Periodic orbit theory of two coupled Tchebyscheff maps

Dettmann, Carl P.

doi:10.1016/s0960-0779(04)00239-5

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…In complement to these results, we would like to mention that estimates on the topological entropy in a CML of 2 sites with Tchebyscheff individual maps have been obtained in [DL05] by using periodic orbit theory.…”

Section: Introductionmentioning

confidence: 67%

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

Coutinho

Fernandez

Guiraud

2008

Physica D: Nonlinear Phenomena

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The bounded dynamics of a system of two coupled piecewise affine and chaotic Lorenz maps is studied over the coupling range, from the uncoupled regime where the entropy is maximal, to the synchronized regime where the entropy is minimal. By formulating the problem in terms of symbolic dynamics, bounds on the set of orbit codes (or the set itself, depending on parameters) are determined which describe the way the dynamics is gradually affected as the coupling increases. Proofs rely on monotonicity properties of bounded orbit coordinates with respect to some partial ordering on the corresponding codes. The estimates are translated in terms of (bounds on the) entropy, which are monotonously decreasing with coupling and which are compared to the numerically computed entropy. A good agreement is found which indicates that these bounds capture the essential features of the transition from the uncoupled regime to synchronisation.

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Section: Introductionmentioning

confidence: 67%

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

Coutinho

Fernandez

Guiraud

2008

Physica D: Nonlinear Phenomena

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“…In this paper we show that the answer is given by Chebyshev maps of N -th order. These are conjugated to the shift of N symbols by means of a cosine function, and have been subject of many previous papers [9,10,11,12,13]. We will show that any other conjugating function produces more higher-order correlations.…”

Section: Introductionmentioning

confidence: 82%

“…We now study spatially coupled systems [11,12,17,26] and investigate how the correlation patterns are modified. As the simplest model, consider a (periodic) lattice with just two sites that consists of two coupled 2nd-order ordinary Chebyshev maps:…”

Section: Cmls Of Two Sitesmentioning

confidence: 99%

Distinguished correlation properties of Chebyshev dynamical systems and their generalisations

Jin

Beck

2020

Preprint

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We show that, among all smooth one-dimensional maps conjugated to an N -ary shift (a Bernoulli shift of N symbols), Chebyshev maps are distinguished in the sense that they have least higher-order correlations. We generalise our consideration and study a family of shifted Chebyshev maps, presenting analytic results for two-point and higher-order correlation functions. We also review results for the eigenvalues and eigenfunctions of the Perron-Frobenius operator of N -th order Chebyshev maps and their shifted generalisations. The spectrum is degenerate for odd N . Finally, we consider coupled map lattices (CMLs) of shifted Chebyshev maps and numerically investigate zeros of the temporal and spatial nearest-neighbour correlations, which are of interest in chaotically quantized field theories.

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“…In Refs. [19,20] weakly coupled N-th order Chebyshev maps [18,24,25,26,27] were studied and it was shown that certain scaling properties with respect to the coupling a can be calculated perturbatively. However, the fine structure of the parameter dependence of important observables of the chaotic string, such as the self energy or vacuum expectation value, could not be explained by these perturbative methods.…”

Section: Introductionmentioning

confidence: 99%

Renormalization group approach to chaotic strings

Groote

Veermäe

Beck

2013

Chaos, Solitons & Fractals

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Coupled map lattices of weakly coupled Chebychev maps, so-called chaotic strings, may have a profound physical meaning in terms of dynamical models of vacuum fluctuations in stochastically quantized field theories. Here we present analytic results for the invariant density of chaotic strings, as well as for the coupling parameter dependence of given observables of the chaotic string such as the vacuum expectation value. A highly nontrivial and selfsimilar parameter dependence is found, produced by perturbative and nonperturbative effects, for which we develop a mathematical description in terms of suitable scaling functions. Our analytic results are in good agreement with numerical simulations of the chaotic dynamics.Comment: 36 pages, 18 figures - v2 contains slightly more than the published versio

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Periodic orbit theory of two coupled Tchebyscheff maps

Cited by 4 publications

References 13 publications

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

Distinguished correlation properties of Chebyshev dynamical systems and their generalisations

Renormalization group approach to chaotic strings

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