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Botvinick-Greenhouse J, Martin R, Yang Y. Learning dynamics on invariant measures using PDE-constrained optimization. CHAOS (WOODBURY, N.Y.) 2023; 33:063152. [PMID: 37368043 DOI: 10.1063/5.0149673] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2023] [Accepted: 06/02/2023] [Indexed: 06/28/2023]
Abstract
We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269-310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.
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Affiliation(s)
| | - Robert Martin
- DEVCOM Army Research Laboratory, Research Triangle Park, Durham, North Carolina 27709, USA
| | - Yunan Yang
- Institute for Theoretical Studies, ETH Zürich, Zürich 8092, Switzerland
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Exploiting deterministic features in apparently stochastic data. Sci Rep 2022; 12:19843. [PMID: 36400910 PMCID: PMC9674651 DOI: 10.1038/s41598-022-23212-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2022] [Accepted: 10/26/2022] [Indexed: 11/19/2022] Open
Abstract
Many processes in nature are the result of many coupled individual subsystems (like population dynamics or neurosystems). Not always such systems exhibit simple stable behaviors that in the past science has mostly focused on. Often, these systems are characterized by bursts of seemingly stochastic activity, interrupted by quieter periods. The hypothesis is that the presence of a strong deterministic ingredient is often obscured by the stochastic features. We test this by modeling classically stochastic considered real-world data from both, the stochastic as well as the deterministic approaches to find that the deterministic approach's results level with those from the stochastic side. Moreover, the deterministic approach is shown to reveal the full dynamical systems landscape, which can be exploited for steering the dynamics into a desired regime.
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Solutions of the Multivariate Inverse Frobenius-Perron Problem. ENTROPY 2021; 23:e23070838. [PMID: 34208901 PMCID: PMC8306100 DOI: 10.3390/e23070838] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/31/2021] [Accepted: 06/18/2021] [Indexed: 11/20/2022]
Abstract
We address the inverse Frobenius–Perron problem: given a prescribed target distribution ρ, find a deterministic map M such that iterations of M tend to ρ in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map; that is, a map under which the uniform distribution on the d-dimensional hypercube is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps.
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High Density Nodes in the Chaotic Region of 1D Discrete Maps. ENTROPY 2018; 20:e20010024. [PMID: 33265118 PMCID: PMC7512224 DOI: 10.3390/e20010024] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/28/2017] [Revised: 12/02/2017] [Accepted: 01/02/2018] [Indexed: 11/16/2022]
Abstract
We report on the definition and characteristics of nodes in the chaotic region of bifurcation diagrams in the case of 1D mono-parametrical and S-unimodal maps, using as guiding example the logistic map. We examine the arrangement of critical curves, the identification and arrangement of nodes, and the connection between the periodic windows and nodes in the chaotic zone. We finally present several characteristic features of nodes, which involve their convergence and entropy.
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Gallas MR, Gallas JAC. Nested arithmetic progressions of oscillatory phases in Olsen's enzyme reaction model. CHAOS (WOODBURY, N.Y.) 2015; 25:064603. [PMID: 26117128 DOI: 10.1063/1.4921178] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We report some regular organizations of stability phases discovered among self-sustained oscillations of a biochemical oscillator. The signature of such organizations is a nested arithmetic progression in the number of spikes of consecutive windows of periodic oscillations. In one of them, there is a main progression of windows whose consecutive number of spikes differs by one unit. Such windows are separated by a secondary progression of smaller windows whose number of spikes differs by two units. Another more complex progression involves a fan-like nested alternation of stability phases whose number of spikes seems to grow indefinitely and to accumulate methodically in cycles. Arithmetic progressions exist abundantly in several control parameter planes and can be observed by tuning just one among several possible rate constants governing the enzyme reaction.
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Affiliation(s)
- Marcia R Gallas
- Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil and Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Brazil
| | - Jason A C Gallas
- Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100-1801, 58039-150 João Pessoa, Brazil and Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Brazil
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Luque B, Lacasa L, Ballesteros FJ, Robledo A. Analytical properties of horizontal visibility graphs in the Feigenbaum scenario. CHAOS (WOODBURY, N.Y.) 2012; 22:013109. [PMID: 22462985 DOI: 10.1063/1.3676686] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [B. Luque et al., PLoS ONE 6, 9 (2011)] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here, we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree distributions, mean distances, clustering coefficients, etc., associated to the bifurcation cascades and their accumulation points. We describe how the resultant families of graphs can be framed into a renormalization group scheme in which fixed-point graphs reveal their scaling properties. These fixed points are then re-derived from an entropy optimization process defined for the graph sets, confirming a suggested connection between renormalization group and entropy optimization. Finally, we provide analytical and numerical results for the graph entropy and show that it emulates the Lyapunov exponent of the map independently of its sign.
