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E-raamat: Replication of Chaos in Neural Networks, Economics and Physics

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  • Formaat: PDF+DRM
  • Sari: Nonlinear Physical Science
  • Ilmumisaeg: 13-Aug-2015
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662475003
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Replication of Chaos in Neural Networks, Economics and PhysicsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Nonlinear Physical Science
  • Ilmumisaeg: 13-Aug-2015
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662475003
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This book presents detailed descriptions of chaos for continuous-time systems. It is the first-ever book to consider chaos as an input for differential and hybrid equations. Chaotic sets and chaotic functions are used as inputs for systems with attractors: equilibrium points, cycles and tori. The findings strongly suggest that chaos theory can proceed from the theory of differential equations to a higher level than previously thought. The approach selected is conducive to the in-depth analysis of different types of chaos. The appearance of deterministic chaos in neural networks, economics and mechanical systems is discussed theoretically and supported by simulations. As such, the book offers a valuable resource for mathematicians, physicists, engineers and economists studying nonlinear chaotic dynamics.

Arvustused

The book presents a systematized and organized exposition of this material, its contextualization within mathematics, science and technology, and a discussion of perspectives for future development. In summary, this monograph provides an overview of results on an aspect of the dynamics of chaotically driven differential equations, such as replication of chaos, which is an interesting issue that goes beyond the research on synchronization and control of chaos that has received so much attention in the last twenty five years. (Jesús M. González-Miranda, Mathematical Reviews, May, 2016)

