-dimensional forms, and then prove the existence of solution to its two-point boundary value problem in the light of prior estimate and Schauder fixed point theorem. We also show the exact solutions for the special equations as an example and plot the numerical solutions by Mathematica. Secondly, we extend the above equation to a general form and obtain the sufficient condition for the existence of the periodic solution. Thirdly, we consider the n-dimensional Duffing type equation with damping term. By virtue of homeomorphism, extension and fixed point method, we discuss the existence of periodic solution under the nonresonance condition. Finally, we give a simple condition for nondegeneracy of symmetric bilinear forms on infinite dimensional vector spaces. We apply this condition and elementary properties of Fourier series to prove a uniqueness theorem for periodic solutions of a class of second order nonlinear ordinary differential systems.