Amplitude response, Melnikov’s criteria, and chaos occurrence in a Duffing’s system subjected to an external periodic excitation with a variable shape

The present study considers the nonlinear dynamics of a Duffing oscillator under a symmetric potential subjected to a pulse-type excitation with a deformable shape. Our attention is focused on the effects of the external excitation shape parameter $r$ and its period. The frequency response of the system is derived by using a semi-analytical approach. Interestingly, the frequency–response curve displays a large number of resonance peaks and anti-resonance peaks as well. Surprisingly, a resonance phenomenon termed here as shape-induced-resonance is noticed as it occurs solely due to the change in the shape parameter of the external periodic force. The system exhibits amplitude jumps and hysteresis depending on $r$⁠. The critical driving magnitude for the chaos occurrence is investigated through Melnikov’s method. Numerical analysis based on bifurcation diagrams and Lyapunov exponent is used to show how chaos occurs in the system. It is shown that the threshold amplitude of the excitation to observe chaotic dynamics decreases/increases for small/large values of $r$⁠. In general, the theoretical estimates match with numerical simulations and electronic simulations as well.