The driven van der Pol equation

Christophe LETELLIER
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Yoshisuke Ueda

There are many versions for the ``van der Pol’’ equation. Among them, there is this one investigated by Yoshisuke Ueda in 1965 in his Ph.D. thesis [1]. It reads

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

This system is semi-conservative 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, and \omega=1.018 combined with the initial conditions

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there is a toroidal chaotic attractor (Fig. 1) which was related with the Curry-Yorke scenario in [3].

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Fig. 1. Toroidal chaotic attractor produced by the driven van der Pol equation.

[1] Y. Ueda, Some problems in the theory of nonlinear oscillations, Ph.D. thesis, (1965) reprinted in The road to chaos, Aerial Press, 1992.

[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.

[3] 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.

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