An Image Cryptography Using Henon Map and Arnold Cat Map
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Approximating Stable and Unstable Manifolds in Experiments
PHYSICAL REVIEW E 67, 037201 ͑2003͒ Approximating stable and unstable manifolds in experiments Ioana Triandaf,1 Erik M. Bollt,2 and Ira B. Schwartz1 1Code 6792, Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375 2Department of Mathematics and Computer Science, Clarkson University, P.O. Box 5815, Potsdam, New York 13699 ͑Received 5 August 2002; revised manuscript received 23 December 2002; published 12 March 2003͒ We introduce a procedure to reveal invariant stable and unstable manifolds, given only experimental data. We assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle. We show that the method is able to capture the fine structure of the manifold, is independent of dimension, and is efficient relative to previous techniques. DOI: 10.1103/PhysRevE.67.037201 PACS number͑s͒: 05.45.Ac, 42.60.Mi Many nonlinear phenomena can be explained by under- We consider a basin saddle of Eq. ͑1͒ which lies on the standing the behavior of the unstable dynamical objects basin boundary between a chaotic attractor and a period-four present in the dynamics. Dynamical mechanisms underlying periodic attractor. The chosen parameters are in a region in chaos may be described by examining the stable and unstable which the chaotic attractor disappears and only a periodic invariant manifolds corresponding to unstable objects, such attractor persists along with chaotic transients due to inter- as saddles ͓1͔. Applications of the manifold topology have secting stable and unstable manifolds of the basin saddle. -
Fractal Growth on the Surface of a Planet and in Orbit Around It
Wilfrid Laurier University Scholars Commons @ Laurier Physics and Computer Science Faculty Publications Physics and Computer Science 10-2014 Fractal Growth on the Surface of a Planet and in Orbit around It Ioannis Haranas Wilfrid Laurier University, [email protected] Ioannis Gkigkitzis East Carolina University, [email protected] Athanasios Alexiou Ionian University Follow this and additional works at: https://scholars.wlu.ca/phys_faculty Part of the Mathematics Commons, and the Physics Commons Recommended Citation Haranas, I., Gkigkitzis, I., Alexiou, A. Fractal Growth on the Surface of a Planet and in Orbit around it. Microgravity Sci. Technol. (2014) 26:313–325. DOI: 10.1007/s12217-014-9397-6 This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. 1 Fractal Growth on the Surface of a Planet and in Orbit around it 1Ioannis Haranas, 2Ioannis Gkigkitzis, 3Athanasios Alexiou 1Dept. of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada 2Departments of Mathematics and Biomedical Physics, East Carolina University, 124 Austin Building, East Fifth Street, Greenville, NC 27858-4353, USA 3Department of Informatics, Ionian University, Plateia Tsirigoti 7, Corfu, 49100, Greece Abstract: Fractals are defined as geometric shapes that exhibit symmetry of scale. This simply implies that fractal is a shape that it would still look the same even if somebody could zoom in on one of its parts an infinite number of times. -
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 -
Robust Transitivity of Singular Hyperbolic Attractors
Robust transitivity of singular hyperbolic attractors Sylvain Crovisier∗ Dawei Yangy January 22, 2020 Abstract Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast to uniform hyperbolicity, singular hyperbolicity does not immediately imply robust topological properties, such as the transitivity. In this paper, we prove that open and densely inside the space of C1 vector fields of a compact manifold, any singular hyperbolic attractors is robustly transitive. 1 Introduction Lorenz [L] in 1963 studied some polynomial ordinary differential equations in R3. He found some strange attractor with the help of computers. By trying to understand the chaotic dynamics in Lorenz' systems, [ABS, G, GW] constructed some geometric abstract models which are called geometrical Lorenz attractors: these are robustly transitive non- hyperbolic chaotic attractors with singularities in three-dimensional manifolds. In order to study attractors containing singularities for general vector fields, Morales- Pacifico-Pujals [MPP] first gave the notion of singular hyperbolicity in dimension 3. This notion can be adapted to the higher dimensional case, see [BdL, CdLYZ, MM, ZGW]. In the absence of singularity, the singular hyperbolicity coincides with the usual notion of uniform hyperbolicity; in that case it has many nice dynamical consequences: spectral decomposition, stability, probabilistic description,... But there also exist open classes of vector fields exhibiting singular hyperbolic attractors with singularity: the geometrical Lorenz attractors are such examples. In order to have a description of the dynamics of arXiv:2001.07293v1 [math.DS] 21 Jan 2020 general flows, we thus need to develop a systematic study of the singular hyperbolicity in the presence of singularity. -
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). -
Hartman-Grobman and Stable Manifold Theorem
THE HARTMAN-GROBMAN AND STABLE MANIFOLD THEOREM SLOBODAN N. SIMIC´ This is a summary of some basic results in dynamical systems we discussed in class. Recall that there are two kinds of dynamical systems: discrete and continuous. A discrete dynamical system is map f : M → M of some space M.A continuous dynamical system (or flow) is a collection of maps φt : M → M, with t ∈ R, satisfying φ0(x) = x and φs+t(x) = φs(φt(x)), for all x ∈ M and s, t ∈ R. We call M the phase space. It is usually either a subset of a Euclidean space, a metric space or a smooth manifold (think of surfaces in 3-space). Similarly, f or φt are usually nice maps, e.g., continuous or differentiable. We will focus on smooth (i.e., differentiable any number of times) discrete dynamical systems. Let f : M → M be one such system. Definition 1. The positive or forward orbit of p ∈ M is the set 2 O+(p) = {p, f(p), f (p),...}. If f is invertible, we define the (full) orbit of p by k O(p) = {f (p): k ∈ Z}. We can interpret a point p ∈ M as the state of some physical system at time t = 0 and f k(p) as its state at time t = k. The phase space M is the set of all possible states of the system. The goal of the theory of dynamical systems is to understand the long-term behavior of “most” orbits. That is, what happens to f k(p), as k → ∞, for most initial conditions p ∈ M? The first step towards this goal is to understand orbits which have the simplest possible behavior, namely fixed and periodic ones. -