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Affiliation(s)
- Bartolo Luque
- Departamento de Matemática Aplicada y Estadística, ETSI Aeronáuticos, Universidad Politécnica de Madrid, Spain
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Oliveira DFM, Robnik M, Leonel ED. Shrimp-shape domains in a dissipative kicked rotator. CHAOS (WOODBURY, N.Y.) 2011; 21:043122. [PMID: 22225359 DOI: 10.1063/1.3657917] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime.
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Affiliation(s)
- Diego F M Oliveira
- CAMTP--Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia.
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11
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Schenke B, Avrutin V, Schanz M. On a bifurcation structure mimicking period adding. Proc Math Phys Eng Sci 2011. [DOI: 10.1098/rspa.2010.0573] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In this work, we investigate a piecewise-linear discontinuous scalar map defined on three partitions. This map is specifically constructed in such a way that it shows a recently discovered bifurcation scenario in its pure form. Owing to its structure on the one hand and the similarities to the nested period-adding scenario on the other hand, we denoted the new bifurcation scenario as nested period-incrementing bifurcation scenario. The new bifurcation scenario occurs in several physical and electronical systems but usually not isolated, which makes the description complicated. By isolating the scenario and using a suitable symbolic description for the asymptotically stable periodic orbits, we derive a set of rules in the space of symbolic sequences that explain the structure of the stable periodic domain in the parameter space entirely. Hence, the presented work is a necessary step for the understanding of the more complicated bifurcation scenarios mentioned above.
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Affiliation(s)
- Björn Schenke
- Institute of Parallel and Distributed Systems, University of Stuttgart, Universitätsstraße 38, 70569 Stuttgart, Germany
| | - Viktor Avrutin
- Institute of Parallel and Distributed Systems, University of Stuttgart, Universitätsstraße 38, 70569 Stuttgart, Germany
| | - Michael Schanz
- Institute of Parallel and Distributed Systems, University of Stuttgart, Universitätsstraße 38, 70569 Stuttgart, Germany
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Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map. Math Biosci 2008; 216:90-9. [DOI: 10.1016/j.mbs.2008.08.012] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2007] [Revised: 08/01/2008] [Accepted: 08/15/2008] [Indexed: 11/17/2022]
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Strelioff CC, Hübler AW. Medium-term prediction of chaos. PHYSICAL REVIEW LETTERS 2006; 96:044101. [PMID: 16486826 DOI: 10.1103/physrevlett.96.044101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2005] [Indexed: 05/06/2023]
Abstract
We study prediction of chaotic time series when a perfect model is available but the initial condition is measured with uncertainty. A common approach for predicting future data given these circumstances is to apply the model despite the uncertainty. In systems with fold dynamics, we find prediction is improved over this strategy by recognizing this behavior. A systematic study of the Logistic map demonstrates prediction of the most likely trajectory can be extended three time steps. Finally, we discuss application of these ideas to the Rössler attractor.
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Affiliation(s)
- Christopher C Strelioff
- Center for Complex Systems Research, Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA
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Anteneodo C. Statistics of finite-time Lyapunov exponents in the Ulam map. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:016207. [PMID: 14995693 DOI: 10.1103/physreve.69.016207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2003] [Indexed: 05/24/2023]
Abstract
The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length n. The variance anomalously decays as 1/n(2). The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting 2(n-1) spikes that narrow and accumulate close to the mean value with increasing n. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small n, where the finiteness of bin size trims the sharp peaks.
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Affiliation(s)
- Celia Anteneodo
- Centro Brasileiro de Pesquisas Físicas, R. Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil
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Schanz M, Pelster A. Analytical and numerical investigations of the phase-locked loop with time delay. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:056205. [PMID: 12786248 DOI: 10.1103/physreve.67.056205] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2002] [Revised: 03/03/2003] [Indexed: 05/24/2023]
Abstract
We derive the normal form for the delay-induced Hopf bifurcation in the first-order phase-locked loop with time delay by the multiple scaling method. The resulting periodic orbit is confirmed by numerical simulations. Further detailed numerical investigations demonstrate exemplarily that this system reveals a rich dynamical behavior. With phase portraits, Fourier analysis, and Lyapunov spectra it is possible to analyze the scaling properties of the control parameter in the period-doubling scenario, both qualitatively and quantitatively. Within the numerical accuracy there is evidence that the scaling constant of the time-delayed phase-locked loop coincides with the Feigenbaum constant delta approximately 4.669 in one-dimensional discrete systems.
