Symmetric and asymmetric bursting oscillations in a hybrid van der Pol-Duffing-Rayleigh system

Inspired by the excited harmonic oscillations in complex dynamical models, this paper addresses the influence of low-frequency amplitude-modulated excitation on oscillating characteristics of a hybrid van der Pol-Duffing-Rayleigh system (HVDRS) theoretically and numerically. First, we analyze the dynamical behaviors of the HVDRS without the low-frequency amplitude-modulated excitation, and only one bursting mode named symmetric compound “fold/supHopf-fold/supHopf” bursting oscillation with two PSEs can be observed. By regarding the slowly time-varying cosine harmonic function as a generalized slow parameter, we find that the amplitude-modulated excitation can significantly influence the equilibrium curve characteristics and bursting oscillations types. Two excited harmonic oscillations can be identified by combining the phase portraits on the (δ, x) space with bifurcation diagrams. The first is asymmetric compound bursting oscillations when the coupling factor n is equal the odd numbers, and we note that the bursting structures become increasingly complex as n increases. The second is compound bursting oscillations with respect to the origin symmetry when the coupling factor n is equal the even numbers. We point out that the generation of the asymmetric and symmetric bursting oscillations are mainly caused by the bifurcation structures and bifurcation delay effect. The validities of the conclusions are verified by numerical simulations.