Attractors and Basins of Dynamical Systems
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Chapter 6: Ensemble Forecasting and Atmospheric Predictability
Chapter 6: Ensemble Forecasting and Atmospheric Predictability Introduction Deterministic Chaos (what!?) In 1951 Charney indicated that forecast skill would break down, but he attributed it to model errors and errors in the initial conditions… In the 1960’s the forecasts were skillful for only one day or so. Statistical prediction was equal or better than dynamical predictions, Like it was until now for ENSO predictions! Lorenz wanted to show that statistical prediction could not match prediction with a nonlinear model for the Tokyo (1960) NWP conference So, he tried to find a model that was not periodic (otherwise stats would win!) He programmed in machine language on a 4K memory, 60 ops/sec Royal McBee computer He developed a low-order model (12 d.o.f) and changed the parameters and eventually found a nonperiodic solution Printed results with 3 significant digits (plenty!) Tried to reproduce results, went for a coffee and OOPS! Lorenz (1963) discovered that even with a perfect model and almost perfect initial conditions the forecast loses all skill in a finite time interval: “A butterfly in Brazil can change the forecast in Texas after one or two weeks”. In the 1960’s this was only of academic interest: forecasts were useless in two days Now, we are getting closer to the 2 week limit of predictability, and we have to extract the maximum information Central theorem of chaos (Lorenz, 1960s): a) Unstable systems have finite predictability (chaos) b) Stable systems are infinitely predictable a) Unstable dynamical system b) Stable dynamical -
A Gentle Introduction to Dynamical Systems Theory for Researchers in Speech, Language, and Music
A Gentle Introduction to Dynamical Systems Theory for Researchers in Speech, Language, and Music. Talk given at PoRT workshop, Glasgow, July 2012 Fred Cummins, University College Dublin [1] Dynamical Systems Theory (DST) is the lingua franca of Physics (both Newtonian and modern), Biology, Chemistry, and many other sciences and non-sciences, such as Economics. To employ the tools of DST is to take an explanatory stance with respect to observed phenomena. DST is thus not just another tool in the box. Its use is a different way of doing science. DST is increasingly used in non-computational, non-representational, non-cognitivist approaches to understanding behavior (and perhaps brains). (Embodied, embedded, ecological, enactive theories within cognitive science.) [2] DST originates in the science of mechanics, developed by the (co-)inventor of the calculus: Isaac Newton. This revolutionary science gave us the seductive concept of the mechanism. Mechanics seeks to provide a deterministic account of the relation between the motions of massive bodies and the forces that act upon them. A dynamical system comprises • A state description that indexes the components at time t, and • A dynamic, which is a rule governing state change over time The choice of variables defines the state space. The dynamic associates an instantaneous rate of change with each point in the state space. Any specific instance of a dynamical system will trace out a single trajectory in state space. (This is often, misleadingly, called a solution to the underlying equations.) Description of a specific system therefore also requires specification of the initial conditions. In the domain of mechanics, where we seek to account for the motion of massive bodies, we know which variables to choose (position and velocity). -
Research on the Digitial Image Based on Hyper-Chaotic And
Research on digital image watermark encryption based on hyperchaos Item Type Thesis or dissertation Authors Wu, Pianhui Publisher University of Derby Download date 27/09/2021 09:45:19 Link to Item http://hdl.handle.net/10545/305004 UNIVERSITY OF DERBY RESEARCH ON DIGITAL IMAGE WATERMARK ENCRYPTION BASED ON HYPERCHAOS Pianhui Wu Doctor of Philosophy 2013 RESEARCH ON DIGITAL IMAGE WATERMARK ENCRYPTION BASED ON HYPERCHAOS A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy By Pianhui Wu BSc. MSc. Faculty of Business, Computing and Law University of Derby May 2013 To my parents Acknowledgements I would like to thank sincerely Professor Zhengxu Zhao for his guidance, understanding, patience and most importantly, his friendship during my graduate studies at the University of Derby. His mentorship was paramount in providing a well-round experience consistent with my long-term career goals. I am grateful to many people in Faculty of Business, Computing and Law at the University of Derby for their support and help. I would also like to thank my parents, who have given me huge support and encouragement. Their advice is invaluable. An extra special recognition to my sister whose love and aid have made this thesis possible, and my time in Derby a colorful and wonderful experience. I Glossary AC Alternating Current AES Advanced Encryption Standard CCS Combination Coordinate Space CWT Continue Wavelet Transform BMP Bit Map DC Direct Current DCT Discrete Cosine Transform DWT Discrete Wavelet Transform -
An Image Cryptography Using Henon Map and Arnold Cat Map
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 05 Issue: 04 | Apr-2018 www.irjet.net p-ISSN: 2395-0072 An Image Cryptography using Henon Map and Arnold Cat Map. Pranjali Sankhe1, Shruti Pimple2, Surabhi Singh3, Anita Lahane4 1,2,3 UG Student VIII SEM, B.E., Computer Engg., RGIT, Mumbai, India 4Assistant Professor, Department of Computer Engg., RGIT, Mumbai, India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - In this digital world i.e. the transmission of non- 2. METHODOLOGY physical data that has been encoded digitally for the purpose of storage Security is a continuous process via which data can 2.1 HENON MAP be secured from several active and passive attacks. Encryption technique protects the confidentiality of a message or 1. The Henon map is a discrete time dynamic system information which is in the form of multimedia (text, image, introduces by michel henon. and video).In this paper, a new symmetric image encryption 2. The map depends on two parameters, a and b, which algorithm is proposed based on Henon’s chaotic system with for the classical Henon map have values of a = 1.4 and byte sequences applied with a novel approach of pixel shuffling b = 0.3. For the classical values the Henon map is of an image which results in an effective and efficient chaotic. For other values of a and b the map may be encryption of images. The Arnold Cat Map is a discrete system chaotic, intermittent, or converge to a periodic orbit. that stretches and folds its trajectories in phase space. Cryptography is the process of encryption and decryption of 3. -