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Affiliation(s)
- Michael Schanz
- Institute of Parallel and Distributed Systems (IPVS), University of Stuttgart, Breitwiesenstrasse 20-22, D-70565 Stuttgart, Germany.
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Stojanovski T, Kocarev L. Chaos-based random number generators-part I: analysis [cryptography]. ACTA ACUST UNITED AC 2001. [DOI: 10.1109/81.915385] [Citation(s) in RCA: 297] [Impact Index Per Article: 12.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Pingel D, Schmelcher P, Diakonos FK. Theory and examples of the inverse Frobenius-Perron problem for complete chaotic maps. CHAOS (WOODBURY, N.Y.) 1999; 9:357-366. [PMID: 12779834 DOI: 10.1063/1.166413] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The general solution of the inverse Frobenius-Perron problem considering the construction of a fully chaotic dynamical system with given invariant density is obtained for the class of one-dimensional unimodal complete chaotic maps. Some interesting connections between this general solution and the special approach via conjugation transformations are illuminated. The developed method is applied to obtain a class of maps having as invariant density the two-parametric beta-probability density function. Varying the parameters of the density a rich variety of dynamics is observed. Observables like autocorrelation functions, power spectra, and Liapunov exponents are calculated for representatives of this family of maps and some theoretical predictions concerning the decay of correlations are tested. (c) 1999 American Institute of Physics.
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Affiliation(s)
- D. Pingel
- Theoretische Chemie, Physikalisch-Chemisches Institut, Universitat Heidelberg, Im Neuenheimer Feld 253, D-69120 Heidelberg, Germany
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Binder PM, Privman V. Collective behaviour in one-dimensional locally coupled map lattices. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/25/13/004] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Diakonos FK, Schmelcher P. Turning point properties as a method for the characterization of the ergodic dynamics of one-dimensional iterative maps. CHAOS (WOODBURY, N.Y.) 1997; 7:239-244. [PMID: 12779652 DOI: 10.1063/1.166249] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Dynamical as well as statistical properties of the ergodic and fully developed chaotic dynamics of iterative maps are investigated by means of a turning point analysis. The turning points of a trajectory are hereby defined as the local maxima and minima of the trajectory. An examination of the turning point density directly provides us with the information of the position of the fixed point for the corresponding dynamical system. Dividing the ergodic dynamics into phases consisting of turning points and nonturning points, respectively, elucidates the understanding of the organization of the chaotic dynamics for maps. The turning point map contains information on any iteration of the dynamical law and is shown to possess an asymptotic scaling behaviour which is responsible for the assignment of dynamical structures to the environment of the two fixed points of the map. Universal statistical turning point properties are derived for doubly symmetric maps. Possible applications of the observed turning point properties for the analysis of time series are discussed in some detail. (c) 1997 American Institute of Physics.
