Control of Chaos: Methods and Applications
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
A General Representation of Dynamical Systems for Reservoir Computing
A general representation of dynamical systems for reservoir computing Sidney Pontes-Filho∗,y,x, Anis Yazidi∗, Jianhua Zhang∗, Hugo Hammer∗, Gustavo B. M. Mello∗, Ioanna Sandvigz, Gunnar Tuftey and Stefano Nichele∗ ∗Department of Computer Science, Oslo Metropolitan University, Oslo, Norway yDepartment of Computer Science, Norwegian University of Science and Technology, Trondheim, Norway zDepartment of Neuromedicine and Movement Science, Norwegian University of Science and Technology, Trondheim, Norway Email: [email protected] Abstract—Dynamical systems are capable of performing com- TABLE I putation in a reservoir computing paradigm. This paper presents EXAMPLES OF DYNAMICAL SYSTEMS. a general representation of these systems as an artificial neural network (ANN). Initially, we implement the simplest dynamical Dynamical system State Time Connectivity system, a cellular automaton. The mathematical fundamentals be- Cellular automata Discrete Discrete Regular hind an ANN are maintained, but the weights of the connections Coupled map lattice Continuous Discrete Regular and the activation function are adjusted to work as an update Random Boolean network Discrete Discrete Random rule in the context of cellular automata. The advantages of such Echo state network Continuous Discrete Random implementation are its usage on specialized and optimized deep Liquid state machine Discrete Continuous Random learning libraries, the capabilities to generalize it to other types of networks and the possibility to evolve cellular automata and other dynamical systems in terms of connectivity, update and learning rules. Our implementation of cellular automata constitutes an reservoirs [6], [7]. Other systems can also exhibit the same initial step towards a general framework for dynamical systems. dynamics. The coupled map lattice [8] is very similar to It aims to evolve such systems to optimize their usage in reservoir CA, the only exception is that the coupled map lattice has computing and to model physical computing substrates. -
EMERGENT COMPUTATION in CELLULAR NEURRL NETWORKS Radu Dogaru
IENTIFIC SERIES ON Series A Vol. 43 EAR SCIENC Editor: Leon 0. Chua HI EMERGENT COMPUTATION IN CELLULAR NEURRL NETWORKS Radu Dogaru World Scientific EMERGENT COMPUTATION IN CELLULAR NEURAL NETWORKS WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Volume 27: The Thermomechanics of Nonlinear Irreversible Behaviors — An Introduction G. A. Maugin Volume 28: Applied Nonlinear Dynamics & Chaos of Mechanical Systems with Discontinuities Edited by M. Wiercigroch & B. de Kraker Volume 29: Nonlinear & Parametric Phenomena* V. Damgov Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L O. Chua Volume 32: From Order to Chaos II L. P. Kadanoff Volume 33: Lectures in Synergetics V. I. Sugakov Volume 34: Introduction to Nonlinear Dynamics* L Kocarev & M. P. Kennedy Volume 35: Introduction to Control of Oscillations and Chaos A. L. Fradkov & A. Yu. Pogromsky Volume 36: Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. Wojewoda Volume 37: Invariant Sets for Windows — Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & O. V. Malysheva Volume 38: Nonlinear Noninteger Order Circuits & Systems — An Introduction P. Arena, R. Caponetto, L Fortuna & D. Porto Volume 39: The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda Volume 40: Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin Volume 41: Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu Volume 42: Chaotic Synchronization: Applications to Living Systems £. -
WHAT IS a CHAOTIC ATTRACTOR? 1. Introduction J. Yorke Coined the Word 'Chaos' As Applied to Deterministic Systems. R. Devane
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties that characterize chaos. We also discuss how this term can be combined with the definition of an attractor. 1. Introduction J. Yorke coined the word `chaos' as applied to deterministic systems. R. Devaney gave the first mathematical definition for a map to be chaotic on the whole space where a map is defined. Since that time, there have been several different definitions of chaos which emphasize different aspects of the map. Some of these are more computable and others are more mathematical. See [9] a comparison of many of these definitions. There is probably no one best or correct definition of chaos. In this paper, we discuss what we feel is one of better mathematical definition. (It may not be as computable as some of the other definitions, e.g., the one by Alligood, Sauer, and Yorke.) Our definition is very similar to the one given by Martelli in [8] and [9]. We also combine the concepts of chaos and attractors and discuss chaotic attractors. 2. Basic definitions We start by giving the basic definitions needed to define a chaotic attractor. We give the definitions for a diffeomorphism (or map), but those for a system of differential equations are similar. The orbit of a point x∗ by F is the set O(x∗; F) = f Fi(x∗) : i 2 Z g. An invariant set for a diffeomorphism F is an set A in the domain such that F(A) = A. -
Adaptive Synchronization of Chaotic Systems and Its Application to Secure Communications
