Ergodic theory of chaos and strange attractors

From Deterministic Chaos to Anomalous Diffusion

“…That this equality is not an artefact due to the simplicity of our chosen model is stated by the following theorem: [2,20,23] For closed C 2 Anosov 13) systems the KS-entropy is equal to the sum of positive Ljapunov exponents.…”

“…This section particularly builds upon the presentations in [2,6]; for a more mathematical approach see [20]. Let us start with a brief motivation outlining the basic idea of entropy production in dynamical systems.…”

“…The last step can be avoided if the partition {J n i } is generating, for which it must hold that diam J n i → 0 (n → ∞) [18,20,22]. 12) It is quite obvious that for the Bernoulli shift the partition chosen above is generating in that sense, hence h KS = ln 2 for this map.…”

Electrochemistry, Emergent Patterns, and Inorganic Intelligent Response

“…The experimental results reported subsequently in the present chapter point to the existence of multiple coexisting strange attractors in, for example, an electrochemical system. These attractors possess positive and negative Lyapunov exponents λ, with |λ − | > |λ + | [25]. These strange attractors, while still squeezing the dimension of input information due to the larger |λ − |, have the capacity to create a specific and nontrivial response based on the emergent pattern of the attractor.…”

Investigation of the Dynamics of a High-Flux CFB Riser Using Chaos Analysis of Pressure Fluctuations

“…This analysis converts the pressure fluctuations from the time-domain into spacedomain, followed by determination of characteristic quantities that characterize the chaotic behavior of the attractor. The parameters derived from the space-domain apply equally well to the time-domain and hence to the signal source (gas-solids flow), based on the ergodic theory [14].…”