By Roberto Livi (auth.), Eric Goles, Servet Martínez (eds.)

ISBN-10: 9048147344

ISBN-13: 9789048147342

ISBN-10: 9401713235

ISBN-13: 9789401713238

This publication includes the classes given on the Fourth tuition on Statistical Physics and Cooperative structures held at Santiago, Chile, from twelfth to sixteenth December 1994. this college brings jointly scientists engaged on topics regarding fresh traits in advanced platforms. a few of these topics take care of dynamical platforms, ergodic thought, mobile automata, symbolic and mathematics dynamics, spatial platforms, huge deviation conception and neural networks. Scientists operating in those topics come from numerous aeras: natural and utilized arithmetic, non linear physics, biology, machine technological know-how, electric engineering and synthetic intelligence. every one contribution is dedicated to 1 or extra of the former topics. often they're established as surveys, offering while an unique standpoint in regards to the subject and displaying normally new effects. The expository textual content of Roberto Livi issues the examine of coupled map lattices (CML) as types of spatially prolonged dynamical platforms. CML is likely one of the so much used instruments for the research of spatially prolonged platforms. The paper emphasizes rigorous effects in regards to the dynamical habit of 1 dimensional CML; i.e. a uniform actual neighborhood functionality outlined within the period [0,1], interacting with its nearest pals in a one dimensional lattice.

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Lemma 8. A v = 0 period four two coordinates interface between He's cannot exist for any e,>.. and a. 25 Proof. Imposing again the condition that the initial point pO reproduces after four time steps one has x~ E II for -2 ~ i ~ 1 and x~ E II for 1 ~ i ~ 4. Therefore, one can derive the values xg different situations = A, = B. 14) Case (a) implies e > e(3 and A-I> a > 2/ A(l + A). In this case one reduces to the period two v = 0 interface. Case (b) implies e > eo if A- 2 < a < 2/A(1+A) ore < e(3 if A-I> a > 2/A(1+A).

Bunimovich, Random and Computational Dynamics 1, 423 (1993). A. Yorke, Z. You, I. Kan, Int. J. Bif. & Chaos 2, 795 [3] [4] [5] [6] (1992). , L. Galgani, A. M. Strelcyn, Meccanica 9, 21 (1980). , private communication. , Ya. G. Sinai, in Theory and Applications of Coupled Map Lattices, K. Kaneko editor, J. Wiley (1993). , Ya. G. Sinai, Nonlinearity 1, 491 (1988). , R. Livi, G. Martinez-Mekler, S. Ruffo, Chaos 2, 283 (1992); A. Politi, R. L. Oppo, R. Kapral, Europhys. Lett. 22, 571 (1993); R. Kapral, R.

Then k t (·) : LN-t{SF) -+ N is a constant denoted by k t and card(At = k t k 2 • • We can state the main theorem of this section. 8. [9] Let F: AN -+ AN be an expansive cellular automaton. Then, (1) (2) (3) (4) (5) (SF,U) is a mixing subshift of finite type, F is topologically mixing, htop(F) = log kt where kt E N, >. ) is a Bernoulli system. Proof. By recoding we can suppose that F is of type (E1). 4) we only have to prove that SF is mixing. WN-2 E LN-t(SF), there exists u E At satisfying wuw E L(SF)' First observe that there is a constant mo, that we choose larger than 3N, such that if x, YEAN satisfy Fi(x)(O) = Fi(y)(O), i = 0, ...

### Dynamics of Complex Interacting Systems by Roberto Livi (auth.), Eric Goles, Servet Martínez (eds.)

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