Quantitative analysis of dynamical bifurcations in a coupled smooth and discontinuous oscillator with high-order nonlinear damping

Abstract The quantitative analysis of the coupled smooth and discontinuous (SD) oscillator remains challenging due to its two irrational nonlinear terms, and few studies have been conducted on it to date. This paper proposes a modified generalized harmonic function perturbation method to investigate the global evolution of limit cycles in a nonlinear damped coupled SD oscillator. This approach enables the derivation of an analytical relationship for the limit cycle amplitude as a function of the system parameters. Furthermore, an analytical stability criterion for the limit cycles is established based on the qualitative theory of ordinary differential equations. To validate the effectiveness of the proposed method, quantitative analyses are carried out for both single-well and triple-well coupled SD oscillators. In particular, critical values of the homo-heteroclinic bifurcation parameters are also predicted for the triple-well system. Additionally, approximate analytical solutions for the limit cycles and homo-heteroclinic orbits are obtained. Based on these results, global evolution of the limit cycles can be quantitatively analyzed. All analytical findings are compared with results from the Runge-Kutta method, confirming the feasibility and reliability of the proposed approach.