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BL
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| Record Number
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863049
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| Main Entry
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Akhmet, Marat.
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| Title & Author
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Dynamics with chaos and fractals /\ Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily.
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| Publication Statement
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Cham :: Springer,, 2020.
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| Series Statement
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Nonlinear Systems and Complexity ;; v. 29
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| Page. NO
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1 online resource (233 pages)
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| ISBN
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3030358542
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: 9783030358549
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3030358534
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9783030358532
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| Notes
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11.5 Dynamics Motivated by Sierpinski Fractals
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro -- Preface -- Contents -- 1 Introduction -- References -- 2 The Unpredictable Point and Poincaré Chaos -- 2.1 Preliminaries -- 2.2 Dynamics with Unpredictable Points -- 2.3 Chaos on the Quasi-Minimal Set -- 2.4 Applications -- 2.5 Notes -- References -- 3 Unpredictability in Bebutov Dynamics -- 3.1 Introduction -- 3.2 Preliminaries -- 3.3 Unpredictable Functions -- 3.4 Unpredictable Solutions of Quasilinear Systems -- 3.5 Examples -- 3.6 Notes -- References -- 4 Nonlinear Unpredictable Perturbations -- 4.1 Preliminaries -- 4.2 An Unpredictable Sequence of the Symbolic Dynamics
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10 Global Weather and Climate in the Light of El Niño-Southern Oscillation -- 10.1 Introduction and Preliminaries -- 10.1.1 Unpredictability of Weather and Deterministic Chaos -- 10.1.2 Ocean-Atmosphere Interaction and Its Effects on Global Weather -- 10.1.3 El Niño Chaotic Dynamics -- 10.1.4 Sea Surface Temperature Advection Equation -- 10.1.5 Unpredictability and Poincaré Chaos -- 10.1.6 The Role of Chaos in Global Weather and Climate -- 10.2 Unpredictable Solution of the Advection Equation -- 10.2.1 Unpredictability Due to the Forcing Source Term
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10.2.2 Unpredictability Due to the Current Velocity -- 10.3 Chaotic Dynamics of the Global Ocean Parameters -- 10.3.1 Extension of Chaos in Coupled Advection Equations -- 10.3.2 Coupling of the Advection Equation with VallisModel -- 10.3.3 Coupling of Vallis Models -- 10.4 Ocean-Atmosphere Unpredictability Interaction -- 10.5 Notes -- References -- 11 Fractals: Dynamics in the Geometry -- 11.1 Introduction -- 11.2 Fatou-Julia Iteration -- 11.3 How to Map Fractals -- 11.4 Dynamics for Julia Sets -- 11.4.1 Discrete Dynamics -- 11.4.2 Continuous Dynamics
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4.3 An Unpredictable Solution of the Logistic Map -- 4.4 An Unpredictable Function -- 4.5 Unpredictable Solutions of Differential Equations -- 4.6 Notes -- References -- 5 Unpredictability in Topological Dynamics -- 5.1 Introduction -- 5.2 Quasilinear Delay Differential Equations -- 5.3 Quasilinear Discrete Equations -- 5.4 A Continuous Unpredictable Function via the Logistic Map -- 5.5 Examples -- 5.6 A Hopfield Neural Network -- 5.7 Notes -- References -- 6 Unpredictable Solutions of Hyperbolic Linear Equations -- 6.1 Preliminaries -- 6.2 Differential Equations with Unpredictable Solutions
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6.3 Discrete Equations with Unpredictable Solutions -- 6.4 Examples -- References -- 7 Strongly Unpredictable Solutions -- 7.1 Preliminaries -- 7.2 Main Results -- 7.3 Examples -- References -- 8 Li-Yorke Chaos in Hybrid Systems on a Time Scale -- 8.1 Introduction -- 8.2 Preliminaries -- 8.3 Bounded Solutions -- 8.4 The Chaotic Dynamics -- 8.5 An Example -- 8.6 Notes -- References -- 9 Homoclinic and Heteroclinic Motions in Economic Models -- 9.1 Introduction -- 9.2 The Model -- 9.3 Homoclinic and Heteroclinic Motions -- 9.4 An Example -- 9.5 Notes -- References
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| Abstract
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The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested. The Book Stands as the first book presenting theoretical background on the unpredictable point and mapping of fractals Introduces the concepts of unpredictable functions, abstract self-similarity, and similarity map Discusses unpredictable solutions of quasilinear ordinary and functional differential equations Illustrates new ways to construct fractals based on the ideas of Fatou and Julia Examines unpredictability in ocean dynamics and neural networks, chaos in hybrid systems on a time scale, and homoclinic and heteroclinic motions in economic models.
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| Subject
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Chaotic behavior in systems.
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| Subject
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Fractals.
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| Subject
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Chaotic behavior in systems.
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| Subject
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Fractals.
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| Dewey Classification
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003/.857
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| LC Classification
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QA614.8
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| Added Entry
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Alejaily, Ejaily Milad.
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Fen, Mehmet Onur.
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