A General Representation of Dynamical Systems for Reservoir Computing
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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 £. -
Thermodynamic Properties of Coupled Map Lattices 1 Introduction
Thermodynamic properties of coupled map lattices J´erˆome Losson and Michael C. Mackey Abstract This chapter presents an overview of the literature which deals with appli- cations of models framed as coupled map lattices (CML’s), and some recent results on the spectral properties of the transfer operators induced by various deterministic and stochastic CML’s. These operators (one of which is the well- known Perron-Frobenius operator) govern the temporal evolution of ensemble statistics. As such, they lie at the heart of any thermodynamic description of CML’s, and they provide some interesting insight into the origins of nontrivial collective behavior in these models. 1 Introduction This chapter describes the statistical properties of networks of chaotic, interacting el- ements, whose evolution in time is discrete. Such systems can be profitably modeled by networks of coupled iterative maps, usually referred to as coupled map lattices (CML’s for short). The description of CML’s has been the subject of intense scrutiny in the past decade, and most (though by no means all) investigations have been pri- marily numerical rather than analytical. Investigators have often been concerned with the statistical properties of CML’s, because a deterministic description of the motion of all the individual elements of the lattice is either out of reach or uninteresting, un- less the behavior can somehow be described with a few degrees of freedom. However there is still no consistent framework, analogous to equilibrium statistical mechanics, within which one can describe the probabilistic properties of CML’s possessing a large but finite number of elements. -
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 -
A Simple Scalar Coupled Map Lattice Model for Excitable Media
This is a repository copy of A simple scalar coupled map lattice model for excitable media. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74667/ Monograph: Guo, Y., Zhao, Y., Coca, D. et al. (1 more author) (2010) A simple scalar coupled map lattice model for excitable media. Research Report. ACSE Research Report no. 1016 . Automatic Control and Systems Engineering, University of Sheffield Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ A Simple Scalar Coupled Map Lattice Model for Excitable Media Yuzhu Guo, Yifan Zhao, Daniel Coca, and S. A. Billings Research Report No. 1016 Department of Automatic Control and Systems Engineering The University of Sheffield Mappin Street, Sheffield, S1 3JD, UK 8 September 2010 A Simple Scalar Coupled Map Lattice Model for Excitable Media Yuzhu Guo, Yifan Zhao, Daniel Coca, and S.A. -
Coupled Map Lattice (CML) Approach
Different mechanisms of synchronization : Coupled Map Lattice (CML) approach – p.1 Outline Synchronization : an universal phenomena Model (Coupled maps/oscillators on Networks) – p.2 Modelling Spatio-temporal dynamics 2- and 3-dimensional fluid flow, Coupled-oscillator models, Cellular automata, Transport models, Coupled map models Model does not have direct connections with actual physical or biological systems – p.3 Modelling Spatio-temporal dynamics 2- and 3-dimensional fluid flow, Coupled-oscillator models, Cellular automata, Transport models, Coupled map models Model does not have direct connections with actual physical or biological systems General results show some universal features – p.3 We select coupled maps(chaotic), oscillators on Networks Models. – p.4 Coupled Dynamics on Networks ε xi(t +1) = f(xi(t)) + Cijg(xi(t),xj(t)) Cij Pj P Periodic orbit Synchronized chaos Travelling wave Spatio-temporal chaos – p.5 Coupled maps Models Coupled oscillators (- Kuramoto ∼ beginning of 1980’s) Coupled maps (- K. Kaneko) - K. Kaneko, ”Simulating Physics with Coupled Map Lattices —— Pattern Dynamics, Information Flow, and Thermodynamics of Spatiotemporal Chaos”, pp1-52 in Formation, Dynamics, and Statistics of Patterns ed. K. Kawasaki et.al., World. Sci. 1990” – p.6 Studying real systems with coupled maps/oscillators models: - Ott, Kurths, Strogatz, Geisel, Mikhailov, Wolf etc . From Kuramoto to Crawford .... (Physica D, Sep 2000) Chaotic Transients in Complex Networks, (Phys Rev Lett, 2004) Turbulence in oscillatory chemical reactions, (J Chem Phys, 1994) Crowd synchrony on the Millennium Bridge, (Nature, 2005) Species extinction - Amritkar and Rangarajan, Phys. Rev. Lett. (2006) – p.7 SYNCHRONIZATION First observation (Pendulum clock (1665)) - discovered in 1665 by the famous Dutch physicists, Christiaan Huygenes. -
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. -
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). -
Predictability: a Way to Characterize Complexity
Predictability: a way to characterize Complexity G.Boffetta (a), M. Cencini (b,c), M. Falcioni(c), and A. Vulpiani (c) (a) Dipartimento di Fisica Generale, Universit`adi Torino, Via Pietro Giuria 1, I-10125 Torino, Italy and Istituto Nazionale Fisica della Materia, Unit`adell’Universit`adi Torino (b) Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38 , 01187 Dresden, Germany (c) Dipartimento di Fisica, Universit`adi Roma ”la Sapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy and Istituto Nazionale Fisica della Materia, Unit`adi Roma 1 Contents 1 Introduction 4 2 Two points of view 6 2.1 Dynamical systems approach 6 2.2 Information theory approach 11 2.3 Algorithmic complexity and Lyapunov Exponent 18 3 Limits of the Lyapunov exponent for predictability 21 3.1 Characterization of finite-time fluctuations 21 3.2 Renyi entropies 25 3.3 The effects of intermittency on predictability 26 3.4 Growth of non infinitesimal perturbations 28 arXiv:nlin/0101029v1 [nlin.CD] 17 Jan 2001 3.5 The ǫ-entropy 32 4 Predictability in extended systems 34 4.1 Simplified models for extended systems and the thermodynamic limit 35 4.2 Overview on the predictability problem in extended systems 37 4.3 Butterfly effect in coupled map lattices 39 4.4 Comoving and Specific Lyapunov Exponents 41 4.5 Convective chaos and spatial complexity 42 4.6 Space-time evolution of localized perturbations 46 4.7 Macroscopic chaos in Globally Coupled Maps 49 Preprint submitted to Elsevier Preprint 30 April 2018 4.8 Predictability in presence of coherent structures 51 5 -
