关键词: chaos, bifurcation, homoclinic orbit, heteroclinic orbit, simple zeros, the Melnikov function

Abstract: In this paper, we use the Melnikov function method to study a kind of soft Duffing equations^[1]x+Af(x,x)+x-x^2k+1=r[M(x,x)cosωt+N(x,x)sinωt](k=1,2,3…) and give the condition that the equations have chaotic motion and bifurcation. The method used in this paper is effective for dealing with the Melnikov function integral of the system whose explict expression of the homoclinic or heteroclinic orbit cannot be given.

Key words: chaos, bifurcation, homoclinic orbit, heteroclinic orbit, simple zeros, the Melnikov function