scholarly journals BIFURCATIONS AND AMPLITUDE DEATH FROM DISTRIBUTED DELAYS IN COUPLED LANDAU STUART OSCILLATORS AND A CHAOTIC PARAMETRICALLY FORCED VAN DER POL-RAYLEIGH SYSTEM

2021 ◽  
Vol 109 (2) ◽  
pp. 121-165
Author(s):  
Sudipto Roy Choudhury
Keyword(s):  
Distributed Delays ◽  
Amplitude Death ◽  
Van Der Pol ◽  
Rayleigh System

Multiple Bifurcations and Complex Responses of Nonlinear Time-Delay Oscillators

2021 ◽  
Author(s):  
Xiaochen Mao ◽  
Fuchen Lei ◽  
Xingyong Li ◽  
Weijie Ding ◽  
Tiantian Shi
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Electronic Circuit ◽  
Dynamic Features ◽  
Nonlinear Circuit ◽  
Amplitude Death ◽  
Van Der Pol ◽  
Dynamical Behaviors

Abstract In this paper, the dynamical properties of multiple van der Pol-Duffing oscillators with time delays are studied. The amplitude death and bifurcation curves in the parameter plane are determined by using the space decomposition method. Different patterns of bifurcated solutions are given on the basis of the symmetric bifurcation theory. The properties of bifurcated solutions are shown by using the norm forms on the center manifold. The interactions of bifurcations are discussed and their dynamical behaviors are shown. An electronic circuit platform is implemented by means of nonlinear circuit and time delay circuit. The revealed behaviors of the circuit reach an agreement with the obtained results. It is shown that the nonlinearity and time delays have great effects on the system performance and can induce interesting and abundant dynamic features.

scholarly journals Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Xiaobing Zhou ◽  
Murong Jiang ◽  
Xiaomei Cai
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Distributed Delays ◽  
Hopf Bifurcations ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Discrete Time Delay ◽  
Form Theory

We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis.

scholarly journals Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3004
Author(s):  
Danjin Zhang ◽  
Youhua Qian
Keyword(s):  
Hopf Bifurcation ◽  
Phase Portrait ◽  
External Excitation ◽  
Fast Subsystem ◽  
Van Der Pol ◽  
Fold Bifurcation ◽  
Modes Of Vibration ◽  
Lyapunov Coefficient ◽  
The Stability ◽  
Rayleigh System

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.

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