Influence of the Coexisting Attractors on the Slow-Fast Behaviors in the Fast Subsystem

This paper is devoted to exploring the effect of the coexisting attractors in the fast subsystem of a full nonlinear system on the complex dynamics. As an example, a modified 3D van der Pol-Duffing circuit is discussed, in which the coexistence of five attractors including a hidden one was reported in 2020. By introducing a parametric excitation, when the exciting frequency is far less than the natural frequency, a slow-fast model with the two-scale coupling is established in the frequency domain. Regarding the exciting term as a slow-varying parameter, all the attractors as well as the bifurcations of the fast subsystem are derived. The coexistence conditions as well as the coexisting attractors, including the equilibrium points and limit cycles, are presented. With the variation of the exciting amplitude, different types of bursting oscillations are observed, the mechanism of which is revealed using a modified slow-fast analysis method, by combining the transformed phase portraits and the attractors as well as the bifurcations. It is found that the coexisting attractors of the fast subsystem may lead to coexisting bursting solutions in the full system, in which a trajectory may visit one of the attracting basins of the attractors, or result in a merged bursting motion, which may visit the attracting basins in turn. The influence of some attractors and the associated bifurcations may vanish in the case that the trajectory passes directly across the parameter interval corresponding to the attractors because of the inertia of the motion; this may also occur in the case that the trajectory does not visit the associated attracting basins, but it starts initially from the basins. It should be pointed out that, when the exciting frequency is small enough, the disappeared effect of stable attractors may reoccur. Furthermore, if the parameter interval between two types of codimension-1 bifurcation points is short enough, the trajectory may behave in a combination of two bifurcations, which is somewhat similar to that caused by a codimension-2 bifurcation.

Automated summary

This paper explores the impact of coexisting attractors in the fast subsystem of a nonlinear system on complex dynamics. By analyzing a modified 3D van der Pol-Duffing circuit, the coexistence of five attractors with a hidden one is reported. The study establishes a slow-fast model with two-scale coupling in the frequency domain by introducing a parametric excitation. The coexistence conditions and coexisting attractors including equilibrium points and limit cycles are derived. Different types of bursting oscillations are observed with varying exciting amplitude, and the mechanism behind it is revealed using a modified slow-fast analysis method. The paper also suggests that the presence of coexisting attractors in the fast subsystem can lead to coexisting bursting solutions in the full system, with trajectories visiting different attracting basins or resulting in merged bursting motions.