On the stability analysis of a pair of van der Pol oscillators with delayed self-connection, position and velocity couplings

In this paper, we perform a stability analysis of a pair of van der Pol oscillators with delayed self-connection, position and velocity couplings. Bifurcation diagram of the damping, position and velocity coupling strengths is constructed, which gives insight into how stability boundary curves come into existence and how these curves evolve from small closed loops into open-ended curves. The van der Pol oscillator has been considered by many researchers as the nodes for various networks. It is inherently unstable at the zero equilibrium. Stability control of a network is always an important problem. Currently, the stabilization of the zero equilibrium of a pair of van der Pol oscillators can be achieved only for small damping strength by using delayed velocity coupling. An interesting question arises naturally: can the zero equilibrium be stabilized for an arbitrarily large value of the damping strength? We prove that it can be. In addition, a simple condition is given on how to choose the feedback parameters to achieve such goal. We further investigate how the in-phase mode or the out-of-phase mode of a periodic solution is related to the stability boundary curve that it emerges from a Hopf bifurcation. Analytical expression of a periodic solution is derived using an integration method. Some illustrative examples show that the theoretical prediction and numerical simulation are in good agreement.