動力系統(tǒng)

內(nèi)容概要

基于對某些重要例子的嚴(yán)格分析，本書概述動力系統(tǒng)及其在復(fù)系統(tǒng)和其他領(lǐng)域研究中的意義。著重于動力系統(tǒng)中的不變量，如系統(tǒng)地論述Morse—Conley理論，本書闡釋了動力系統(tǒng)中的基本數(shù)學(xué)概念。本書討論了熵及拓?fù)?、度量、測度、光滑假設(shè)中與動力系統(tǒng)有關(guān)的概念，以及動力系統(tǒng)與信息論的某些聯(lián)系。本書還論述了細(xì)胞自動機(jī)和隨機(jī)布爾網(wǎng)絡(luò)等特殊例子。

書籍目錄

1 Introduction2 Stability of dynamical systems, bifurcations, and generic properties  2.1 Some general notions  2.2 Autonomous systems of ODEs  2.3 Examples: Bifurcation depending on a parameter  2.4 Chaos in differential and difference equations. The concept of an attractor  2.5 Interaction, or the interplay between concentration or reaction and diffusion  2.6 Discrete and continuous systems. The Poincare return map  2.7 Stability and bifurcations; generic properties  2.8 The Hopf bifurcation  2.9 Lotka-Volterra equations  2.10 Stable, unstable, and center manifolds3 Discrete invariants of dynamical systems  3.1 The topology of graphs  3.2 Floer homology  3.3 Conley theory: examples and results  3.4 Cohomological conley index  3.5 Homotopical invariants  3.6 Continuation properties of the Conley index  3.7 The discrete Conley index4 Entropy and topological aspects of dynamical systems  4.1 The entropy of a process as an asymptotic quantity  4.2 Positive entropy and chaos  4.3 Symbolic dynamics5 Entropy and metric aspects of dynamical systems  5.1 the metric approach to topological entropy  5.2 Complexity and intrinsic scales6 Entropy and measure theoretic aspects of dynamical systems  6.1 Probabolity spaces and measure preserving maps  6.2 Ergodicity  6.3 Entropy and information  6.4 Invariant measures  6.5 Stochastic processes  6.6 Stochastic bifurcations7 Smooth dynamical systems  7.1 Lyapunov exponents  7.2 Hyperbolicity  7.3 Information loss8 Cellular automata and Boolean networks as examples of discrete dynamical systems  8.1 Cellular automata  8.2 boolean networksReferencesIndex

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