BURSTING OSCILLATIONS AS WELL AS THE CLASSIFICATION IN THE FIELD WITH CODIMENSION-3 FOLD-FOLD-HOPF BIFURCATION

Abstract

The dynamical systems with the coupling of different scales observed widly in engineering problems often behave in the bursting oscilltions, characterized by the alternations between large-amplitude oscilltations and small-amplitude oscillations, the generation mechanism of which has been one of the hot topics in nonlinear science at home and abroad. The traditional geometric pertubation method can be employed to explore the mechanism of the oscillations only in the systems with two scales in time domain, which can not be directly used to investigate the interaction between different scales in frequency domain. Meanwhile, most of the results are obtained in the vector fields with codimension-1 fold or Hopf bifurcations. Here we focus on the complicated behaviors in the vector field with codimension-3 fold-folfd-Hopf bifurcation when two scales in frequency domain exist. Based on the normal form as well as its universal unfolding with the nonlinear terms up to the third order, all the possible bifurcations are derived, which divide the two unfolding parameter plane into several regions with different dynamics. By introducing a slow-varying periodic excitation instead of one of the unfolding parameters, two types of routes for the tarjectory visiting those regions can be observed, which may result in four classes of bursting oscillations, i.e., periodic Hopf/LPC, Hopf/LPC/Hopf/LPC, fold/LPC/Hopf/Homoclinic and fold/LPC bursting attractors with the variation of the exciting term. It is found that there may exist delay between the locations of the theoretical bifurcation points and the real bifurcation points on the trajectory. The delay may increase with the exciting amplitude, since the inertia of the movement along the equilibrium states may increase. Especiallty, when the exciting amplitude increases to an extent, the trajectory may pass acorss the corresponding regions before the related bifurcation occurs, which leads to the qualitative change of the oscillations. It is shown that, the slow-fast effect with local bifurcations can be investigated by using the periodic perturbation on the unfolding parameters in the normal form of the vector field, which can therefore to present all possible types of bursting patterns as well as the mechanism.