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