Journal Article•
Correlation dimension and topological entropy in discrete maps
About: This article is published in Applied mathematical sciences.The article was published on 2012-01-01 and is currently open access. It has received 1 citations till now. The article focuses on the topics: Topological entropy & Correlation dimension.
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TL;DR: Time-Frequency Analysis and Poincare Surface of Section are considered for the study of the phase space structure of nonlinear dynamical system and with the help of ridge-plots, the phenomenon of transient chaos is visualize.
Abstract: In this paper, we have considered Time-Frequency Analysis (TFA) and Poincare Surface of Section (PSS) for the study of the phase space structure of nonlinear dynamical system. We have examined a sample of orbits taken in the framework of Circular Restricted Three-Body Problem (CRTBP). We have computed ridge-plots (i.e. time-frequency landscape) using the phase of the continuous wavelet transform. Clear visualization of resonance trappings and the transitions is an important feature of this method, which is presented using ridge-plots. The identification between periodic and quasi-periodic, chaotic sticky and nonsticky and regular and chaotic orbits are done in comparatively less time and with less computational effort. The spatial case of Circular Restricted Three-Body problem is considered to show the strength of Time-Frequency Analysis to higher dimensional systems. Also, with the help of ridge-plots, we can visualize the phenomenon of transient chaos.
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References
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TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.
5,239 citations
TL;DR: In this article, a topological entropy for affine maps of Lie groups and certain homogeneous spaces is defined and compared with measure theoretic entropy for Haar measure and affine map of compact metrizable groups.
Abstract: Topological entropy há(T) is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare hd(T) with measure theoretic entropy h(T); in particular h(T) = hd(T) for Haar measure and affine maps Ton compact metrizable groups. A particular case of this yields the wellknown formula for h(T) when T is a toral automorphism. Introduction. We shall study topological entropy, concentrating on its relation to measure theoretic entropy and algebraic examples. Our topological entropy hd(T) is defined (in §2) for a uniformly continuous map F on a metric space (X, d). In [1] a topological entropy h(T) was defined for a continuous map on a compact topological space; if the space is compact metric then h(T)—hd(T). An essential part of this paper is the computation of hd(T) for certain maps on noncompact spaces. Suppose p is a Borel measure on p(X) = l, and p is F-invariant (i.e. p(T~x(A)) =p(A) for every Borel set A). One can then define a measure theoretic entropy hu(T) as follows: Call a={Au ..., Ar} a (finite) measurable partition of A\" if the A¡ are disjoint measurable subsets of X covering X. Now set Hja) = 2 -4mc\\ T-kAik) iogp(mn t-*a\\ Then the limit hß(T, a) = limm_00 (\\/m)HJa) exists and one defines hu(T) = sup {hu(T, a) : a is a finite measurable partition of X}. (See [6] for details about measure theoretic entropy.) Two points in X are separated by a = {A±,..., Ar} provided they lie in different ^i's. We shall use the following fact to compute entropy : Fact (see [6]). Let {ak}k = 0 be a sequence of measurable partitions of X satisfying the following property: If x, ye X are distinct there is an n(x, y) such that ak separates x and y whenever k S: n(x, y). Then hu(f) = supfc h(T, ak). As is generally known, if T: G -> G is a surjective endomorphism of a compact metrizable group, then F preserves Haar measure p. For such a F we show that the Received by the editors October 17, 1969 and, in revised form, February 2, 1970. AMS 1969 subject classifications. Primary 2870, 2875; Secondary 5482.
1,105 citations
TL;DR: In this article, it was shown that the transmitted light from a ring cavity containing a nonlinear dielectric medium undergoes transition from a stationary state to periodic and nonperiodic states, when the intensity of the incident light is increased.
Abstract: It is theoretically shown that the transmitted light from a ring cavity containing a nonlinear dielectric medium undergoes transition from a stationary state to periodic and nonperiodic states, when the intensity of the incident light is increased. The nonperiodic state is characterized by a chaotic variation of the light intensity and associated broadband noise in the power spectrum. The experimental possibility of observing such a transition is also discussed.
662 citations
TL;DR: The physical system is described; equations of motion and iterative maps are reviewed and computed behavior is compared to data, with reasonable agreement for Poincare sections, bifurcation diagrams, and phase diagrams in parameter space (drive voltage, drive frequency).
Abstract: The nonlinear charge storage property of driven Si p-n junction passive resonators gives rise to chaotic dynamics: period doubling, chaos, periodic windows, and an extended period-adding sequence corresponding to entrainment of the resonator by successive subharmonics of the driving frequency. The physical system is described; equations of motion and iterative maps are reviewed. Computed behavior is compared to data, with reasonable agreement for Poincar\'e sections, bifurcation diagrams, and phase diagrams in parameter space (drive voltage, drive frequency). N=2 symmetrically coupled resonators are found to display period doubling, Hopf bifurcations, entrainment horns (``Arnol'd tongues''), breakup of the torus, and chaos. This behavior is in reasonable agreement with theoretical models based on the characteristics of single-junction resonators. The breakup of the torus is studied in detail, by Poincar\'e sections and by power spectra. Also studied are oscillations of the torus and cyclic crises. A phase diagram of the coupled resonators can be understood from the model. Poincar\'e sections show self-similarity and fractal structure, with measured values of fractal dimension d=2.03 and d=2.23 for N=1 and N=2 resonators, respectively. Two line-coupled resonators display first a Hopf bifurcation as the drive parameter is increased, in agreement with the model. For N=4 and N=12 line-coupled resonators complex quasiperiodic behavior is observed with up to 3 and 4 incommensurate frequencies, respectively.
127 citations
Book•
03 Sep 1999
TL;DR: In this article, the authors present an analysis of four different types of dynamical systems: one-dimensional, linear, nonlinear, and discrete linear systems, and nonlinear Dynamical Systems.
Abstract: Discrete Dynamical Systems. One-Dimensional Dynamical Systems. R ~ q, Matrices, and Functions. Discrete Linear Dynamical Systems. Nonlinear Dynamical Systems. Chaotic Behavior. Analysis of Four Dynamical Systems. Appendices. Index.
101 citations