Probability characteristics of transitions for a Rayleigh oscillator driven by Lévy noise propagated via the RKCK method

This paper investigates the probabilistic characteristics of the dynamical transitions of a Rayleigh oscillator driven by α-stable Lévy noise through an improved numerical method. Lévy noise exhibits a heavy-tailed distribution and may undergo sudden jumps over short timescales, enabling it to accurately characterize random loads in complex environments, and thus it is applied to a sixth-power configuration Rayleigh oscillator to effectively evaluate the laws governing the dynamic response of the oscillator. The data- and equation-driven adaptive-step Runge-Kutta-Cash-Karp (RKCK) method is introduced to numerically evaluate the stationary probability density functions, joint probability distributions, and stochastic P-bifurcation characteristics of systems driven by Lévy noise. Initial simulations compare the results of conventional Runge-Kutta (RK) and RKCK methods for Rayleigh oscillator under Gaussian white noise, and indicate that RKCK improves both accuracy and efficiency of relevant research metrics while maintaining numerical solution reliability. In addition, owing to the characteristic properties of Lévy noise in capturing extreme events, the adaptive RKCK method is employed to investigate the dynamical transitions of the Rayleigh oscillator under Lévy noise excitation. Simultaneously, the key parameters such as Rayleigh damping coefficients and noise parameters are adjusted to identify the multi-rhythmic oscillation modes manifested through the amplitude probability distribution of system states, which reveals the essential causes of the stochastic P-bifurcation phenomenon, and the modulation relationship among the oscillator vibrational modes, bifurcation phenomenon and critical parameters source are verified through the joint probability distribution and its contours. Finally, it is concluded that adjusting the Rayleigh damping variant coefficients and the stability index of Lévy noise provides an effective way to regulate the vibrational stability of the system. This research provides new insights for the vibration control and the design and simulation of complex environmental loads in related cross-application scenarios, while also holding scientific significance for addressing extreme events in engineering scenarios.