Dynamical Systems, Graphs, and AlgorithmsThe modern theory and practice of dynamical systems requires the study of structures that fall outside the scope of traditional subjects of mathematical analysis. An important tool to investigate such complicated phenomena as chaos and strange attractors is the method of symbolic dynamics. This book describes a family of the algorithms to study global structure of systems. By a finite covering of the phase space we construct a directed graph (symbolic image) with vertices corresponding to cells of the covering and edges corresponding to admissible transitions. The method is used to localize the periodic orbits and the chain recurrent set, to construct the attractors and their basins, to estimate the entropy, Lyapunov exponents and the Morse spectrum, to verify the hyperbolicity and the structural stability. Considerable information can be obtained thus, and more techniques may be discovered in future research. |
Contents
Introduction | 1 |
Symbolic Image | 15 |
Periodic Trajectories | 27 |
Copyright | |
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A₁ admissible path algorithm approximation C₁ cell M(i chain recurrent set chaotic characteristic exponent class H compact complementary differential component computation construction coordinates corresponding covering defined Definition Denote diffeomorphism discrete domain of attraction dynamical system eigenvalues entropy equivalent recurrent vertices Example exists finite follows G(PF G₁ graph G H₁ Hence homeomorphism homoclinic points hyperbolic fixed point Ikeda map image G intersect iterates labeled Let us consider linear Lyapunov exponents M₁ mapping f mapping ƒ matrix maximal diameter maximal invariant set method Möbius band Morse spectrum non-leaving nonwandering obtain p-periodic parameter Per(p periodic orbits periodic trajectory phase space projective space Proof Proposition pseudo-orbit repellor separatrices set of vertices set Q simple periodic paths solution stable and unstable strongly connected components structural graph subdivision symbolic dynamics symbolic image topological entropy U₁ UM(i unstable manifold V₁ vector vertex