Chaos, Solitons and Fractals 11 (2000) 1387±1396 www.elsevier.nl/locate/chaos Adaptive synchronization of chaotic systems and its application to secure communications Teh-Lu Liao *, Shin-Hwa Tsai Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan, ROC Accepted 15 March 1999 Abstract This paper addresses the adaptive synchronization problem of the drive±driven type chaotic systems via a scalar transmitted signal. Given certain structural conditions of chaotic systems, an adaptive observer-based driven system is constructed to synchronize the drive system whose dynamics are subjected to the systemÕs disturbances and/or some unknown parameters. By appropriately selecting the observer gains, the synchronization and stability of the overall systems can be guaranteed by the Lyapunov approach. Two well-known chaotic systems: Rossler-like and Chua's circuit are considered as illustrative examples to demonstrate the eectiveness of the proposed scheme. Moreover, as an application, the proposed scheme is then applied to a secure communication system whose process consists of two phases: the adaptation phase in which the chaotic transmitterÕs disturbances are estimated; and the communication phase in which the information signal is transmitted and then recovered on the basis of the estimated parameters. Simulation results verify the proposed schemeÕs success in the communication application. Ó 2000 Elsevier Science Ltd. All rights reserved. 1. Introduction Synchronization in chaotic systems has received increasing attention [1±6] with several studies on the basis of theoretical analysis and even realization in laboratory having demonstrated the pivotal role of this phenomenon in secure communications [7±10]. The preliminary type of chaos synchronization consists of drive±driven systems: a drive system and a custom designed driven system. -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Role of Nonlinear Dynamics and Chaos in Applied Sciences
v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings. -
Instructional Experiments on Nonlinear Dynamics & Chaos (And
Bibliography of instructional experiments on nonlinear dynamics and chaos Page 1 of 20 Colorado Virtual Campus of Physics Mechanics & Nonlinear Dynamics Cluster Nonlinear Dynamics & Chaos Lab Instructional Experiments on Nonlinear Dynamics & Chaos (and some related theory papers) overviews of nonlinear & chaotic dynamics prototypical nonlinear equations and their simulation analysis of data from chaotic systems control of chaos fractals solitons chaos in Hamiltonian/nondissipative systems & Lagrangian chaos in fluid flow quantum chaos nonlinear oscillators, vibrations & strings chaotic electronic circuits coupled systems, mode interaction & synchronization bouncing ball, dripping faucet, kicked rotor & other discrete interval dynamics nonlinear dynamics of the pendulum inverted pendulum swinging Atwood's machine pumping a swing parametric instability instabilities, bifurcations & catastrophes chemical and biological oscillators & reaction/diffusions systems other pattern forming systems & self-organized criticality miscellaneous nonlinear & chaotic systems -overviews of nonlinear & chaotic dynamics To top? Briggs, K. (1987), "Simple experiments in chaotic dynamics," Am. J. Phys. 55 (12), 1083-9. Hilborn, R. C. (2004), "Sea gulls, butterflies, and grasshoppers: a brief history of the butterfly effect in nonlinear dynamics," Am. J. Phys. 72 (4), 425-7. Hilborn, R. C. and N. B. Tufillaro (1997), "Resource Letter: ND-1: nonlinear dynamics," Am. J. Phys. 65 (9), 822-34. Laws, P. W. (2004), "A unit on oscillations, determinism and chaos for introductory physics students," Am. J. Phys. 72 (4), 446-52. Sungar, N., J. P. Sharpe, M. J. Moelter, N. Fleishon, K. Morrison, J. McDill, and R. Schoonover (2001), "A laboratory-based nonlinear dynamics course for science and engineering students," Am. J. Phys. 69 (5), 591-7. http://carbon.cudenver.edu/~rtagg/CVCP/Ctr_dynamics/Lab_nonlinear_dyn/Bibex_nonline.. -
Edge of Spatiotemporal Chaos in Cellular Nonlinear Networks
1998 Fifth IEEE International Workshop on Cellular Neural Networks and their Applications, London, England, 14-17 April, 1998 EDGE OF SPATIOTEMPORAL CHAOS IN CELLULAR NONLINEAR NETWORKS Alexander S. Dmitriev and Yuri V. Andreyev Institute of Radioengineering and Electronics of the Russian Academy of Sciences Mokhovaya st., 11, Moscow, GSP-3, 103907, Russia Email: [email protected] ABSTRACT: In this report, we investigate phenomena on the edge of spatially uniform chaotic mode and spatial temporal chaos in a lattice of chaotic 1-D maps with only local connections. It is shown that in autonomous lattice with local connections, spatially uni- form chaotic mode cannot exist if the Lyapunov exponent l of the isolated chaotic map is greater than some critical value lcr > 0. We proposed a model of a lattice with a pace- maker and found a spatially uniform mode synchronous with the pacemaker, as well as a spatially uniform mode different from the pacemaker mode. 1. Introduction A number of arguments indicates that along with the three well-studied types of motion of dynamic systems — stationary state, stable periodic and quasi-periodic oscillations, and chaos — there is a special type of the system behavior, i.e., the behavior at the boundary between the regular motion and chaos. As was noticed, pro- cesses similar to evolution or information processing can take place at this boundary, as is often called, on the "edge of chaos". Here are some remarkable examples of the systems demonstrating the behavior on the edge of chaos: cellu- lar automata with corresponding rules [1], self-organized criticality [2], the Earth as a natural system with the main processes taking place on the boundary between the globe and the space. -
