DOCSLIB.ORG

The Effect of Explosive Divergence in a Coupled Map Lattice of Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Effect of Explosive Divergence in a Coupled Map Lattice of Matrices The Effect of Explosive Divergence in a Coupled Map Lattice of Matrices ∗ Guangqing Lu a, Rasa Smidtaite b,c, Zenonas Navickas b, Minvydas Ragulskis a,b, a School of Electrical and Information Engineering of Jinan University, 206 Qianshan Road, Zhuhai, Guangdong 519070, P.R. China b Centre of Nonlinear Systems, Kaunas University of Technology, Studentu 50-146, Kaunas LT-51368, Lithuania c Department of Applied Mathematics, Kaunas University of Technology, Studentu 50-318, Kaunas LT-51368, Lithuania a r t i c l e i n f o a b s t r a c t Article history: The extension of the Coupled map lattice (CML) model by replacing scalar nodal variables by matrix Received 16 April 2018 variables is investigated in this paper. The dynamics of the extended CML is investigated using formal Revised 11 June 2018 analytical and computational techniques. Necessary conditions for the occurrence of the effect of explo- Accepted 12 June 2018 sive divergence in the extended CML are derived. It is demonstrated that the extended CML can generate complex fractal patterns representing spatiotemporal divergence which can be controlled by the coupling Keywords: parameter between the nodes. The logistic map Coupled map lattices Kaneko model 1. Introduction plications in physics, cryptography, steganography, economics, bi- ology, etc. CMLs play an important role in image encryption The Logistic map is a paradigmatic model exhibiting chaotic be- algorithms [13–16] . Chaotic CML based stream ciphers are pro- havior [1]: posed by Li et al. [17] and Wang et al. [18] . CMLs are introduced to the market maker framework in order to investigate price changes ( + ) ( ) ( ) ( ) x t 1 = ax t (1 − x t ) ; t = 0 , 1 , 2 , . ; 0 ≤ a ≤ 4 ; 0 ≤ x 0 ≤ 1 . in a multi-market system [19] . A multiple phenotype predator– (1) prey model with mutation modelled via a CML [20] is proposed by Abernethy et al. CMLs are used to study neuronal dynamics [21] , A variety of extensions of the discrete logistic map has been coherence resonance phenomena [22] , and even stochastic reso- proposed during the last decades. The phase-modulated logistic nance on excitable small-world networks via a pacemaker [23] . map [2] is introduced by Nandi et al. Logistic map with a squared CMLs are also used as models for chaotic variations in spatiotem- sine perturbation [3] is presented by de Carvalho et al. Alternat- poral systems [24] . ing sign generalized logistic map with three parameters [4] is dis- CMLs exhibit a variety of spatiotemporal phenomena [25–32] . cussed by Sayed et al. Three bi- and tri-parameter generalized lo- Frozen random pattern, defect chaotic diffusion pattern, pattern gistic maps with arbitrary powers [5] are presented by Radwan. competition intermittency and fully developed turbulence in logis- Generalized fractional order logistic map suitable for pseudoran- tic map lattice with mixed linear-nonlinear couplings and its possi- dom number key generators [6] is presented by Ismail et al. Logis- ble applications in cryptography [25] are reviewed by Zhang. Two tic map in two dimensions is used to encrypt digital images [7– qualitative classes spatiotemporal intermittency and spatial inter- 9] . Chaotic encryption algorithm based on 3D logistic map [10] is mittency [26,27] are analyzed in the sine circle map lattice by proposed by Khade et al. Modified three-dimensional logistic map Jabeen et al. Sufficient conditions for synchronization of coupled is applied in steganography [11] . Symmetric image encryption maps on certain graphs [28] are derived Medvedev et al. An an- method