Slow–fast dynamics in a perturbation model of double pendulum system with singularity of triple zero eigenvalues.

Slow–fast dynamics such as bursting behaviors are common in many physical and engineering systems. In the previous study, some focused on the bursting behavior caused by the codimension-1 bifurcations, and others focus on the bursting behavior due to the particular structures. However, systems under critical conditions may exhibit many complicated dynamics due to the high co-dimensional bifurcations. Our research aims to investigate the bursting oscillations near a triple zero eigenvalues singularity. A perturbation model of the double pendulum system with external excitation is taken as an example and investigated the dynamical mechanism of bursting behaviors. Because of the order gap between the exciting frequency and natural frequency, the perturbation model with the external excitation can be regarded as a generalized autonomous system. By overlapping the transformed phase portrait and the equilibrium branch, four types of bursting oscillations are determined: fold/fold type, zero-Hopf/zero-Hopf type, symmetrical zero-Hopf/sup-Hopf/fold-cycle type, and symmetrical zero-Hopf/sup-Hopf/fold-cycle/sub-Hopf type. Primarily, we find that due to the singularity of compound fold conditions, many bifurcations of the limit cycle occur, which cause many complex dynamics such as 2-D tori bursting, breaking of symmetric structure, and chaotic bursting. These results play an essential role in understanding the stability of a system with high co-dimensional bifurcation conditions. They can be expected to provide a theoretical basis for formulating a control strategy. [ABSTRACT FROM AUTHOR]