Abstract
In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem.
Highlights
Nonlinear oscillators appear in many applied sciences, involving mechanical engineering, mechanics, chemistry, biology and physics, and have a wide range of engineering backgrounds
The modified hybrid van der Pol-Rayleigh (MHVR) system proposed by Erlicher et al [7] is a kind of typical self-excited oscillation system, which can be suitably applied to the pedestrian walking lateral force model
The dynamic behavior of the van der Pol-Rayleigh system with the double Hopf bifurcations generated by external excitation is studied by the fast–slow analysis method
Summary
Nonlinear oscillators appear in many applied sciences, involving mechanical engineering, mechanics, chemistry, biology and physics, and have a wide range of engineering backgrounds. Tang et al [8] studied the vibration response and its generation mechanism in the van der Pol-Rayleigh system under slowvarying periodic excitation, and analyzed the excitation hysteresis behavior and its generation mechanism of the system. Chen et al [15] used the multi-scale method to decouple the van der Pol-Rayleigh equation and applied the discussion results to the dynamic model of the side-coupling system of a footbridge—a flexible footbridge. We will analyze the dynamic response behavior of the van der PolRayleigh system, reveal the mechanism of vibration generation in different modes and discuss the transition process among the different types of attractors in the process of rapid change. Bifurcation Analysis To further reveal the mixed-mode vibration behavior of Equation (2), its bifurcation behavior with respect to the slow variable f will be discussed
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