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Defining complete and observable chaos

Víctor Jiménez López (1996)

Annales Polonici Mathematici

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which l i m i n f n | f n ( x ) - f n ( y ) | = 0 and l i m s u p n | f n ( x ) - f n ( y ) | > 0 . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions...

Dense chaos

Ľubomír Snoha (1992)

Commentationes Mathematicae Universitatis Carolinae

According to A. Lasota, a continuous function f from a real compact interval I into itself is called generically chaotic if the set of all points ( x , y ) , for which lim inf n | f n ( x ) - f n ( y ) | = 0 and lim sup n | f n ( x ) - f n ( y ) | > 0 , is residual in I × I . Being inspired by this definition we say that f is densely chaotic if this set is dense in I × I . A characterization of the generically chaotic functions is known. In the paper the densely chaotic functions are characterized and it is proved that in the class of piecewise monotone maps with finite number of pieces the...

Determination of phase-space reconstruction parameters of chaotic time series

Wei-Dong Cai, Yi-Qing Qin, Bing Ru Yang (2008)

Kybernetika

A new method called C-C-1 method is suggested, which can improve some drawbacks of the original C-C method. Based on the theory of period N, a new quantity S(t) for estimating the delay time window of a chaotic time series is given via direct computing a time-series quantity S(m,N,r,t), from which the delay time window can be found. The optimal delay time window is taken as the first period of the chaotic time series with a local minimum of S(t). Only the first local minimum of the average of a...

Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation

Giuseppe Da Prato, Arnaud Debussche (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup P t φ does exist for any bounded φ ; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

For the real quadratic map P a ( x ) = x 2 + a and a given ϵ > 0 a point x has good expansion properties if any interval containing x also contains a neighborhood  J of x with P a n | J univalent, with bounded distortion and B ( 0 , ϵ ) P a n ( J ) for some n . The ϵ -weakly expanding set is the set of points which do not have good expansion properties. Let α denote the negative fixed point and M the first return time of the critical orbit to [ α , - α ] . We show there is a set of parameters with positive Lebesgue measure for which the Hausdorff dimension of...

Dimensions des spirales

Yves Dupain, Michel Mendès France, Claude Tricot (1983)

Bulletin de la Société Mathématique de France

Distributional chaos on tree maps: the star case

Jose S. Cánovas (2001)

Commentationes Mathematicae Universitatis Carolinae

Let 𝕏 = { z : z n [ 0 , 1 ] } , n , and let f : 𝕏 𝕏 be a continuous map having the branching point fixed. We prove that f is distributionally chaotic iff the topological entropy of f is positive.

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