Journal ArticleDOI
An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)
TLDR
In this paper, the quadratic family has been used to define hyperbolicity in linear algebra and advanced calculus, including the Julia set and the Mandelbrot set.Abstract:
Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Functionread more
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Dissertation
On non-archimedean dynamical systems
Abstract: A discrete dynamical system is a pair (X, cf;) comprising a non-empty set X and a map cf; : X ---+ X. A study is made of the effect of repeated application of cf; on X, whereby points and subsets of X are classified according to their behaviour under iteration. These subsets include the JULIA and FATOU sets of the map and the sets of periodic and preperiodic points, and many interesting questions arise in the study of their properties. Such questions have been extensively studied in the case of complex dynamics, but much recent work has focussed on non-archimedean dynamical systems, when X is projective space over some field equipped with a non-archimedean metric. This work has uncovered many parallels to complex dynamics alongside more striking differences. In this thesis, various aspects of the theory of non-archimedean dynamics are presented, with particular reference to JULIA and FATOU sets and the relationship between good reduction of a map and the empty JULIA set. We also discuss questions of the finiteness of the sets of periodic points in special contexts. ii Stellenbosch University http://scholar.sun.ac.za
New nonlinear mechanisms of midlatitude atmospheric low-frequency variability
TL;DR: In this paper, a low-order model derived from the 2-layer shallow-water equations on a β-plane channel is studied, which is characterised by recurrent non-propagating and temporally persistent flow patterns, with typical spatial and temporal scales of 6000-10,000 km and 10-50 days, respectively.
Journal Article
An Improved Chaotic Flower Pollination Algorithm for Solving LargeInteger Programming Problems
TL;DR: An improved version of Flower pollination Meta-heuristic Algorithm, (FPPSO), for solving integer programming problems, which combines the standard flower pollination algorithm (FP) with the particle swarm optimization (PSO) algorithm to improve the searching accuracy.
Journal ArticleDOI
Chaotic Behavior of Deterministic Dissipative Systems (Milos Marek and Igor Schreiber)
TL;DR: This volume is compulsory reading for all who are interested in mathematics in the real world; its contribution to the continuing development of a synergetic relationship between its readers and the IMA is Very welcome and would surely have met with approval of the late editor.
References
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Journal ArticleDOI
Metaheuristics in combinatorial optimization: Overview and conceptual comparison
Christian Blum,Andrea Roli +1 more
TL;DR: A survey of the nowadays most important metaheuristics from a conceptual point of view and introduces a framework, that is called the I&D frame, in order to put different intensification and diversification components into relation with each other.
Book
Chaos: An Introduction to Dynamical Systems
TL;DR: One-dimensional maps, two-dimensional map, fractals, and chaotic attraction attractors have been studied in this article for state reconstruction from data, including the state of Washington.
Journal ArticleDOI
Symmetric Ciphers Based on Two-Dimensional Chaotic Maps
TL;DR: Methods are shown how to adapt invertible two-dimensional chaotic maps on a torus or on a square to create new symmetric block encryption schemes to encrypt an N×N image.
Journal ArticleDOI
The dynamics of coherent structures in the wall region of a turbulent boundary layer.
TL;DR: In this article, the wall region of a turbulent boundary layer is modelled by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem.
Journal ArticleDOI
Adaptive greedy approximations
TL;DR: A notion of the coherence of a signal with respect to a dictionary is derived from the characterization of the approximation errors of a pursuit from their statistical properties, which can be obtained from the invariant measure of the pursuit.