1 Introduction 1(32)
1.1 Synchronization of Chaotic Systems
6(2)
1.2 Control of Chaos
8(2)
1.3 Neural Networks and Chaos
10(1)
1.4 Extension of Chaos
10(2)
1.5 Ordering Chaos
12(2)
1.6 Self-organization of Chaos
14(4)
1.7 Morphogenesis of Chaos
18(2)
1.8 Chaos and Cellular Automata
20(1)
1.9 Synergetics and Chaos
21(1)
1.10 Mathematics in Chaos Theory
22(1)
1.11 Chaos Theory and Real World
23(3)
1.12 Organization of the Book
26(1)
References
27(6)
2 Replication of Continuous Chaos About Equilibria 33(68)
2.1 Introduction
33(4)
2.2 Preliminaries
37(4)
2.3 Chaotic Sets of Functions
41(4)
2.3.1 Devaney Set of Functions
41(3)
2.3.2 Li-Yorke Set of Functions
44(1)
2.4 Hyperbolic Set of Functions
45(3)
2.5 Replication of Devaney's Chaos
48(14)
2.6 Extension of Li-Yorke Chaos
62(7)
2.7 Morphogenesis of Chaos
69(6)
2.8 Period-Doubling Cascade
75(7)
2.9 Control by Replication
82(4)
2.10 Miscellany
86(12)
2.10.1 Intermittency
87(1)
2.10.2 Shilnikov Orbits
88(3)
2.10.3 Morphogenesis of the Double-Scroll Chua's Attractor
91(1)
2.10.4 Quasiperiodicity in Chaos
92(4)
2.10.5 Replicators with Nonnegative Eigenvalues
96(2)
2.11 Notes
98(1)
References
98(3)
3 Chaos Extension in Hyperbolic Systems 101(26)
3.1 Introduction
101(3)
3.2 Preliminaries
104(3)
3.3 Extension of Chaos
107(13)
3.4 Simulations
120(4)
3.5 Notes
124(1)
References
124(3)
4 Entrainment by Chaos 127(30)
4.1 Introduction
127(3)
4.2 Preliminaries
130(2)
4.3 Sensitivity
132(3)
4.4 Unstable Periodic Solutions
135(1)
4.5 Main Result
136(3)
4.6 Examples
139(4)
4.7 Miscellany
143(8)
4.7.1 Chaotic Tori
143(1)
4.7.2 Entrainment in Chua's Oscillators
144(1)
4.7.3 Controlling Chaos
145(2)
4.7.4 Entrainment and Synchronization
147(4)
4.8 The Regular Motion Near the Limit Cycle
151(2)
4.9 Notes
153(1)
References
154(3)
5 Chaotification of Impulsive Systems 157(26)
5.1 Introduction
157(3)
5.2 Preliminaries
160(3)
5.3 Chaotic Dynamics
163(10)
5.4 An Example
173(6)
5.5 Notes
179(1)
References
179(4)
6 Chaos Generation in Continuous/Discrete-Time Models 183(82)
6.1 Devaney's Chaos of a Relay System
183(12)
6.1.1 Introduction and Preliminaries
183(4)
6.1.2 The Chaos
187(4)
6.1.3 The Chaos on the Attractor
191(2)
6.1.4 The Period-Doubling Cascade and Intermittency: An Example
193(2)
6.2 Li-Yorke Chaos in Systems with Impacts
195(10)
6.2.1 Introduction and Preliminaries
195(4)
6.2.2 Main Results
199(6)
6.3 Li-Yorke Chaos in the System with Relay
205(4)
6.3.1 Introduction and Preliminaries
205(2)
6.3.2 The Li-Yorke Chaos
207(2)
6.4 Dynamical Synthesis of Quasi-Minimal Sets
209(7)
6.4.1 Introduction
209(2)
6.4.2 Main Result
211(2)
6.4.3 A Simulation Result
213(1)
6.4.4 Appendix
214(2)
6.5 Hyperbolic Sets of Impact Systems
216(3)
6.6 Chaos and Shadowing
219(10)
6.6.1 Introduction and Preliminaries
219(2)
6.6.2 The Devaney's Chaos
221(5)
6.6.3 Shadowing Property
226(2)
6.6.4 Simulations
228(1)
6.7 Chaos in the Forced Duffing Equation
229(28)
6.7.1 Introduction and Preliminaries
229(4)
6.7.2 The Chaos Emergence
233(10)
6.7.3 Controlling Results
243(8)
6.7.4 Morphogenesis and the Logistic Map
251(2)
6.7.5 Miscellany
253(4)
6.8 Notes
257(1)
References
258(7)
7 Economic Models with Exogenous Continuous/Discrete Shocks 265(46)
7.1 Chaos in Economic Models with Equilibria
265(21)
7.1.1 Introduction
265(4)
7.1.2 Modeling the Exogenous Shock
269(3)
7.1.3 Mathematical Investigation of System (7.1.5)
272(7)
7.1.4 Chaos in a Kaldor-Kalecki Model
279(7)
7.2 Chaotic Business Cycles
286(17)
7.2.1 Introduction
286(3)
7.2.2 The Input-Output Mechanism and Applications
289(2)
7.2.3 Economic Models: The Base Systems
291(2)
7.2.4 Chaos in a Stellar of Economical Models
293(7)
7.2.5 Kaldor-Kalecki Model with Time Delay
300(2)
7.2.6 Chaos Extension Versus Synchronization
302(1)
7.3 The Global Unpredictability, Self-organization and Synergetics
303(2)
7.4 Notes
305(2)
References
307(4)
8 Chaos by Neural Networks 311(96)
8.1 SICNNs with Chaotic External Inputs
312(13)
8.1.1 Introduction
312(1)
8.1.2 Preliminaries
313(4)
8.1.3 Chaotic Dynamics
317(5)
8.1.4 Examples
322(3)
8.2 Attraction of Chaos by Retarded SICNNs
325(25)
8.2.1 Introduction
326(5)
8.2.2 Preliminaries
331(2)
8.2.3 Li-Yorke Chaos
333(7)
8.2.4 An Example
340(6)
8.2.5 Synchronization of Chaos
346(4)
8.3 Impulsive SICNNs with Chaotic Postsynaptic Currents
350(28)
8.3.1 Introduction
350(5)
8.3.2 Preliminaries
355(6)
8.3.3 The Existence of Chaos
361(12)
8.3.4 Examples
373(5)
8.4 Cyclic/Toroidal Chaos in Hopfield Neural Networks
378(16)
8.4.1 Introduction
379(3)
8.4.2 Entrainment by Chaos in HNNs
382(6)
8.4.3 Control of Cyclic/Toroidal Chaos in Neural Networks
388(6)
8.5 Notes
394(3)
References
397(10)
9 The Prevalence of Weather Unpredictability 407(34)
9.1 Introduction
407(5)
9.2 Coupling Mechanism for Unpredictability
412(1)
9.3 Extension of Lorenz Unpredictability
413(6)
9.4 Period-Doubling Cascade
419(3)
9.5 Cyclic Chaos in Lorenz Systems
422(2)
9.6 Intermittency in the Weather Dynamics
424(1)
9.7 Self-Organization and Synergetics
425(2)
9.8 The Mathematical Background
427(9)
9.8.1 Bounded Positively Invariant Region
428(3)
9.8.2 Unpredictability Analysis
431(4)
9.8.3 Unstable Cycles and Unpredictability
435(1)
9.9 Notes
436(1)
References
437(4)
10 Spatiotemporal Chaos in Glow Discharge-Semiconductor Systems 441
10.1 Introduction
442(1)
10.2 Preliminaries
442(5)
10.2.1 Description of the GDS Model
444(2)
10.2.2 The Model in Dimensionless Form
446(1)
10.3 Chaotically Coupled GDS Systems
447(5)
10.4 The Chaos in the Drive GDS System
452(2)
10.5 Notes
454(1)
References
455
Prof. Dr. Marat Akhmet is a professor at the Department of Mathematics, Middle East Technical University, Ankara, Turkey. He is a specialist in dynamical models, chaos theory and differential equations. In the last several years, he has been investigating dynamics of neural networks, economic models and mechanical systems.

Dr. Mehmet Onur Fen is a postdoctoral researcher at the Department of Mathematics, Middle East Technical University, Ankara, Turkey. His research interests are differential equations, chaos theory and applications to neural networks, economics and mechanical systems.
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