based on intertwining logistic map [12] is proposed by Ye alytical study of traveling waves in CML [29,30] is developed by et al. Herrera et al. Properties of rotating pulse waves and oscillations Coupled map lattices are widely used to study the dynam- in a unidirectionally CML of ring structure [31] are discussed by ics of spatially extended logistic map systems. CMLs have ap- Horikawa. Method of adaptive control (based on the quasi sliding mode control design) of spatiotemporal chaos in CML [32] is pro- posed by Rahmani. ∗ Corresponding author at: Centre of Nonlinear Systems, Kaunas University of The main objective of this manuscript as well as necessary pre- Technology, Studentu 50-146, Kaunas LT-51368, Lithuania. liminary information is depicted in the following section. E-mail address: [email protected] (M. Ragulskis). 2. Preliminaries and motivation 2.4. Motivation 2.1. Idempotents and nilpotents The replacement of the scalar variable in the Logistic map by a second order square matrix yields unexpected and non-trivial phe- Let us consider such second order real matrices X ∈ R 2 × 2 that nomena (explosive divergence, temporary divergence, etc.) [33,34] . λ λ both eigenvalues of X (denoted as 1 , 2 ) are real. Note that X is A natural question is: what happens if scalar nodal variables in the not necessarily a symmetric matrix. CML nodes are replaced by matrix variables? In other words, this λ = λ = 1 − λ , = , ; = If 1 2 , then matrices Dk : λ −λ ( X l I); k l 1 2 k replacement increases the complexity of the spatially extended Lo- k l = gistic map at the nodes. How this increased complexity is repre- l are conjugate idempotents (I is the identity matrix): det Dk ; + = · = δ δ sented in the patterns generated by the lattice? The main objective 0 D1 D2 I; Dk Dl kl Dk ( kl is the Dirac delta function). Then X can be decomposed into the idempotent matrix form: of this paper is to seek answers to these questions. X = λ1 D 1 + λ2 D 2 (2) 3. The Coupled map lattice of matrices λ = λ = λ , = − λ = , If 1 2 0 then N : X 0 I is a nilpotent: det N 0 As stated previously, the main objective of this paper is to study 2 = N ( is a zero matrix). Then X can be expressed in the form a CML where scalar nodal variables x (t)( i ) are replaced by matrix of a nilpotent matrix: variables X ( t ) ( i ) ∈ R 2 × 2 : X = λ I + N . (3) ( + ) ( ) ε ( ) ( ) 0 X t 1 ( i ) = ( 1 − ε ) f (X t ( i ) ) + f (X t ( i + 1 ) ) + f (X t ( i − 1 ) ) 2 (7) 2.2. The Logistic map of matrices Further on, this model is entitled as the Coupled map lattice of The extension of the Logistic map [33] by replacing the scalar matrices (CMLM). variable x ( t ) by a second order square matrix variable X ( t ) is dis- cussed by Navickas et al. It is shown [33] that if the matrix of ini- 3.1. Idempotent matrices of initial conditions with identical tial conditions X (0) is an idempotent matrix, then the evolution of idempotents the Logistic map of matrices is described by two separate scalar Logistic maps of eigenvalues: Let us assume that all matrices of initial conditions are idem- potent matrices with the same idempotents D1 and D2 : (t +1 ) (t ) (t ) λ = aλ 1 − λ ; (0) ( 0 ) ( 0 ) 1 1 1 ( ) = λ ( ) + λ ( ) X i 1 i D1 2 i D2 (8) ( + ) ( ) ( ) (4) λ t 1 = λ t − λ t , 2 a 2 1 2 Note that ( ) ( ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) 0 ( ) + 0 ( ) = λ ( ) + λ ( ) + λ ( ) + λ ( ) λ λ (0) X k X l 1 k 1 l D1 2 k 2 l D2 where 1 and 2 are two different distinct eigenvalues of X . In other words, the dynamics of the map is fully determined by (9) the initial eigenvalues and the parameter a . (0) ( ) ( ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) However, if the matrix of initial conditions X is a nilpotent 0 ( ) 0 ( ) = λ ( )λ ( ) + λ ( )λ ( ) X k X l 1 k 1 l D1 2 k 2 l D2 (10) matrix, then the Logistic map of matrices can be divided into two Then, different scalar maps. The first one is a scalar Logistic map of the ( 0 ) ( ) ( ) ( ) λ (0) 0 = 0 − 0 eigenvalue ( 0 is the eigenvalue of X ). The other scalar map is f X ( i) aX ( i) I X ( i) μ( t ) μ(0) = the map incorporating the supplementary parameter ( ( 0 ) ( 0 ) ( 0 ) ( 0 ) = aλ ( i ) 1 − λ ( i ) D + aλ ( i ) 1 − λ ( i ) D , 1 ) [33] : 1 1 1 2 2 2 (11) ( + ) ( ) ( ) λ t 1 = λ t − λ t ; 0 a 0 1 0 (5) where I is the second order identity matrix. ( + ) ( ) (t ) μ t 1 = μ t − λ . Therefore, a 1 2 0 (1) ( 1 ) ( 1 ) X ( i ) = L ( i )D + L ( i )D (12) It is demonstrated [34] that the Logistic map of matrices can 1 1 2 2 exhibit the effect of explosive divergence if and only if the ma- where trix of initial conditions is a nilpotent matrix and the Lyapunov ex- ( 1 ) ( 1 ) ε ( 1 ) ( 1 ) ( ) = ( − ε )λ ( ) + λ ( + ) + λ ( − ) ; L1 i 1 1 i 2 1 i 1 1 i 1 ponent of the complementary scalar iterative map is greater than (13) ( 1 ) ( 1 ) ε ( 1 ) ( 1 ) zero. ( ) = ( − ε )λ ( ) + λ ( + ) + λ ( − ) , L2 i 1 2 i 2 2 i 1 2 i 1 ( ) ( ) λ 1 (s ) = f (λ 0 (s )) ; k = 1 , 2 ; s = 1 , . , n ; n is the number of k k 2.3. Coupled map lattices nodes in the CMLM with periodic boundary conditions ( (i − 1) = n when i = 1 ; (i + 1) = 1 when i = n ). The paradigmatic model of coupled map lattices is the Kaneko Thus, finally, model with periodic boundary conditions [35] : (t+1) (t +1 ) (t +1 ) X ( i ) = L ( i )D + L ( i )D (14) 1 1 2 2 ( + ) ( ) ε ( ) ( ) x t 1 ( i ) = ( 1 − ε ) f (x t ( i ) ) + f (x t ( i + 1 ) ) + f (x t ( i − 1 ) ) where 2 (t +1 ) (t ) ε (t ) (t ) (6) L ( i ) = ( 1 − ε ) f L ( i ) + f L ( i + 1 ) + f L ( i − 1 ) ; 1 1 2 1 1 (t +1 ) (t ) ε (t ) (t ) = , , .
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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • SPATIOTEMPORAL CHAOS in COUPLED MAP LATTICE Itishree
    SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE By Itishree Priyadarshini Under the Guidance of Prof. Biplab Ganguli Department of Physics National Institute of Technology, Rourkela CERTIFICATE This is to certify that the project thesis entitled ” Spatiotemporal chaos in Cou- pled Map Lattice ” being submitted by Itishree Priyadarshini in partial fulfilment to the requirement of the one year project course (PH 592) of MSc Degree in physics of National Institute of Technology, Rourkela has been carried out under my super- vision. The result incorporated in the thesis has been produced by developing her own computer codes. Prof. Biplab Ganguli Dept. of Physics National Institute of Technology Rourkela - 769008 1 ACKNOWLEDGEMENT I would like to acknowledge my guide Prof. Biplab Ganguli for his help and guidance in the completion of my one-year project and also for his enormous moti- vation and encouragement. I am also very much thankful to research scholars whose encouragement and support helped me to complete my project. 2 ABSTRACT The sensitive dependence on initial condition, which is the essential feature of chaos is demonstrated through simple Lorenz model. Period doubling route to chaos is shown by analysis of Logistic map and other different route to chaos is discussed. Coupled map lattices are investigated as a model for spatio-temporal chaos. Diffusively coupled logistic lattice is studied which shows different pattern in accordance with the coupling constant and the non-linear parameter i.e. frozen random pattern, pattern selection with suppression of chaos , Brownian motion of the space defect, intermittent collapse, soliton turbulence and travelling waves. 3 Contents 1 Introduction 3 2 Chaos 3 3 Lorenz System 4 4 Route to Chaos 6 4.1 PeriodDoubling.............................
