Investigations on the bifurcation of a noisy Duffing–Van der Pol oscillator

• P and D bifurcation analysis is carried out for a Duffing–Van der Pol oscillator.
• D-bifurcation analysis is investigated using Largest Lyapunov exponents using existing algorithms in the literature.
• P-bifurcation analysis is primarily qualitative.
• A new quantitative measure based on Shannon entropy is proposed that enables identifying P-bifurcations.
• A comparison of the bifurcation domains identified in the parameter space is presented using the three approaches.

This study carried out investigations on the bifurcation characteristics of a Duffing–Van der Pol (DVDP) oscillator subjected to white noise excitations. Dynamical (or D-) bifurcations are characterised by dramatic changes in the dynamical behaviour leading to topological changes in the phase portrait. An additional mode of bifurcation - phenomenological (or P-) bifurcation, is observed in stochastically excited systems when the stationary joint probability density function of the state variables undergo topological changes in the probability space. While D-bifurcation analysis is quantified in terms of the sign changes in the largest Lyapunov exponent, P-bifurcation analysis is usually qualitative and through visual inspection. In this study, a new quantitative measure for P-bifurcations based on the Shannon entropy is proposed. A comparison of the parameter regimes of the noisy DVDP oscillator identified by the three bifurcation methodologies have been presented.