Relaxation oscillation patterns induced by amplitude-modulated excitation in the Duffing system

Relaxation oscillations are typical fast–slow behaviors in the excited Duffing system. The present paper focuses on the effects of amplitude-modulated excitation on relaxation oscillations. We show that the quasi-static process of the relaxation oscillations can be prolonged in the presence of amplitude-modulated excitation. Then, a two-parameter bifurcation set is plotted to investigate the transition of the relaxation oscillations. In particular, we find that the hysteresis with four fold points can be split into the one with eight fold points. Based on this, dynamical mechanism underlying the appearance of a compound relaxation oscillation patterns is revealed. Our study shows that amplitude-modulated excitation has great effects on the fast–slow dynamics. •The effect of periodic excitation and amplitude-modulated excitation are compared.•Transition of compound bursting is explored by two-parameter bifurcation analysis.•Dynamical mechanism of the relaxation oscillations with prolonged quasi-static processes is investigated.•Dynamical mechanism of the relaxation oscillations multiple jumps is investigated.