    [Show full text]
  • Analysis of the Coupled Logistic Map
    Theoretical Population Biology TP1365 Theoretical Population Biology 54, 1137 (1998) Article No. TP981365 Spatial Structure, Environmental Heterogeneity, and Population Dynamics: Analysis of the Coupled Logistic Map Bruce E. Kendall* Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721, and National Center for Ecological Analysis and Synthesis, University of California, Santa Barbara, California 93106 and Gordon A. Fox- Department of Biology 0116, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0116, and Department of Biology, San Diego State University, San Diego, California 92182 Spatial extent can have two important consequences for population dynamics: It can generate spatial structure, in which individuals interact more intensely with neighbors than with more distant conspecifics, and it allows for environmental heterogeneity, in which habitat quality varies spatially. Studies of these features are difficult to interpret because the models are complex and sometimes idiosyncratic. Here we analyze one of the simplest possible spatial population models, to understand the mathematical basis for the observed patterns: two patches coupled by dispersal, with dynamics in each patch governed by the logistic map. With suitable choices of parameters, this model can represent spatial structure, environmental heterogeneity, or both in combination. We synthesize previous work and new analyses on this model, with two goals: to provide a comprehensive baseline to aid our understanding of more complex spatial models, and to generate predictions about the effects of spatial structure and environmental heterogeneity on population dynamics. Spatial structure alone can generate positive, negative, or zero spatial correlations between patches when dispersal rates are high, medium, or low relative to the complexity of the local dynamics.
    [Show full text]
  • Novel Coupling Scheme to Control Dynamics of Coupled Discrete Systems
    Novel coupling scheme to control dynamics of coupled discrete systems Snehal M. Shekatkar, G. Ambika1 Indian Institute of Science Education and Research, Pune 411008, India Abstract We present a new coupling scheme to control spatio-temporal patterns and chimeras on 1-d and 2-d lattices and random networks of discrete dynamical systems. The scheme involves coupling with an external lattice or network of damped systems. When the system network and external network are set in a feedback loop, the system network can be controlled to a homogeneous steady state or synchronized periodic state with suppression of the chaotic dynamics of the individual units. The control scheme has the advantage that its design does not require any prior information about the system dynamics or its parameters and works effectively for a range of parameters of the control network. We analyze the stability of the controlled steady state or amplitude death state of lattices using the theory of circulant matrices and Routh-Hurwitz’s criterion for discrete systems and this helps to isolate regions of effective control in the relevant parameter planes. The conditions thus obtained are found to agree well with those obtained from direct numerical simulations in the specific context of lattices with logistic map and Henon map as on-site system dynamics. We show how chimera states developed in an experimentally realizable 2-d lattice can be controlled using this scheme. We propose this mechanism can provide a phenomenological model for the control of spatio-temporal patterns in coupled neurons due to non-synaptic coupling with the extra cellular medium.
    [Show full text]
  • Coupled Logistic Maps and Non-Linear Differential Equations
    Coupled logistic maps and non-linear differential equations Eytan Katzav1 and Leticia F. Cugliandolo1,2 1Laboratoire de Physique Th´eorique de l’Ecole´ Normale Sup´erieure, 24 rue Lhomond, 75231 Paris Cedex 05, France 2Laboratoire de Physique Th´eorique et Hautes Energies,´ Jussieu, 5`eme ´etage, Tour 25, 4 Place Jussieu, 75252 Paris Cedex 05, France Abstract. We study the continuum space-time limit of a periodic one dimensional array of deterministic logistic maps coupled diffusively. First, we analyse this system in connection with a stochastic one dimensional Kardar-Parisi-Zhang (KPZ) equation for confined surface fluctuations. We compare the large-scale and long-time behaviour of space-time correlations in both systems. The dynamic structure factor of the coupled map lattice (CML) of logistic units in its deep chaotic regime and the usual d = 1 KPZ equation have a similar temporal stretched exponential relaxation. Conversely, the spatial scaling and, in particular, the size dependence are very different due to the intrinsic confinement of the fluctuations in the CML. We discuss the range of values of the non-linear parameter in the logistic map elements and the elastic coefficient coupling neighbours on the ring for which the connection with the KPZ-like equation holds. In the same spirit, we derive a continuum partial differential equation governing the evolution of the Lyapunov vector and we confirm that its space-time behaviour becomes the one of KPZ. Finally, we briefly discuss the interpretation of the continuum limit of the CML as a Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) non-linear diffusion equation with an additional KPZ non-linearity and the possibility of developing travelling wave configurations.