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. -
Control of Chaos: Methods and Applications
Automation and Remote Control, Vol. 64, No. 5, 2003, pp. 673{713. Translated from Avtomatika i Telemekhanika, No. 5, 2003, pp. 3{45. Original Russian Text Copyright c 2003 by Andrievskii, Fradkov. REVIEWS Control of Chaos: Methods and Applications. I. Methods1 B. R. Andrievskii and A. L. Fradkov Institute of Mechanical Engineering Problems, Russian Academy of Sciences, St. Petersburg, Russia Received October 15, 2002 Abstract|The problems and methods of control of chaos, which in the last decade was the subject of intensive studies, were reviewed. The three historically earliest and most actively developing directions of research such as the open-loop control based on periodic system ex- citation, the method of Poincar´e map linearization (OGY method), and the method of time- delayed feedback (Pyragas method) were discussed in detail. The basic results obtained within the framework of the traditional linear, nonlinear, and adaptive control, as well as the neural network systems and fuzzy systems were presented. The open problems concerned mostly with support of the methods were formulated. The second part of the review will be devoted to the most interesting applications. 1. INTRODUCTION The term control of chaos is used mostly to denote the area of studies lying at the interfaces between the control theory and the theory of dynamic systems studying the methods of control of deterministic systems with nonregular, chaotic behavior. In the ancient mythology and philosophy, the word \χαωσ" (chaos) meant the disordered state of unformed matter supposed to have existed before the ordered universe. The combination \control of chaos" assumes a paradoxical sense arousing additional interest in the subject. -
A Robust Image-Based Cryptology Scheme Based on Cellular Non-Linear Network and Local Image Descriptors
PRE-PRINT IJPEDS A robust image-based cryptology scheme based on cellular non-linear network and local image descriptors Mohammad Mahdi Dehshibia, Jamshid Shanbehzadehb, and Mir Mohsen Pedramb aDepartment of Computer Engineering, Faculty of Computer and Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran; bDepartment of Electrical and Computer Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran ARTICLE HISTORY Compiled August 14, 2018 ABSTRACT Cellular nonlinear network (CNN) provides an infrastructure for Cellular Automata to have not only an initial state but an input which has a local memory in each cell with much more complexity. This property has many applications which we have investigated it in proposing a robust cryptology scheme. This scheme consists of a cryptography and steganography sub-module in which a 3D CNN is designed to produce a chaotic map as the kernel of the system to preserve confidentiality and data integrity in cryptology. Our contributions are three-fold including (1) a feature descriptor is applied to the cover image to form the secret key while conventional methods use a predefined key, (2) a 3D CNN is used to make a chaotic map for making cipher from the visual message, and (3) the proposed CNN is also used to make a dynamic k-LSB steganography. Conducted experiments on 25 standard images prove the effectiveness of the proposed cryptology scheme in terms of security, visual, and complexity analysis. KEYWORDS Cryptography; Steganography; Chaotic map; Cellular Non-linear Network; Image Descriptor; Complexity 1. Introduction Sensitive and personal data are parts of data transmission over the Internet which have always been suspicious to be intercepted. -
Mitra S, Kulkarni S, Stanfield J. Learning at the Edge of Chaos: Self-Organising Systems in Education
Mitra S, Kulkarni S, Stanfield J. Learning at the Edge of Chaos: Self-Organising Systems in Education. In: Lees, HE; Noddings, N, ed. The Palgrave International Handbook of Alternative Education. London, UK: Palgrave Macmillan, 2016, pp.227-239. Copyright: This extract is taken from the author's original manuscript and has not been edited. The definitive, published, version of record is available here: http://www.palgrave.com/us/book/9781137412904 for product on www.palgrave.com and www.palgraveconnect.com. Please be aware that if third party material (e.g. extracts, figures, tables from other sources) forms part of the material you wish to archive you will need additional clearance from the appropriate rights holders. Date deposited: 13/10/2016 Embargo release date: 19 September 2019 Newcastle University ePrints - eprint.ncl.ac.uk Learning at the edge of chaos - self-organising systems in education Abstract: This chapter takes stock of the evolution of the current primary education model and the potential afforded by current technology and its effects on children. It reports the results of experiments with self-organising systems in primary education and introduces the concept of a Self- Organised Learning Environment (SOLEs). It then describes how SOLEs operate and discusses the implications of the physics of complex systems and their possible connection with self-organised learning amongst children. The implications for certification and qualifications in an internet- immersive world are also discussed. Key words: SOLE, self-organised learning environments, edge of chaos Authors: Sugata Mitra is Professor of Educational Technology and Director of SOLE Central at Newcastle University.