Stochastic bifurcations in a vibro-impact Duffing-Van der Pol oscillator

The stochastic bifurcations in a vibro-impact Duffing-Van der Pol oscillator, subjected to white noise excitations, are investigated. Bifurcations in noisy systems occur either due to topological changes in the phase space-known as D-bifurcations-or due to topological changes associated with the stochastic attractors-known as P-bifurcations. In either case, the singularities in the phase space near the grazing orbits due to impact lead to inherent difficulties in bifurcation analysis. Loss of dynamic stability-or D-bifurcations-is analyzed through computation of the largest Lyapunov exponent using the Nordmark-Poincare mapping that enables bypassing the problems associated with discontinuities. For P-bifurcation analysis, the steady-state solution of the Fokker-Planck equation is computed after applying suitable non-smooth coordinate transformations and mapping the problem into a continuous domain. A quantitative measure for P-bifurcations has been carried out using a newly developed measure based on Shannon entropy. A comparison of the stability domains obtained from P-bifurcation and D-bifurcation analyses is presented which reveals that these bifurcations need not occur in same regimes.