A Markov chain approximation of a segment description of chaos

DOI10.4310/DPDE.2010.V7.N1.A4zbMATH Open1196.37089arXiv1002.0843MaRDI QIDQ983135

J. Martínez

Publication date: 30 July 2010

Published in: Dynamics of Partial Differential Equations (Search for Journal in Brave)

Abstract: We test a Markov chain approximation to the segment description (Li, 2007) of chaos (and turbulence) on a tent map, the Minea system, the H'enon map, and the Lorenz system. For the tent map, we compute the probability transition matrix of the Markov chain on the segments for segment time length (iterations) $T=1,2,3,100$. The matrix has $1,2,4$ tents corresponding to $T=1,2,3$; and is almost uniform for $T=100$. As $Ta+infty$, our conjecture is that the matrix will approach a uniform matrix (i.e. every entry is the same). For the simple fixed point attractor in the Minea system, the Reynolds average performs excellently and better than the maximal probability Markov chain and segment linking. But for the strange attractors in the H'enon map, and the Lorenz system, the Reynolds average performs very poorly and worse than the maximal probability Markov chain and segment linking.

Full work available at URL: https://arxiv.org/abs/1002.0843

zbMATH Keywords

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Mathematics Subject Classification ID

Attractors (35B41) Statistical turbulence modeling (76F55) Complex behavior and chaotic systems of ordinary differential equations (34C28) Dynamical aspects of attractors and their bifurcations (37G35)

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