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. -
A Cell Dynamical System Model for Simulation of Continuum Dynamics of Turbulent Fluid Flows A
A Cell Dynamical System Model for Simulation of Continuum Dynamics of Turbulent Fluid Flows A. M. Selvam and S. Fadnavis Email: [email protected] Website: http://www.geocities.com/amselvam Trends in Continuum Physics, TRECOP ’98; Proceedings of the International Symposium on Trends in Continuum Physics, Poznan, Poland, August 17-20, 1998. Edited by Bogdan T. Maruszewski, Wolfgang Muschik, and Andrzej Radowicz. Singapore, World Scientific, 1999, 334(12). 1. INTRODUCTION Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with inverse power-law form for power spectra of temporal fluctuations of all scales ranging from turbulence (millimeters-seconds) to climate (thousands of kilometers-years) (Tessier et. al., 1996) Long-range spatiotemporal correlations are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality (Bak et. al., 1988) Standard models for turbulent fluid flows in meteorological theory cannot explain satisfactorily the observed multifractal (space-time) structures in atmospheric flows. Numerical models for simulation and prediction of atmospheric flows are subject to deterministic chaos and give unrealistic solutions. Deterministic chaos is a direct consequence of round-off error growth in iterative computations. Round-off error of finite precision computations doubles on an average at each step of iterative computations (Mary Selvam, 1993). Round- off error will propagate to the mainstream computation and give unrealistic solutions in numerical weather prediction (NWP) and climate models which incorporate thousands of iterative computations in long-term numerical integration schemes. A recently developed non-deterministic cell dynamical system model for atmospheric flows (Mary Selvam, 1990; Mary Selvam et. -
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 -
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. -
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. -
The Dynamics of Runge–Kutta Methods Julyan H
The Dynamics of Runge±Kutta Methods Julyan H. E. Cartwright & Oreste Piro School of Mathematical Sciences Queen Mary and West®eld College University of London Mile End Road London E1 4NS U.K. The ®rst step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dy- namics of the most commonly used family of numerical integration schemes, Runge±Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis. QMW preprint DYN #91-9, Int. J. Bifurcation and Chaos, 2, 427±449, 1992 ¡ A CONICET (Consejo Nacional de Investigaciones Cienti®cas y Tecnicas de Argentina) Fellow. Present address: Institut Nonlineaire de Nice, Universite de NiceÐSophia Antipo- lis, Parc Valrose, 06034 Nice Cedex, France. 1 1. Introduction UMERICAL solution of ordinary differential equations is the most important technique in continuous time dynamics. Since most or- N dinary differential equations are not soluble analytically, numeri- cal integration is the only way to obtain information about the trajectory. Many different methods have been proposed and used in an attempt to solve accurately various types of ordinary differential equations. However there are a handful of methods known and used universally (i.e., Runge± Kutta, Adams±Bashforth±Moulton and Backward Differentiation Formulae methods). All these discretize the differential system to produce a differ- ence equation or map. The methods obtain different maps from the same differential equation, but they have the same aim; that the dynamics of the map should correspond closely to the dynamics of the differential equa- tion. -
Bifurcations, Mode Locking, and Chaos in a Dissipative System
The Bogdanov Map: Bifurcations, Mode Locking, and Chaos in a Dissipative System David K. Arrowsmith, School of Mathematical Sciences Queen Mary and Westfield College University of London Mile End Road London E1 4NS, UK Julyan H. E. Cartwright, Departament de F´ısica Universitat de les Illes Balears 07071 Palma de Mallorca, Spain Alexis N. Lansbury, Department of Physics Brunel, The University of West London Uxbridge, Middlesex UB8 3PH, UK &ColinM.Place.∗ Int. J. Bifurcation and Chaos, 3,803–842,1993 We investigate the bifurcations and basins of attraction in the Bog- danov map, a planar quadratic map which is conjugate to the Henon´ area-preserving map in its conservative limit. It undergoesaHopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homoclinic tangency of the manifolds of aremotesaddlepoint.TheBogdanovmapistheEulermapofatwo- dimensional system of ordinary differential equations firstconsidered by Bogdanov and Arnold in their study of the versal unfolding of the double-zero-eigenvalue singularity, and equivalently of avectorfield invariant under rotation of the plane by an angle 2π.Itisauseful system in which to observe the effect of dissipative perturbations on Hamiltonian structure. In addition, we argue that the Bogdanov map provides a good approximation to the dynamics of the Poincaremaps´ of periodically forced oscillators. ∗Formerly: Department of Mathematics, Westfield College, University of London, UK. 1 1. Introduction Tisnowwellknownthatthestudyofsystemsofordinarydifferential equations, which commonly arise in dynamical systems investigated in I many fields of science, can be aided by utilizing the surface-of-section technique of Poincaremaps.The´ Poincar´e or return map of a system of or- dinary differential equations reduces the dimension of the problem, replac- ing an n-dimensional set of ordinary differential equations with an (n 1)- dimensional set of difference equations.