Stochastic Hopf bifurcation and random chaos of a multi-stable rotational energy harvesting system

Abstract

This study examines stochastic Hopf bifurcation and random chaos in a multi-stable rotational vibration energy harvester (VEH) for automotive tire applications. The system is modeled as a strongly nonlinear Duffing-van der Pol (DVP) oscillator subject to forced and stochastic Gaussian white noise excitations. Analytical methods, including incomplete elliptic integrals, are used to derive exact solutions for eight possible homoclinic and heteroclinic orbits. Stochastic averaging and three-exponential techniques are employed to analyze Hopf bifurcation, identifying D- and P-bifurcation points and stationary probability density functions (PDFs). The stochastic Melnikov method is applied to derive chaos thresholds for six types of orbital entanglement and establish parameter criteria for four distinct chaos types. Numerical simulations validate the analytical results, demonstrating noise-induced transitions between multiple attractors and intermittent chaotic behavior. The findings provide insights for optimizing VEH performance through controlled chaotic dynamics.