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On the validity of the 0-1 test for chaos. (English) Zbl 1171.65091
The authors extend and add some background to previous results derived by themselves on a binary test to detect chaos in deterministic dynamical systems. For a given map f:XX with invariant ergodic probability measure μ and a scalar square integrable observable v:X, the authors define the function p c (n)= j=0 n-1 e ijc v(f j x), c(0,2π), xX and the limit K c =lim n lim N N -1 j=1 N |p c (j+n)-p c (j)| 2 (logn) -1 · In this context the main claim of the authors is that under suitable assumptions, with probability 1, the limit K c exists, K c {0,1} and K c =0 implies a regular dynamics whereas K c =1 implies chaotic dynamics.
MSC:
65P20Numerical chaos
37C40Smooth ergodic theory, invariant measures
37A25Ergodicity, mixing, rates of mixing
37D45Strange attractors, chaotic dynamics
37M25Computational methods for ergodic theory