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 he 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.