1978 : The Curry-Yorke map

Christophe LETELLIER

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

GIF - 1.1 ko

and the second is defined in Cartesian coordinates according to

GIF - 912 octets

The Curry-Yorke map is

GIF - 360 octets

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.

JPG - 14.1 ko
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 R2, 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.

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