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Introduction 1
Dynamics 1
Order and Disorder 3
Orbit Coding 6
Dynamical Systems 9
Discrete Dynamical Systems 10
Continuous Dynamical Systems 11
Symbolic Image 15
Construction of a Symbolic Image 15
Symbolic Image Parameters 17
Pseudo-orbits and Admissible Paths 19
Transition Matrix 21
Subdivision Process 22
Sequence of Symbolic Images 23
Periodic Trajectories 27
Periodic [epsilon]-Trajectories 27
Localization Algorithm 31
Newton's Method 35
Basic Results 35
Component of Periodic [epsilon]-Trajectories 38
Component of Periodic Vertices 40
Invariant Sets 43
Definitions and Examples 43
Symbolic Image and Invariant Sets 46
Construction of Non-leaving Vertices 50
A Set-oriented Method 52
Chain Recurrent Set 55
Definitions and Examples 55
Neighborhood of Chain Recurrent Set 59
Algorithm for Localization 61
Attractors 65
Definitions and Examples 65
Attractor on Symbolic Image 72
Attractors of a System and its Symbolic Image 74
Transition Matrix and Attractors 77
The Construction of the Attractor-Repellor Pair 78
Filtration 85
Definition and Properties 85
Filtration on a Symbolic Image 90
Fine Sequence of Filtrations 93
Structural Graph 97
Symbolic Image and Structural Graph 97
Sequence of Symbolic Images 100
Structural Graph of the Symbolic Image 101
Construction of the Structural Graph 103
Entropy 107
Definitions and Properties 107
Entropy of the Space of Sequences 110
Entropy and Symbolic Image 113
The Entropy of a Label Space 115
Computation of Entropy 118
The Entropy of Henon Map 119
The Entropy of Logistic Map 119
Projective Space and Lyapunov Exponents 123
Definitions and Examples 123
Coordinates in the Projective Space 125
Linear Mappings 126
Base Sets on the Projective Space 128
Lyapunov Exponents 129
Morse Spectrum 137
Linear Extension 137
Definition of the Morse Spectrum 139
Labeled Symbolic Image 140
Computation of the Spectrum 141
Spectrum of the Symbolic Image 144
Estimates for the Morse Spectrum 147
Localization of the Morse Spectrum 150
Exponential Estimates 151
Chain Recurrent Components 154
Linear Programming 156
Hyperbolicity and Structural Stability 161
Hyperbolicity 161
Structural Stability 168
Complementary Differential 169
Structural Stability Conditions 171
Verification Algorithm 172
Controllability 175
Global and Local Control 175
Symbolic Image of a Control System 177
Test for Controllability 178
Invariant Manifolds 181
Stable and Unstable Manifolds 181
Local Invariant Manifolds 185
Global Invariant Manifolds 186
Separatrices for a Hyperbolic Point 188
Two-dimensional Invariant Manifolds 193
Ikeda Mapping Dynamics 197
Analytical Results 197
Numerical Results 198
R = 0.3 199
R = 0.4 199
R = 0.5 199
R = 0.6 200
R = 0.7 203
R = 0.8 204
R = 0.9 204
R = 1.0 205
R = 1.1 207
Modified Ikeda Mappings 209
Mappings Preserving Orientation 210
Mappings Reversing Orientation 212
A Dynamical System of Mathematical Biology 219
Analytical Results 219
Numerical Results 221
M[subscript 0] = 3.000 221
M[subscript 0] = 3.300 222
M[subscript 0] = 3.3701 223
M[subscript 0] = 3.4001 224
M[subscript 0] = 3.480 225
M[subscript 0] = 3.532 226
M[subscript 0] = 3.540 227
M[subscript 0] = 3.570 227
M[subscript 0] = 3.571 229
Chaos 231
Conclusion 231
References 233
Double Logistic Map 241
Introduction 241
Hopf Bifurcation 242
The Application to Double Logistic Map 244
Construction of Periodic Orbits 247
Construction of the First Approximation 248
Refinement of Periodic Orbits 249
References 252
Implementation of the Symbolic Image 253
Implementation Details 254
Box and Cell Objects 254
Construction of the Symbolic Image 255
Subdivision Process 258
Basic Investigations on the Graph 259
Localization of the Chain Recurrent Set 259
Localization of Periodic Points 260
Performance Analysis 262
Accuracy of the Computations 263
Extensions for the Graph Construction 264
Dynamical Systems Continuous in Time 264
Error Tolerance for Box Images 265
Tunings for the Graph Investigation 266
Use of Higher Iterated Functions 267
Reconstruction of Fragmented Solutions 268
Numerical Case Studies 269
Ikeda Map 270
Coupled Logistic Map 273
Discrete Food Chain Model 275
Lorenz System 276
References 278
Index 281
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Add Dynamical Systems, Graphs, and Algorithms, This book describes a family of algorithms for studying the global structure of systems. By a finite covering of the phase space we construct a directed graph with vertices corresponding to cells of the covering and edges corresponding to admissible trans, Dynamical Systems, Graphs, and Algorithms to the inventory that you are selling on WonderClubX
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Add Dynamical Systems, Graphs, and Algorithms, This book describes a family of algorithms for studying the global structure of systems. By a finite covering of the phase space we construct a directed graph with vertices corresponding to cells of the covering and edges corresponding to admissible trans, Dynamical Systems, Graphs, and Algorithms to your collection on WonderClub |