International Journal of Engineering and Advanced Technology (IJEAT)
IInntteerrnnaattiioonnaall JJoouurrnnaall ooff EEnnggiinneeeerriinngg aanndd AAddvvaanncceedd TTeecchhnnoollooggyy ISSN : 2249 - 8958 Website: www.ijeat.org Volume-2 Issue-5, June 2013 Published by: Blue Eyes Intelligence Engineering and Sciences Publication Pvt. Ltd. ced Te van ch d no A l d o n g y a g n i r e e IJEat n i I E n g X N t P e L IO n O E T r R A I V n NG NO f IN a o t l i o a n n r a u l J o www.ijeat.org Exploring Innovation Editor In Chief Dr. Shiv K Sahu Ph.D. (CSE), M.Tech. (IT, Honors), B.Tech. (IT) Director, Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., Bhopal (M.P.), India Dr. Shachi Sahu Ph.D. (Chemistry), M.Sc. (Organic Chemistry) Additional Director, Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., Bhopal (M.P.), India Vice Editor In Chief Dr. Vahid Nourani Professor, Faculty of Civil Engineering, University of Tabriz, Iran Prof.(Dr.) Anuranjan Misra Professor & Head, Computer Science & Engineering and Information Technology & Engineering, Noida International University, Noida (U.P.), India Chief Advisory Board Prof. (Dr.) Hamid Saremi Vice Chancellor of Islamic Azad University of Iran, Quchan Branch, Quchan-Iran Dr. Uma Shanker Professor & Head, Department of Mathematics, CEC, Bilaspur(C.G.), India Dr. Rama Shanker Professor & Head, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea Dr. Vinita Kumari Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., India Dr. Kapil Kumar Bansal Head (Research and Publication), SRM University, Gaziabad (U.P.), India Dr. -
The Edge of Chaos – an Alternative to the Random Walk Hypothesis
THE EDGE OF CHAOS – AN ALTERNATIVE TO THE RANDOM WALK HYPOTHESIS CIARÁN DORNAN O‟FATHAIGH Junior Sophister The Random Walk Hypothesis claims that stock price movements are random and cannot be predicted from past events. A random system may be unpredictable but an unpredictable system need not be random. The alternative is that it could be described by chaos theory and although seem random, not actually be so. Chaos theory can describe the overall order of a non-linear system; it is not about the absence of order but the search for it. In this essay Ciarán Doran O’Fathaigh explains these ideas in greater detail and also looks at empirical work that has concluded in a rejection of the Random Walk Hypothesis. Introduction This essay intends to put forward an alternative to the Random Walk Hypothesis. This alternative will be that systems that appear to be random are in fact chaotic. Firstly, both the Random Walk Hypothesis and Chaos Theory will be outlined. Following that, some empirical cases will be examined where Chaos Theory is applicable, including real exchange rates with the US Dollar, a chaotic attractor for the S&P 500 and the returns on T-bills. The Random Walk Hypothesis The Random Walk Hypothesis states that stock market prices evolve according to a random walk and that endeavours to predict future movements will be fruitless. There is both a narrow version and a broad version of the Random Walk Hypothesis. Narrow Version: The narrow version of the Random Walk Hypothesis asserts that the movements of a stock or the market as a whole cannot be predicted from past behaviour (Wallich, 1968). -
Transformations)
TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof. -
Attractors of Dynamical Systems in Locally Compact Spaces
Open Math. 2019; 17:465–471 Open Mathematics Research Article Gang Li* and Yuxia Gao Attractors of dynamical systems in locally compact spaces https://doi.org/10.1515/math-2019-0037 Received February 7, 2019; accepted March 4, 2019 Abstract: In this article the properties of attractors of dynamical systems in locally compact metric space are discussed. Existing conditions of attractors and related results are obtained by the near isolating block which we present. Keywords: Flow, attraction neighborhood, near isolating block MSC: 34C35 54H20 1 Introduction The dynamical system theory studies the rules of changes in the state which depends on time. In the investigation of dynamical systems, one of very interesting topics is the study of attractors (see [1 − 4] and the references therein). In [1 − 3], the authors gave some denitions of attractors and also made further investigations about the properties of them. In [5], the limit set of a neighborhood was used in the denition of an attractor, and in [6] Hale and Waterman also emphasized the importance of the limit set of a set in the analysis of the limiting behavior of a dynamical system. In [7 − 9], the authors dened intertwining attractor of dynamical systems in metric space and obtained some existing conditions of intertwining attractor. In [10], the author studied the properties of limit sets of subsets and attractors in a compact metric space. In [11], the author studied a positively bounded dynamical system in the plane, and obtained the conditions of compactness of the set of all bounded solutions. In [12], the uniform attractor was dened as the minimal compact uniformly pullback attracting random set, several existence criteria for uniform attractors were given and the relationship between uniform and co-cycle attractors was carefully studied.