05-429 A. Jorba, J.C. Tatjer
A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps (7746K, PostScript, gzipped and uuencoded) Dec 20, 05
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Abstract. We focus on the continuation with respect to parameters of smooth invariant curves of quasi-periodically forced 1-D systems. In particular, we are interested in mechanisms leading to the destruction of the curve. One of these mechanisms is the so-called fractalization: the curve gets increasingly wrinkled until it stops being a smooth curve. Here we show that this situation can appear when the Lyapunov exponent of a smooth non reducible curve (a curve whose linear normal behaviour cannot be reduced to constant coefficients) goes from a strictly negative value to zero. More concretely, using the Implicit Function Theorem (IFT) we show that an attracting curve can always be locally continued w.r.t. parameters inside its differentiability class, and that a zero Lyapunov exponent implies a failure of the IFT. In our scenario, the curve can only become fractal when the Lyapunov exponent vanishes. We illustrate these phenomena with some examples, including the quasi-periodically forced logistic map and an example based on the one used by G.~Keller to prove the existence of Strange Non-chaotic Attractors.

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