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Statistical Properties of Deterministic Systems
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rgodic theory is a mathematical subject that studies the statistical proper- ties of deterministic dynamical systems. It is a combination of several branches of pure mathematics, such as measure theory, functional analysis, topology, and geometry, and it also has applications in a variety of fields in science and engineering, as a branch of applied mathematics. In the past decades, the ergodic theory of chaotic dynamical systems has found more and more appli- cations in mathematics, physics, engineering, biology and various other fields. For example, its theory and methods have played a major role in such emerging interdisciplinary subjects as computational molecular dynamics, drug designs, and third generation wireless communications in the past decade.
Many problems in science and engineering are often reduced to studying the asymptotic behavior of discrete dynamical systems. We know that in neural net- works, condensed matter physics, turbulence in flows, large scale laser arrays, convection-diffusion equations, coupled mapping lattices in phase transition, and molecular dynamics, the asymptotic property of the complicated dynamical system often exhibits chaotic phenomena and is unpredictable. However, if we study chaotic dynamical systems from the statistical point of view, we find that chaos in the deterministic sense usually possesses some kind of regularity in the probabilistic sense. In this textbook, which is written for the upper level undergraduate students and graduate students, we study chaos from the statis- tical point of view. From this viewpoint, we mainly investigate the evolution process of density functions governed by the underlying deterministic dynam- ical system. For this purpose, we employ the concept of density functions in the study of the statistical properties of sequences of iterated measurable trans- formations. These statistical properties often depend on the existence and the properties of those probability measures which are absolutely continuous with respect to the Lebesgue measure and which are invariant under the transforma- tion with respect to time. The existence of absolutely continuous invariant finite measures is equivalent to the existence of nontrivial fixed points of a class of stochastic operators (or Markov operators), called Frobenius-Perron operators by the great mathematician Stanislaw Ulam, who pioneered the exploration of nonlinear science, in his famous book “A Collection of Mathematical Problems” [120] in 1960.
Jiu Ding and Aihui Zhou - Personal Name
1st Edtion
978-7-302-18296-1
NONE
Statistical Properties of Deterministic Systems
Management
English
Tsinghua University Press
2009
Beijing
1-248
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