Asymmetric coupling and nonlinear resonance in coupled anharmonic oscillators

Bidirectional coupling with unequal strengths in a nonlinear system leads to complex and diverse dynamics. In this work, we investigate the emergence of resonance in a system of bidirectionally coupled Duffing oscillators with different coupling strengths in each direction, treating these asymmetric coupling strengths as control parameters. Using a theoretical approach, we derive frequency response relations for a system of two Duffing oscillators coupled bidirectionally with distinct coupling strengths. Notably, both oscillators exhibit a resonance peak in the response amplitude curve under a weak periodic input signal. The validity of our theoretical expressions is confirmed through numerical simulations. Furthermore, we examine the effects of asymmetric interactions between the two oscillators. Our results show that the resonance behavior of each oscillator emerges with a monotonic increase in either of the coupling strengths. Additionally, we analyze how system parameters, along with the two coupling strengths, influence the response amplitude. Finally, we find that as the coupling strengths vary, the system exhibits hysteresis phenomena in both oscillators.