Permutation Complexity in Dynamical Systems: Ordinal by José Amigó

By José Amigó

The learn of permutation complexity may be predicted as a brand new type of symbolic dynamics whose simple blocks are ordinal styles, that's, diversifications outlined via the order kin between issues within the orbits of dynamical systems.

Since its inception in 2002 the concept that of permutation entropy has sparked a brand new department of analysis specifically concerning the time sequence research of dynamical structures that capitalizes at the order constitution of the nation house. certainly, on one hand ordinal styles and periodic issues are heavily similar, but ordinal styles are amenable to numerical tools, whereas periodicity is not.

Another fascinating function is that because it will be proven that random (unconstrained) dynamics has no forbidden styles with chance one, their lifestyles can be utilized as a fingerprint to spot any deterministic foundation of orbit generation.

This e-book is essentially addressed to researchers operating within the box of nonlinear dynamics and intricate structures, but can be appropriate for graduate scholars attracted to those matters. The presentation is a compromise among mathematical rigor and pedagogical process. consequently, a number of the extra mathematical heritage wanted for extra intensive realizing has been shifted into the appendices.

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Extra info for Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That (Springer Series in Synergetics)

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When p(x0 , . . 9) by p(π), π ∈ SL (or estimated by the relative frequency pˆ (π )), one speaks of the Rényi permutation entropy and the Tsallis permutation entropy, respectively. To complete the picture, let us add that the situation reverses when the alphabet comprises few symbols. But in this case, Lempel–Ziv complexity (specifically, LZ-76) can be a better choice than block counting [6]; see [82] for the entropy estimation in binary sequences. 11) N(L) being the number of distinct ordinal patterns defined by sliding windows xnn+L−1 of size L.

Some power of its transition matrix is strictly positive) has a unique ergodic invariant measure, which furthermore is absolutely continuous with respect to the Lebesgue measure [134]. This measure can be calculated or numerically estimated by a variety of methods (Perron–Frobenius operator, Ulam’s method, or just computation of long time averages) [105]. 02 1 0 Fig. 5 Ordinal pattern frequencies of the tent map family using itineraries. Here L = 4, W = 100, M = 10104, and (a) π = 0, 1, 2, 3 , (b) π = 0, 1, 3, 2 , (c) π = 0, 3, 1, 2 , and (d) π = 3, 0, 1, 2 not necessary, since the relative frequencies of the ordinal patterns can be calculated with numerical simulations.

This expectation can be rigorously proved under the proviso that f is ergodic with respect to the invariant measure μ. In this case [39], k(f , x) = hμ (f ) μ-almost everywhere. 5 Cellular Automata A cellular automaton is a discrete-time dynamical system with discrete space and discrete states. The state variables are defined on the sites of a D-dimensional regular lattice (ZD )—the cells of the D-dimensional automaton—taking on values in a finite alphabet S = {0, 1, . . , k − 1}. The set of all possible states (formally the set of all possible mappings ZD → S) is called the configuration space.

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