A. Jorba, J.C. Tatjer
A mechanism for the fractalization of invariant curves in
quasi-periodically forced 1-D maps
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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.