Intro -- Preface -- Nomenclature -- 1 Dynamical systems -- 1.1 Main definitions -- 1.2 Embedding of a discrete dynamical system into a flow -- 1.3 Local Poincaré diffeomorphism -- 1.4 Time-periodic systems of differential equations -- 1.5 Action of an Abelian group -- 2 Topologies on spaces of dynamical systems -- 2.1 C0-topology -- 2.2 C1-topology -- 2.3 Metrics on the space of systems of differential equations -- 2.4 Generic properties -- 2.5 Immersions and embeddings -- 3 Equivalence relations -- 3.1 Topological conjugacy -- 3.2 Topological equivalence of flows -- 3.3 Nonwandering set -- 3.4 Local equivalence -- 4 Hyperbolic fixed point -- 4.1 Hyperbolic linear mapping -- 4.2 The Grobman-Hartman theorem -- 4.3 Neighborhood of a hyperbolic fixed point -- 4.4 The stable manifold theorem -- 4.5 Hyperbolic periodic point -- 5 Hyperbolic rest point and hyperbolic closed trajectory -- 5.1 Hyperbolic rest point -- 5.2 Hyperbolic closed trajectory -- 6 Transversality -- 6.1 Transversality of mappings and submanifolds -- 6.2 Transversality condition -- 6.3 Palis lemma -- 6.4 Transversality and hyperbolicity for one-dimensional mappings -- 7 Hyperbolic sets -- 7.1 Definition of a hyperbolic set -- 7.2 Examples of hyperbolic sets -- 7.3 Basic properties of hyperbolic sets -- 7.4 Stable manifold theorem -- 7.5 Axiom A -- 7.6 Hyperbolic sets of flows -- 8 Anosov diffeomorphisms -- 9 Smale's horseshoe and chaos -- 9.1 Smale's horseshoe -- 9.2 Chaotic sets -- 9.3 Homoclinic points -- 10 Closing Lemma -- 11 C0-generic properties of dynamical systems -- 11.1 Hausdorff metric -- 11.2 Semicontinuous mappings -- 11.3 Tolerance stability and Takens' theory -- 11.4 Attractors of dynamical systems -- 12 Shadowing of pseudotrajectories in dynamical systems -- 12.1 Definitions and results -- 12.2 Proof of Theorem 12.1 -- 12.3 Proof of Theorem 12.2.

12.4 Proof of Theorem 12.3 -- A Scheme of the proof of the Mane theorem -- B Lectures on the history of differential equations and dynamical systems -- B.1 Differential equations and Newton's anagram -- B.2 Development of the general theory -- B.3 Linear equations and systems -- B.4 Stability -- B.5 Nonlocal qualitative theory. Dynamical systems -- B.6 Structural stability -- B.7 Dynamical systems with chaotic behavior -- Bibliography -- Index.

Dynamical systems are abundant in theoretical physics and engineering. Their understanding, with sufficient mathematical rigor, is vital to solving many problems. This work conveys the modern theory of dynamical systems in a didactically developed fashion. In addition to topological dynamics, structural stability and chaotic dynamics, also generic properties and pseudotrajectories are covered, as well as nonlinearity. The author is an experienced book writer and his work is based on years of teaching.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.