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Affiliation(s)
- F. K. Diakonos
- Theoretische Chemie, Physikalisch-Chemisches Institut, Im Neuenheimer Feld 253, 69120 Heidelberg, Germany
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Alonso D, MacKernan D, Gaspard P, Nicolis G. Statistical approach to nonhyperbolic chaotic systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:2474-2478. [PMID: 9965356 DOI: 10.1103/physreve.54.2474] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Binder P, Campos DH. Direct calculation of invariant measures for chaotic maps. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:R4259-R4262. [PMID: 9964898 DOI: 10.1103/physreve.53.r4259] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Chaudhuri S, Gangopadhyay G, Ray DS. Fluctuations and decoherence in classical chaos: A model study of a Kubo oscillator generated by a chaotic system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:2262-2267. [PMID: 9963666 DOI: 10.1103/physreve.52.2262] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Németh A, Szépfalusy P. Properties of border states of transient chaos. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:1544-1549. [PMID: 9963575 DOI: 10.1103/physreve.52.1544] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Eckhardt B, Grossmann S. Correlation functions in chaotic systems from periodic orbits. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:4571-4576. [PMID: 9962536 DOI: 10.1103/physreve.50.4571] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Reimann P. Suppression of deterministic diffusion by noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:727-735. [PMID: 9962031 DOI: 10.1103/physreve.50.727] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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MacKernan D, Nicolis G. Generalized Markov coarse graining and spectral decompositions of chaotic piecewise linear maps. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:988-999. [PMID: 9962057 DOI: 10.1103/physreve.50.988] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Neiman A, Anishchenko V, Kurths J. Period-doubling bifurcations in the presence of colored noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3801-3806. [PMID: 9961666 DOI: 10.1103/physreve.49.3801] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Pérez G, Cerdeira HA. Static parametric fluctuations give nonstatistical behavior in uncoupled chaotic systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:R15-R18. [PMID: 9961298 DOI: 10.1103/physreve.49.r15] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Sacher J, Baums D, Panknin P, Elsässer W, Göbel EO. Intensity instabilities of semiconductor lasers under current modulation, external light injection, and delayed feedback. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:1893-1905. [PMID: 9907177 DOI: 10.1103/physreva.45.1893] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Reick C. Universal corrections to parameter scaling in period-doubling systems: Multiple scaling and crossover. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:777-792. [PMID: 9907043 DOI: 10.1103/physreva.45.777] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Fick E, Fick M, Hausmann G. Logistic equation with memory. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:2469-2473. [PMID: 9906228 DOI: 10.1103/physreva.44.2469] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Prakash S, Peng CK, Alstrom P. Deterministic diffusion generated by a chaotic map. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:6564-6571. [PMID: 9905007 DOI: 10.1103/physreva.43.6564] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Szépfalusy P, Tél T, Vattay G. Thermodynamics of Lorenz-type maps. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:681-692. [PMID: 9905085 DOI: 10.1103/physreva.43.681] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Christiansen F, Paladin G, Rugh HH. Determination of correlation spectra in chaotic systems. PHYSICAL REVIEW LETTERS 1990; 65:2087-2090. [PMID: 10042450 DOI: 10.1103/physrevlett.65.2087] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Peng CK, Prakash S, Herrmann HJ, Stanley HE. Randomness versus deterministic chaos: Effect on invasion percolation clusters. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:4537-4542. [PMID: 9904560 DOI: 10.1103/physreva.42.4537] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Auerbach D, Procaccia I. Grammatical complexity of strange sets. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:6602-6614. [PMID: 9903073 DOI: 10.1103/physreva.41.6602] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Theiler J. Statistical precision of dimension estimators. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:3038-3051. [PMID: 9903455 DOI: 10.1103/physreva.41.3038] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
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Abarbanel HD, Brown R, Kadtke JB. Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:1782-1807. [PMID: 9903289 DOI: 10.1103/physreva.41.1782] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/09/2023]
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Kovács Z, Tél T. Scaling in multifractals: Discretization of an eigenvalue problem. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 40:4641-4646. [PMID: 9902708 DOI: 10.1103/physreva.40.4641] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Solari HG, Gilmore R. Relative rotation rates for driven dynamical systems. PHYSICAL REVIEW. A, GENERAL PHYSICS 1988; 37:3096-3109. [PMID: 9900045 DOI: 10.1103/physreva.37.3096] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Badii R, Heinzelmann K, Meier PF, Politi A. Correlation functions and generalized Lyapunov exponents. PHYSICAL REVIEW. A, GENERAL PHYSICS 1988; 37:1323-1328. [PMID: 9899791 DOI: 10.1103/physreva.37.1323] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Chen C, Györgyi G, Schmidt G. Rapid convergence to the universal dissipation sequence in dynamical systems. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 36:5502-5504. [PMID: 9898837 DOI: 10.1103/physreva.36.5502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Richetti P, Roux JC, Argoul F, Arneodo A. From quasiperiodicity to chaos in the Belousov–Zhabotinskii reaction. II. Modeling and theory. J Chem Phys 1987. [DOI: 10.1063/1.451992] [Citation(s) in RCA: 75] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Chen C, Györgyi G, Schmidt G. Universal scaling in dissipative systems. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 35:2660-2668. [PMID: 9898455 DOI: 10.1103/physreva.35.2660] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Argoul F, Arneodo A, Richetti P, Roux JC. From quasiperiodicity to chaos in the Belousov–Zhabotinskii reaction. I. Experiment. J Chem Phys 1987. [DOI: 10.1063/1.452751] [Citation(s) in RCA: 105] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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