Christophe LETELLIER
27/08/2018

James H. Curry and James Yorke proposed in 1978 a two-dimensional map for a illustrating one of the routes to chaos from a quasi-periodic behavior. [1] This map results from the composition of two simple homeomorphisms. The first homeomorphism is defined in polar coordinates by

and the second is defined in Cartesian coordinates according to

The Curry-Yorke map is

The numerical invstigation starts with three typical behaviors solutions to the Curry-Yorke map, namely a quasi-periodic regime (Fig. 1a), an intermittent toroidal chaotic behavior (Fig. 1b) and a fully developed toroidal chaos (Fig. 1c). These three behaviors are characterized by a toroidal structure, the first being not sensitive to initial conditions. The intermittent toroidal chaos corresponds to a weakly developed chaos, in the sense that it is only slightly sensitive to initial conditions. Moreover, it exhibit an intermittency since located just at the end of the period-3 window. The laminar phases could be seen as phase during which the behavior is purely quasi-periodic. Theses phases are interupted by chaotic bursts resulting from the slight wrinkles already observed on the section (Fig. 1b). Far from the period-3 window, the behavior corresponds to a fully developed toroidal chaos, that is, a chaotic regime organized around a torus.

**Fig. 1. Different behaviors produced by the Curry -Yorke map.**

This route to toroidal chaos was observed in the driven van der Pol system. [2]

[1] **J. H. Curry & J. A. Yorke**, A transition from Hopf bifurcation to chaos : computer experiments with maps on R^{2}, *Lecture Notes in Mathematics*, **668**, 48-66, 1978.

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