The driven van der Pol equation

Christophe LETELLIER

There are many versions for the ``van der Pol’’ equation. Among them, there is this one, involving the cubic term, which reads

  \dot{x} &=& y\\
  \dot{y} &=& \mu (1-\gamma x^2)y -x^3+u \\
  \dot{u} &=& v \\
  \dot{v} &=& -\omega^2 u \, . 

and which was investigated in details in [1]. This is sstem is a semi-conservative system which means that there is a continuum of attractors, that is, many many different attractors co-exist in the state space [2]. When parameter values are \mu=0.2, \gamma=11.03, \omega=1.018 and \beta=0.35, and initial conditions are

    x_0 =-0.36 \\        
          y_0=-0.28 \\
          u_0=0.1 \\

there is a toroidal chaotic attractor as shown in Fig. 1.

JPG - 21.6 ko
Fig. 1. Toroidal chaotic attractor produced by the driven van der Pol equation.

[1] C. Letellier, V. Messager & R. Gilmore, From quasi-periodicity to toroidal chaos : analogy between the Curry-Yorke map and the van der Pol system, Physical Review E, 77 (4), 046203, 2008.

[2] O. Ménard, C. Letellier, J. Maquet, L. Le Sceller & G. Gouesbet, Analysis of a non synchronized sinusoidally driven dynamical system, International Journal of Bifurcation & Chaos, 10 (7), 1759-1772, 2000.

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