On the occurrence of bursting oscillations in the damping Helmholtz–Rayleigh–Duffing oscillator with slow-changing parametrical and external forcings

Abstract

Multiple timescale effects can be reflected bursting oscillations in many classical nonlinear oscillators. In this work, we are concerned about the bursting oscillations induced by two timescale effects in the damped Helmholtz-Rayleigh-Duffing oscillator (written as DHRDO for short) excited by slow-changing parametrical and external forcings. By using trigonometric function variation and authenticating the slow excitations as a slowly varying state variable, the time-varying DHRDO can be rewritten as a new time-invariant system. Then, the critical conditions of some typical bifurcations are presented by bifurcation theory. With the help of bifurcation analyses, six bursting patterns, i.e., ‘Hopf/Hopf-Hopf/Hopf’ bursting, ‘fold/Homoclinic-Hopf/Hopf’ bursting, ‘fold/Homoclinic/Hopf’ bursting, ‘Hopf/fold/Homoclinic/Hopf’ bursting, ‘Hopf/Homoclinic/Homoclinic/Hopf’ bursting and ‘Hopf/Homoclinic/Hopf-Hopf/Homoclinic/Hopf’ bursting, are explored by the slow/fast decomposition method and the other techniques. Our findings provide different forms of the excited state oscillation modes as well as the bursting patterns. In addition, we use the numerical simulation to prove the correctness of the theoretical analyses.