Decidability of chaos for some families of dynamical systems

Details Decidability of chaos for some families of dynamical systems Decidability of chaos for some families of dynamical systems

2002-12-06

arxiv.org/abs/math/0212101v2

Mathematics

Dynamical Systems

7 pages

Scientific paper

We show that existence of positive Lyapounov exponents and/or SRB measures are undecidable (in the algorithmic sense) properties within some parametrized families of interesting dynamical systems: quadratic family and H\'enon maps. Because the existence of positive exponents (or SRB measures) is, in a natural way, a manifestation of ``chaos'', these results may be understood as saying that the chaotic character of a dynamical system is undecidable. Our investigation is directly motivated by questions asked by Carleson and Smale in this direction.

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