Global evolution of limit cycles and homoclinic bifurcation of smooth and discontinuous oscillator with quartic nonlinear damping

Abstract

As a typical model of irrational oscillator, the smooth and discontinuous (SD) oscillator has been researched intensively in these few years. However, the researches on nonlinear damped SD oscillator are still rare. Therefore, this work is devoted to the quantitative analysis of certain SD oscillator with quartic nonlinear damping. First, by introducing the Padé approximation method into the modified generalized harmonic function perturbation method, the latter one has been further improved. Via this method, the limit cycle’s amplitude-system parameter relationship, as well as the stability criterion about limit cycle, are derived analytically. By utilizing these relationships, the global evolution of each limit cycle and its homoclinic bifurcation are analyzed quantitatively and analytically with respect to single parameter and multi-parameters. These analyses answer the questions such as when a limit cycle emerges, how it bifurcates, and where it converges. Additionally, the analytical approximate solution of limit cycle and homoclinic orbits are also obtained. To show the effectiveness, several calculation examples are presented and analyzed elaborately. To demonstrate the accuracy, all results obtained in this paper are confirmed by Runge–Kutta method. The above results are of great significance in analyzing the global dynamic behavior of nonlinear damped SD oscillator. Thus, the presented work can be considered as an important supplement to the researches on SD oscillator. And the proposed method can be also utilized in study other oscillators.