Homoclinic bifurcation of the strongly nonlinear oscillation system by the normal form method

Abstract

The available range of the homoclinic bifurcation criterions are extended from the weakly nonlinear oscillation system to the strongly nonlinear oscillation system. It combines the analysis method of the strongly nonlinear oscillation system with the former criterions based on the improved complex normal form method. The periodic solution of this kind of system with a single degree of freedom is obtained by introducing the fundamental frequency under determination into the complex normal form computation. Then two different analytical criteria to predict the critical values of homoclinic bifurcation are adapted to the new system. It includes the undertermined fundamental frequency approaching zero and the collision of the periodic orbit with the saddle point. The results derived from different methods are compared in the specific systems with numerical simulation to testify the correctness and efficiency of the theoretical results.