Systems of van der pol equations in the resonance case

1983 ◽  
Vol 34 (5) ◽  
pp. 536-538
Author(s):  
G. P. Khoma
Keyword(s):  
Resonance Case ◽  
Van Der Pol ◽  
Van Der Pol Equations

scholarly journals Asymptotic Behavior of the Stochastic Rayleigh-van der Pol Equations with Jumps

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
ZaiTang Huang ◽  
ChunTao Chen
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Stochastic Bifurcation ◽  
Brownian Motion Process ◽  
Van Der Pol ◽  
Limit Circle ◽  
Random Attractors ◽  
Motion Process ◽  
Van Der Pol Equations ◽  
The Stability

We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.

scholarly journals Comment on the Chaotic Behaviour of van der Pol Equations with an External Periodic Excitation

1989 ◽  
Vol 44 (2) ◽  
pp. 160-162
Author(s):  
W.-H. Steeb ◽  
Jeun Chyuan Huang ◽  
Yih Shun Gou
Keyword(s):  
Limit Cycle ◽  
Chaotic Motion ◽  
Chaotic Behaviour ◽  
Surprising Result ◽  
Periodic Excitation ◽  
Van Der Pol ◽  
Periodic Force ◽  
Cycle System ◽  
Van Der Pol Equations

Abstract The limit cycle system with an external periodic force d2u/dt2 - a( 1 - u2)du/dt + un = kcos(Ωf) (n = 1, 3, 5,...) can show chaotic behaviour for certain values of a, k and Ω. We study the influence of n on the chaotic behaviour. For n = 1 we select values which result in chaotic motion of the system. Then we investigate the behaviour of the system for n = 3, 5 and 7. Introducing the nonlinearity un(n - 3, 5, 7) gives the surprising result that the chaotic motion ceases to exist.

scholarly journals On investigation of resonances in system of two Duffing–van der Pol equations

2010 ◽  
pp. 241-254 ◽  
Author(s):  
R. E. Kondrashov ◽  
◽  
A. D. Morozov ◽  
Keyword(s):  
Van Der Pol ◽  
Van Der Pol Equations
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