Book•
Global Stability of Dynamical Systems
01 Mar 2013-
TL;DR: In this article, the authors present a Potpourri of stability results for Hyperbolic Sets and Markov Partitions for stable manifolds, as well as a list of notations.
Abstract: 1 Generalities.- 2 Filtrations.- 3 Sequences of Filtrations.- 4 Hyperbolic Sets.- 5 Stable Manifolds.- 6 Stable Manifolds for Hyperbolic Sets.- 7 More Consequences of Hyperbolicity.- 8 Stability.- 9 A Potpourri of Stability Results.- 10 Markov Partitions.- List of Notation.
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Book•
19 Aug 1998
TL;DR: This chapter establishes the framework of random dynamical systems and introduces the concept of random attractors to analyze models with stochasticity or randomness.
Abstract: I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.
2,663 citations
Proceedings Article•
06 Jun 2016TL;DR: The authors showed that gradient descent converges to a local minimizer almost surely with random initialization by applying the Stable Manifold Theorem from dynamical systems theory, which is proved by applying stable manifold theorem to gradient descent.
Abstract: We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.
475 citations
01 Jan 1999
TL;DR: These notes were written for a D.E.A. course given at Ecole Normale Superieure de Cachan and University Toulouse III to introduce the reader to the dynamical system aspects of the theory of stochastic approximations.
Abstract: These notes were written for a D.E.A. course given at Ecole Normale Superieure de Cachan during the 1996–97 and 1997–98 academic years and at University Toulouse III during the 1997–98 academic year. Their aim is to introduce the reader to the dynamical system aspects of the theory of stochastic approximations.
462 citations
Book•
17 Aug 2011TL;DR: The theory of non-autonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this paper, where the focus is on dissipative systems and nonautonomous attractors.
Abstract: The theory of nonautonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this book. The focus is on dissipative systems and nonautonomous attractors, in particular the recently introduced concept of pullback attractors. Linearization theory, invariant manifolds, Lyapunov functions, Morse decompositions and bifurcations for nonautonomous systems and set-valued generalizations are also considered as well as applications to numerical approximations, switching systems and synchronization. Parallels with corresponding theories of control and random dynamical systems are briefly sketched. With its clear and systematic exposition, many examples and exercises, as well as its interesting applications, this book can serve as a text at the beginning graduate level. It is also useful for those who wish to begin their own independent research in this rapidly developing area.
440 citations
TL;DR: In this article, the authors studied the dynamical behavior of polynomial mappings with polynomials inverse from the real or complex plane to itself, and showed that the inverse can be represented as
Abstract: This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.
387 citations
Cites methods from "Global Stability of Dynamical Syste..."
...Standard proofs of this theorem for the real case, as described for example in [Sh], extend easily to the complex case....
[...]
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357 citations