Nonlinear Dynamics and Coexistence of Attractors of a New Jerk-Modified Rayleigh Oscillator

Jerk models (the time derivative of acceleration) in nonlinear dynamics provide the mathematical framework necessary to generate deterministic chaos in autonomous systems. This article investigates deterministic chaos in the simplified Selkov model in the absence of external excitation. Specifically, this work studies the nonlinear dynamics, the transition to chaos, and the coexistence of attractors in a new jerk-modified Rayleigh oscillator derived from a Rayleigh oscillator originating from the Selkov system. By applying Tikhonov’s theorem to the Selkov model, the equation of the modified autonomous Rayleigh oscillator is determined. Through conversion, the equation of a new jerk-modified Rayleigh oscillator is obtained, and the fixed points of this new system are determined and then analyzed. The emergence of complex phenomena is studied through numerical simulations based on the direct integration of the system via the fourth-order Runge–Kutta method. The influence of system parameters on the transition to chaos and the coexistence of attractors is studied using parametric diagrams, attraction basins, bifurcation diagrams, the Lyapunov exponent, time series, and phase portraits. Thus, the parameters that promote and those that prevent the appearance of chaotic dynamics and the coexistence of attractors are identified. The autonomous-modified jerk-Rayleigh oscillator is implemented in OrCAD PSpice software to confirm the numerical simulation results. Finally, a linear controller is proposed to eliminate chaotic dynamics and the coexistence of attractors from this new jerk-modified Rayleigh oscillator.