By Milos Marek, Igor Schreiber
Surveying either theoretical and experimental facets of chaotic habit, this booklet offers chaos as a version for plenty of probably random approaches in nature. simple notions from the idea of dynamical platforms, bifurcation thought and the homes of chaotic recommendations are then defined and illustrated by means of examples. A evaluation of numerical equipment used either in stories of mathematical versions and within the interpretation of experimental info is usually supplied. furthermore, an intensive survey of experimental statement of chaotic habit and techniques of its research are used to emphasize common gains of the phenomenon.
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Additional resources for Chaotic Behaviour of Deterministic Dissipative Systems (Cambridge Nonlinear Science S.)
G. Demko, New York, Academic Press, 1985. , Kaplan J. , Yorke E. D. and Yorke J. A. The Lyapunov dimension of strange attractors. /. Diff. Eqs. 49 (1983) 185. Grassberger P. Information aspects of strange attractors. Preprint, Wuppertal, Univ. of Wuppertal, 1984. Grassberger P. Generalizations of the Hausdorff dimension of fractal measures. Phys. Lett. 107A (1985) 101. Grassberger P. and Procaccia I. Measuring the strangeness of strange attractors. Physica 9D (1983) 189. GrebogiC, McDonald S. , OttE.
In other words, we need to generalize the idea of an attractor to more complex invariant sets other than stationary points or periodic orbits. 1 above does not suffice, because it can be expected that a general attractor will contain saddle periodic orbits which themselves are co-limit sets of orbits located in their stable manifolds. On the other hand, most orbits asymptotic to such an attractor will have far more complex co-limits. Hence we have to consider entire sets of orbits at the same time.
Different ways of an effective description of the randomness of a chaotic trajectory are the subject of the ergodic theory of dynamical systems2*9'2-n~2-14'2'16'2-47'2-56'2-58, which is in many aspects closely connected with probability theory 214 ' 2 ' 46 . In a certain sense there is not much difference between a chaotic trajectory and the realization of a random process. Let us consider a finite-dimensional dynamical system defined by a pair (X, g*). e. a countably additive nonnegative function JU defined on the a-algebra E of subsets of the state space X2 46.