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. -
The Transition Between the Complex Quadratic Map and the Hénon Map
The Transition Between the Complex Quadratic Map and the Hénon Map by Sarah N. Kabes Technical Report Department of Mathematics and Statistics University of Minnesota Duluth, MN 55812 July 2012 The Transition Between the Complex Quadratic Map and the Hénon map A PROJECT SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Sarah Kabes in partial fulfillment of the requirements for the degree of Master of Science July 2012 Acknowledgements Thank you first and foremost to my advisor Dr. Bruce Peckham. Your patience, encouragement, excitement, and support while teaching have made this experience not only possible but enjoyable as well. Additional thanks to my committee members, Dr. Marshall Hampton and Dr. John Pastor for reading and providing suggestions for this project. Furthermore, without the additional assistance from two individuals my project would not be as complete as it is today. Thank you Dr. Harlan Stech for finding the critical value , and thank you Scot Halverson for working with me and the open source code to produce the movie. Of course none of this would be possibly without the continued support of my family and friends. Thank you all for believing in me. i Abstract This paper investigates the transition between two well known dynamical systems in the plane, the complex quadratic map and the Hénon map. Using bifurcation theory, an analysis of the dynamical changes the family of maps undergoes as we follow a “homotopy” from one map to the other is presented. Along with locating common local bifurcations, an additional un-familiar bifurcation at infinity is discussed. -
Chapter 3 Hyperbolicity
Chapter 3 Hyperbolicity 3.1 The stable manifold theorem In this section we shall mix two typical ideas of hyperbolic dynamical systems: the use of a linear approxi- mation to understand the behavior of nonlinear systems, and the examination of the behavior of the system along a reference orbit. The latter idea is embodied in the notion of stable set. Denition 3.1.1: Let (X, f) be a dynamical system on a metric space X. The stable set of a point x ∈ X is s ∈ k k Wf (x)= y X lim d f (y),f (x) =0 . k→∞ ∈ s In other words, y Wf (x) if and only if the orbit of y is asymptotic to the orbit of x. If no confusion will arise, we drop the index f and write just W s(x) for the stable set of x. Clearly, two stable sets either coincide or are disjoint; therefore X is the disjoint union of its stable sets. Furthermore, f W s(x) W s f(x) for any x ∈ X. The twin notion of unstable set for homeomorphisms is dened in a similar way: Denition 3.1.2: Let f: X → X be a homeomorphism of a metric space X. Then the unstable set of a ∈ point x X is u ∈ k k Wf (x)= y X lim d f (y),f (x) =0 . k→∞ u s In other words, Wf (x)=Wf1(x). Again, unstable sets of homeomorphisms give a partition of the space. On the other hand, stable sets and unstable sets (even of the same point) might intersect, giving rise to interesting dynamical phenomena. -
MA427 Ergodic Theory
MA427 Ergodic Theory Course Notes (2012-13) 1 Introduction 1.1 Orbits Let X be a mathematical space. For example, X could be the unit interval [0; 1], a circle, a torus, or something far more complicated like a Cantor set. Let T : X ! X be a function that maps X into itself. Let x 2 X be a point. We can repeatedly apply the map T to the point x to obtain the sequence: fx; T (x);T (T (x));T (T (T (x))); : : : ; :::g: We will often write T n(x) = T (··· (T (T (x)))) (n times). The sequence of points x; T (x);T 2(x);::: is called the orbit of x. We think of applying the map T as the passage of time. Thus we think of T (x) as where the point x has moved to after time 1, T 2(x) is where the point x has moved to after time 2, etc. Some points x 2 X return to where they started. That is, T n(x) = x for some n > 1. We say that such a point x is periodic with period n. By way of contrast, points may move move densely around the space X. (A sequence is said to be dense if (loosely speaking) it comes arbitrarily close to every point of X.) If we take two points x; y of X that start very close then their orbits will initially be close. However, it often happens that in the long term their orbits move apart and indeed become dramatically different. -
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. -
Attractors and Orbit-Flip Homoclinic Orbits for Star Flows
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 8, August 2013, Pages 2783–2791 S 0002-9939(2013)11535-2 Article electronically published on April 12, 2013 ATTRACTORS AND ORBIT-FLIP HOMOCLINIC ORBITS FOR STAR FLOWS C. A. MORALES (Communicated by Bryna Kra) Abstract. We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be C1 approximated by vector fields with orbit-flip homoclinic orbits. 1. Introduction The notion of attractor deserves a fundamental place in the modern theory of dynamical systems. This assertion, supported by the nowadays classical theory of turbulence [27], is enlightened by the recent Palis conjecture [24] about the abun- dance of dynamical systems with finitely many attractors absorbing most positive trajectories. If confirmed, such a conjecture means the understanding of a great part of dynamical systems in the sense of their long-term behaviour. Here we attack a problem which goes straight to the Palis conjecture: The fini- tude of the number of attractors for a given dynamical system. Such a problem has been solved positively under certain circunstances. For instance, we have the work by Lopes [16], who, based upon early works by Ma˜n´e [18] and extending pre- vious ones by Liao [15] and Pliss [25], studied the structure of the C1 structural stable diffeomorphisms and proved the finitude of attractors for such diffeomor- phisms. His work was largely extended by Ma˜n´e himself in the celebrated solution of the C1 stability conjecture [17]. On the other hand, the Japanesse researchers S.