Abstract
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.
Highlights
Many important physical, chemical and biological systems such as semiconductor lasers,[1,2] coupled Brusselator models[3,4] and neural networks for circadian pacemakers[5] are composed of coupled nonlinear oscillators
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
Bifurcation diagram on stability analysis of the damping, position and velocity coupling strengths is constructed, which consists of three types of regions: (i) absolutely unstable region, (ii) regions where closed stability boundary curves exist, and (iii) regions where the stability boundary curves are open-ended
Summary
With delayed velocity coupling due to its relevance to coupled laser oscillators. They found that both the in-phase and out-of-phase modes were stable for delays of about a quarter of the uncoupled period of the oscillators. For the pair of van der Pol oscillators with delay velocity coupling studied in Ref. 16, amplitude death is possible only when the damping strength is less than 0.5. Complex dynamics such as periodic-doubling sequences leading to chaos occur for strongly nonlinear situation.[38] An interesting question naturally arises: can the strongly nonlinear van der Pol oscillators be stabilized using delay coupling? Our goal is to derive a delay feedback control strategy for the amplitude death of nonlinear van der Pol oscillators with arbitrary large damping strength and investigate periodic solutions of the in-phase and out-of-phase modes arising from Hopf bifurcation.
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