An Alternative Poincaré Section for Steady-State Responses and Bifurcations of a Duffing-Van der Pol Oscillator

To apply effective methods to analyze and distinguish all kinds of response patterns is an important issue of nonlinear dynamics. This paper contributes to develop an alternative Poincaré section method to analyze and identify the whirl responses of a nonlinear oscillator. This integration for analyzing high-order harmonic and chaotic motions is used to integrate the distance between state trajectory and the origin in the phase plane during a specific period. This response integration process is based on the fact that the integration value would be constant if the integration interval is equal to the response period. It provides a quantitative characterization of system responses as the role of the traditional stroboscopic technique (Poincaré section method) to observe bifurcations and chaos of the nonlinear oscillators. However, due to the signal response contamination of system, the section points on Poincaré maps might be too close to distinguish, especially for a high-order subharmonic vibration. Combining the capability of precisely identifying period and constructing bifurcation diagrams, the advantages of the proposed method are shown by the simulations of a Duffing-Van der Pol oscillator. The simulation results show that the high-order subharmonic and chaotic responses and their bifurcations can be effectively observed. Key-Words: Poincaré section method, Chaotic motion, Response integration, Bifurcation, Duffing-Van der Pol oscillator