Slow-Fast Dynamics in a Non-smooth Vector Field with Zero-Hopf Bifurcation

Purpose The purpose of this paper is to study the influence of non-smoothness on the slow-fast dynamics of a vector field with codimension-2 zero-Hopf bifurcation at the origin. Upon the analysis of the equilibrium branches and their bifurcations of the generalized autonomous subsystems in two regions, with the increase of the exciting amplitude, different equilibrium branches and bifurcations may involve the vector field of the full system, which may lead to different types of bursting oscillations. Four special cases are considered, where the bursting oscillations may vary from two coexisted asymmetric attractors to an enlarged symmetric attractor, the mechanism of which is explored by overlapping the transformed phase portrait with the equilibrium branches as well as the bifurcations of the fast subsystem. Methods By regarding the whole exciting term as a slow-varying parameter, the bifurcations of the generalized autonomous non-smooth fast subsystem can be derived, which are used to account for the mechanism of the bursting attractors in the full system by explore the influence of the bifurcations on the alternations between the quiescence and the repetitive spiking oscillations. Results and conclusion In the study, the influence of smooth and non-smooth bifurcations on the slow-fast dynamics of a vector field with zero-Hopf bifurcation at the origin is investigated. It is found that sliding phenomenon along the boundary can be observed on the trajectory of bursting attractor because of the non-smoothness. Because of the coexistence of two stable equilibrium branches of the generalized autonomous fast subsystem, the trajectory may have two choices, which may lead to symmetric-breaking bifurcation between two coexisted asymmetric non-smooth bursting attractor and an enlarged symmetric bursting attractor. The dependent and independent relationship between two groups of state variables may result in synchronized and non-synchronized oscillations between the associated state variables.