Transition to Chaos and Vibrational Resonance of a Nonlinear Electromagnetic Force Model Under an Amplitude-Modulated Perturbation and Driven by Double-Well Potential

In this work, the issues of vibrational resonance and transition to chaos of the nonlinear electromagnetic model are addressed. By applying the fundamental relation of dynamics, the dynamics equation is obtained and the fixed points and their stability are analyzed. The nature and the depth of the potential wells of the unperturbed system are determined. The analytical study of vibrational resonance is carried out by considering slow and fast motions. The parameters of the electromagnetic module that are unfavorable for the occurrence of vibrational resonance are identified through the complete study of the effects of all parameters. The transition to chaos is studied analytically by determining the Melnikov integral and numerically through basins of attraction, bifurcation diagrams, Lyapunov exponents, phase portraits and the Poincaré section by solving the obtained dynamic equation using a Runge–Kutta algorithm of order 4, programmed using Fortran. The influence of amplitude-modulated disturbances on the system dynamics is then numerically studied. The different motions are investigated when the modulation frequency is rational and irrational, respectively. The anti-monotonicity obtained proves that amplitude-modulated excitation reveals the rich dynamics of the electromagnetic module.