Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors
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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 -
Complexity” Makes a Difference: Lessons from Critical Systems Thinking and the Covid-19 Pandemic in the UK
systems Article How We Understand “Complexity” Makes a Difference: Lessons from Critical Systems Thinking and the Covid-19 Pandemic in the UK Michael C. Jackson Centre for Systems Studies, University of Hull, Hull HU6 7TS, UK; [email protected]; Tel.: +44-7527-196400 Received: 11 November 2020; Accepted: 4 December 2020; Published: 7 December 2020 Abstract: Many authors have sought to summarize what they regard as the key features of “complexity”. Some concentrate on the complexity they see as existing in the world—on “ontological complexity”. Others highlight “cognitive complexity”—the complexity they see arising from the different interpretations of the world held by observers. Others recognize the added difficulties flowing from the interactions between “ontological” and “cognitive” complexity. Using the example of the Covid-19 pandemic in the UK, and the responses to it, the purpose of this paper is to show that the way we understand complexity makes a huge difference to how we respond to crises of this type. Inadequate conceptualizations of complexity lead to poor responses that can make matters worse. Different understandings of complexity are discussed and related to strategies proposed for combatting the pandemic. It is argued that a “critical systems thinking” approach to complexity provides the most appropriate understanding of the phenomenon and, at the same time, suggests which systems methodologies are best employed by decision makers in preparing for, and responding to, such crises. Keywords: complexity; Covid-19; critical systems thinking; systems methodologies 1. Introduction No one doubts that we are, in today’s world, entangled in complexity. At the global level, economic, social, technological, health and ecological factors have become interconnected in unprecedented ways, and the consequences are immense. -
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). -
Dissipative Structures, Complexity and Strange Attractors: Keynotes for a New Eco-Aesthetics
Dissipative structures, complexity and strange attractors: keynotes for a new eco-aesthetics 1 2 3 3 R. M. Pulselli , G. C. Magnoli , N. Marchettini & E. Tiezzi 1Department of Urban Studies and Planning, M.I.T, Cambridge, U.S.A. 2Faculty of Engineering, University of Bergamo, Italy and Research Affiliate, M.I.T, Cambridge, U.S.A. 3Department of Chemical and Biosystems Sciences, University of Siena, Italy Abstract There is a new branch of science strikingly at variance with the idea of knowledge just ended and deterministic. The complexity we observe in nature makes us aware of the limits of traditional reductive investigative tools and requires new comprehensive approaches to reality. Looking at the non-equilibrium thermodynamics reported by Ilya Prigogine as the key point to understanding the behaviour of living systems, the research on design practices takes into account the lot of dynamics occurring in nature and seeks to imagine living shapes suiting living contexts. When Edgar Morin speaks about the necessity of a method of complexity, considering the evolutive features of living systems, he probably means that a comprehensive method should be based on deep observations and flexible ordering laws. Actually designers and planners are engaged in a research field concerning fascinating theories coming from science and whose playground is made of cities, landscapes and human settlements. So, the concept of a dissipative structure and the theory of space organized by networks provide a new point of view to observe the dynamic behaviours of systems, finally bringing their flowing patterns into the practices of design and architecture. However, while science discovers the fashion of living systems, the question asked is how to develop methods to configure open shapes according to the theory of evolutionary physics. -
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. -
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. -
Fractal Curves and Complexity
Perception & Psychophysics 1987, 42 (4), 365-370 Fractal curves and complexity JAMES E. CUTI'ING and JEFFREY J. GARVIN Cornell University, Ithaca, New York Fractal curves were generated on square initiators and rated in terms of complexity by eight viewers. The stimuli differed in fractional dimension, recursion, and number of segments in their generators. Across six stimulus sets, recursion accounted for most of the variance in complexity judgments, but among stimuli with the most recursive depth, fractal dimension was a respect able predictor. Six variables from previous psychophysical literature known to effect complexity judgments were compared with these fractal variables: symmetry, moments of spatial distribu tion, angular variance, number of sides, P2/A, and Leeuwenberg codes. The latter three provided reliable predictive value and were highly correlated with recursive depth, fractal dimension, and number of segments in the generator, respectively. Thus, the measures from the previous litera ture and those of fractal parameters provide equal predictive value in judgments of these stimuli. Fractals are mathematicalobjectsthat have recently cap determine the fractional dimension by dividing the loga tured the imaginations of artists, computer graphics en rithm of the number of unit lengths in the generator by gineers, and psychologists. Synthesized and popularized the logarithm of the number of unit lengths across the ini by Mandelbrot (1977, 1983), with ever-widening appeal tiator. Since there are five segments in this generator and (e.g., Peitgen & Richter, 1986), fractals have many curi three unit lengths across the initiator, the fractionaldimen ous and fascinating properties. Consider four. sion is log(5)/log(3), or about 1.47. -
Measures of Complexity a Non--Exhaustive List
