A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

Abstract

Abstract Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed RayleighLiénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.