Amplitude death, oscillation death, and stable coexistence in a pair of VDP oscillators with direct–indirect coupling

In this paper, we investigated the dynamics of a pair of VDP (Van der Pol) oscillators with direct-indirect coupling, which is described by five first-order differential equations. The system presented three types of equilibria including HSS (homogeneous steady state), IHSS (inhomogeneous steady state) and NPSS (no-pattern steady state). Employing the corresponding characteristic equations of the linearized system, we obtained the necessary conditions for the pitchfork and Hopf bifurcations of the equilibria. Further, we illustrated one-dimensional bifurcation and phase diagrams to verify theoretical results. The results show that the system exhibited two types of oscillation quenching, i.e., amplitude death (AD) for HSS equilibria and oscillation death (OD) for IHSS equilibria. In some special regions of the parameters, the system proposed multiple types of stable coexistence including HSS and IHSS equilibria, periodic orbits or quasi-periodic oscillations.