Dynamical Systems, Graphs, and Algorithms Book

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