Chaos and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation.

Abstract

Chaotic dynamics and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and a harmonic excitation are investigated analytically in this paper. With the Fourier series, the system is expanded to the equivalent smooth system, and chaos arising from heteroclinic intersections is studied with the Melnikov method. The chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena, such as chaotic bands for some excitation frequencies, are presented. The relationship between the frequency range of chaotic bands and the amplitude of the excitation as well as the damping is obtained analytically. Particularly, for some system parameters satisfying a particular relationship, chaos cannot occur for any excitation amplitudes or frequencies. Subharmonic bifurcations are investigated with a subharmonic Melnikov method. It is analytically proved that the system may undergo chaotic motions through infinite or finite odd order subharmonic bifurcations. Numerical simulations are given to verify the chaos threshold and revolution from subharmonic bifurcations to chaos obtained by analytical methods.