Abstract
There exist rich patterns in nonlinear dynamical processes, but they merge into averages in traditional statistics-based time series analysis. Herein the multi-scale transition matrix is adopted to display the patterns and their evolutions in several typical chaotic systems, including the Logistic Map, the Tent Map, and the Lorentz System. Compared with Markovian processes, there appear rich non-trivial patterns. The unpredictability of transitions matches almost exactly with the Lyapunov exponent. The eigenvalues decay exponentially with respect to the time scale, whose decaying exponents give us the details in the curves of Lyapunov exponent versus dynamical parameters. The evolutionary behaviors differ with each other and do not saturate to the ones for the corresponding shuffled series.
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