Hopf bifurcation of hybrid Van der Pol oscillators

Abstract

The nonlinear Van der Pol oscillator is long recognized for the generation of a stable limit cycle, independent of initial conditions. This feature explains its popular use in the model reference control. A specific modification of the Van der Pol oscillator is motivated by the practical need to reshape the limit cycle to the one with harmonic behavior whose amplitude and frequency would straightforwardly rely on the oscillator parameters. In this work, such a modification is numerically analyzed in the hybrid setting under unilateral constraints. An equilibrium of the constrained modification of the Van der Pol oscillator is revealed to bifurcate to a stable limit cycle under certain parameter variations. This phenomenon, otherwise known as a Hopf bifurcation, is further studied via the corresponding Poincaré map to numerically carry out the set of the bifurcation parameter values and to verify the asymptotic stability of the generated hybrid limit cycle. The knowledge of the bifurcation parameter values is then utilized to illustrate the capability of the constrained oscillator to degenerate its limit cycle into an asymptotically stable equilibrium once the oscillator parameters are properly modified online.