Bifurcation Structures and Periodicity Behaviour of an Excited Modified Hybrid van der Pol–Rayleigh Oscillator: A Novel Methodology

Abstract

This study presents a novel analytical–numerical approach to reveal bifurcation structures of an excited hybrid van der Pol–Rayleigh oscillator (HVR), driven by the necessity of enhanced understanding and prediction of intricate nonlinear behaviour of a hybrid self-excited system. The study used the non-perturbative approach (NPA)to analyze the existing scheme and estimate its efficiency. The non-perturbative approach serves as primary basis of He’s frequency formula (HFF). This procedure aims to produce a linear representation of a weakly nonlinear oscillator categorized by a nonlinear ordinary differential equation (ODE). The study aims to diverge from conventional perturbation techniques. The parametric equation is corroborated via Mathematica Software (MS), demonstrating substantial concordance with the original problem. Therefore, the study of current oscillator is considered as a new possibility in using established approaches. The stability enactment is assessed under various parameters. The current approach is categorized by specific concepts, exceptional numerical accuracy, suitability, and user-friendliness. Furthermore, Arnold tongues as well as Floquet multipliers are also examined. The dynamic of the nonlinear model is examined through Poincaré maps, phase portraits, and bifurcation diagrams, which are essential analytical elements affecting system behaviour. The lack of chaos and periodic oscillations are explained by the greatest Lyapunov exponent, which also sheds light on long-term stability.