    [Show full text]
  • Synchronization and Periodic Windows in Globally Coupled Map Lattice
    The Fourteenth International Symposium on Artificial Life and Robotics 2009 (AROB 14th ’09), B-Con Plaza, Beppu, Oita, Japan, February 5 - 7, 2009 Synchronization and Periodic Windows in Globally Coupled Map Lattice Tokuzo Shimada, Kazuhiro Kubo, Takanobu Moriya Department of Physics, Meiji University Higashi-Mita 1-1, Kawasaki, Kanagawa 214, Japan A globally coupled map lattice (GCML) is an extension of a spin glass model. It consists of a large number of maps with a high nonlinearity and evolves iteratively under averaging interaction via their mean field. It exhibits various interesting phases under the conflict between randomness and coherence. We have found that even in its weak coupling regime, effects of the periodic windows of element maps dominate the dynamics of the system, and the system forms periodic cluster attractors. This may give a clue to the efficient pattern recognition by the brain. We analyze how the effect systematically depends on the distance from the periodic windows in the parameter space. Key words: synchronization, cluster attractors, peri- which oscillate in period p around the mean field with odicity manifestation, globally coupled maps phases (exp(2¼j=p); j = 0; 1; ¢ ¢ ¢ ; p ¡ 1). Such a MSCA is in general associated by a sequence of cluster attrac- tors; (p; c = p ¡ 1) ! (p; c = p ¡ 2) ! ¢ ¢ ¢ , which are I. INTRODUCTION produced in order with the increase of the coupling ".A basic tool to detect the GCML state is the mean square GCML devised by Kaneko [1, 2] is an unfailing source deviation of the mean field h(n) in time defined by of ideas on the behavior of a complex system composed of many chaotic elements.
    [Show full text]
  • Collective Dynamics and Homeostatic Emergence in Complex Adaptive Ecosystem
    ECAL - General Track Collective Dynamics and Homeostatic Emergence in Complex Adaptive Ecosystem Dharani Punithan and RI (Bob) McKay Structural Complexity Laboratory, College of Engineering, School of Computer Science and Engineering, Seoul National University, South Korea punithan.dharani,rimsnucse @gmail.com { } Downloaded from http://direct.mit.edu/isal/proceedings-pdf/ecal2013/25/332/1901807/978-0-262-31709-2-ch049.pdf by guest on 26 September 2021 Abstract 1. Life-environment feedback via the daisyworld model We investigate the behaviour of the daisyworld model on 2. State-topology feedback via an adaptive network model an adaptive network, comparing it to previous studies on a fixed topology grid, and a fixed small-world (Newman-Watts 3. Density-growth feedback via a logistic growth model (NW)) network. The adaptive networks eventually generate topologies with small-world effect behaving similarly to the The topology of the our ecosystem evolves with a simple NW model – and radically different from the grid world. Un- der the same parameter settings, static but complex patterns local rule – a frozen habitat is reciprocally linked to an ac- emerge in the grid world. In the NW model, we see the tive habitat – and self-organises to complex topologies with emergence of completely coherent periodic dominance. In small-world effect. In this paper, we focus on the emergent the adaptive-topology world, the systems may transit through collective phenomena and properties that arise in egalitarian varied behaviours, but can self-organise to a small-world small-world ecosystems, constructed from a large number of network structure with similar cyclic behaviour to the NW model.
    [Show full text]