Hopf bifurcation of Van der Pol oscillators operating under unilateral constraints

摘要

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 adaptive control. A modification of the Van der Pol oscillator, proposed in [1], was 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 presence of unilateral constraints. It is revealed that the limit cycle of the modified Van der Pol oscillator, operating under unilateral constraints, occurs when the transient speed parameter ε exceeds a bifurcation value and it disappears for smaller, yet positive values ε when just an asymptotically stable equilibrium is presented. Using a Poincaré map, the numerical analysis is then performed to carry out the Hopf bifurcation value of the transient speed parameter and to guarantee the local asymptotic stability of the hybrid limit cycle thus generated.