Christophe LETELLIER
12/04/2017

**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

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 , , and combined with the initial conditions

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

**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.