Measures of Complexity a non--exhaustive list Seth Lloyd d'Arbeloff Laboratory for Information Systems and Technology Department of Mechanical Engineering Massachusetts Institute of Technology [email protected] The world has grown more complex recently, and the number of ways of measuring complexity has grown even faster. This multiplication of measures has been taken by some to indicate confusion in the field of complex systems. In fact, the many measures of complexity represent variations on a few underlying themes. Here is an (incomplete) list of measures of complexity grouped into the corresponding themes. An historical analog to the problem of measuring complexity is the problem of describing electromagnetism before Maxwell's equations. In the case of electromagnetism, quantities such as electric and magnetic forces that arose in different experimental contexts were originally regarded as fundamentally different. Eventually it became clear that electricity and magnetism were in fact closely related aspects of the same fundamental quantity, the electromagnetic field. Similarly, contemporary researchers in architecture, biology, computer science, dynamical systems, engineering, finance, game theory, etc., have defined different measures of complexity for each field. Because these researchers were asking the same questions about the complexity of their different subjects of research, however, the answers that they came up with for how to measure complexity bear a considerable similarity to eachother. Three questions that researchers frequently ask to quantify the complexity of the thing (house, bacterium, problem, process, investment scheme) under study are 1. How hard is it to describe? 2. How hard is it to create? 3. What is its degree of organization? Here is a list of some measures of complexity grouped according to the question that they try to answer. -
Lecture Notes on Descriptional Complexity and Randomness
Lecture notes on descriptional complexity and randomness Peter Gács Boston University [email protected] A didactical survey of the foundations of Algorithmic Information The- ory. These notes introduce some of the main techniques and concepts of the field. They have evolved over many years and have been used and ref- erenced widely. “Version history” below gives some information on what has changed when. Contents Contents iii 1 Complexity1 1.1 Introduction ........................... 1 1.1.1 Formal results ...................... 3 1.1.2 Applications ....................... 6 1.1.3 History of the problem.................. 8 1.2 Notation ............................. 10 1.3 Kolmogorov complexity ..................... 11 1.3.1 Invariance ........................ 11 1.3.2 Simple quantitative estimates .............. 14 1.4 Simple properties of information ................ 16 1.5 Algorithmic properties of complexity .............. 19 1.6 The coding theorem....................... 24 1.6.1 Self-delimiting complexity................ 24 1.6.2 Universal semimeasure.................. 27 1.6.3 Prefix codes ....................... 28 1.6.4 The coding theorem for ¹Fº . 30 1.6.5 Algorithmic probability.................. 31 1.7 The statistics of description length ............... 32 2 Randomness 37 2.1 Uniform distribution....................... 37 2.2 Computable distributions..................... 40 2.2.1 Two kinds of test..................... 40 2.2.2 Randomness via complexity............... 41 2.2.3 Conservation of randomness............... 43 2.3 Infinite sequences ........................ 46 iii Contents 2.3.1 Null sets ......................... 47 2.3.2 Probability space..................... 52 2.3.3 Computability ...................... 54 2.3.4 Integral.......................... 54 2.3.5 Randomness tests .................... 55 2.3.6 Randomness and complexity............... 56 2.3.7 Universal semimeasure, algorithmic probability . 58 2.3.8 Randomness via algorithmic probability........ -
Chaos: the Mathematics Behind the Butterfly Effect
Chaos: The Mathematics Behind the Butterfly E↵ect James Manning Advisor: Jan Holly Colby College Mathematics Spring, 2017 1 1. Introduction A butterfly flaps its wings, and a hurricane hits somewhere many miles away. Can these two events possibly be related? This is an adage known to many but understood by few. That fact is based on the difficulty of the mathematics behind the adage. Now, it must be stated that, in fact, the flapping of a butterfly’s wings is not actually known to be the reason for any natural disasters, but the idea of it does get at the driving force of Chaos Theory. The common theme among the two is sensitive dependence on initial conditions. This is an idea that will be revisited later in the paper, because we must first cover the concepts necessary to frame chaos. This paper will explore one, two, and three dimensional systems, maps, bifurcations, limit cycles, attractors, and strange attractors before looking into the mechanics of chaos. Once chaos is introduced, we will look in depth at the Lorenz Equations. 2. One Dimensional Systems We begin our study by looking at nonlinear systems in one dimen- sion. One of the important features of these is the nonlinearity. Non- linearity in an equation evokes behavior that is not easily predicted due to the disproportionate nature of inputs and outputs. Also, the term “system”isoftenamisnomerbecauseitoftenevokestheideaof asystemofequations.Thiswillbethecaseaswemoveourfocuso↵ of one dimension, but for now we do not want to think of a system of equations. In this case, the type of system we want to consider is a first-order system of a single equation. -
Analysis of Chaotic Time Series Mathematical Theory of Chaotic
Analysis of chaotic Mathematical theory time series of chaotic systems Dimension Identification Lyapunov exponents Approximation and Modelling Prediction Control Use of chaotic properties in system control Chaotic time series Part I Estimation of invariant prop eries in state space y D Kugiumtzis B Lillekjendlie N Christophersen y January Abs tra ct Certain deterministic nonlinear systems may show chaotic b ehaviour Time series de rived from such systems seem sto chastic when analyzed with linear techniques However uncovering the deterministic structure is important because it allows for construction of more realistic and better models and thus improved predictive capabilities This pap er describ es key features of chaotic systems including strange attractors and Lyapunov expo nents The emphasis is on state space reconstruction techniques that are used to estimate these properties given scalar observations Data generated from equations known to display chaotic behaviour are used for illustration A compilation of applications to real data from widely dierent elds is given If chaos is found to b e present one may proceed to build nonlinear models which is the topic of the second pap er in this series Intro du ct ion Chaotic behaviour in deterministic dynamical systems is an intrinsicly nonlinear phenomenon A characteristic feature of chaotic systems is an extreme sensitivity to changes in initial con ditions while the dynamics at least for socalled dissipative systems is still constrained to a nite region of state space called a strange