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The book surveys how chaotic behaviors can be described with topological tools and how this approach occurred in chaos theory. Some modern applications are included. The contents are mainly devoted to topology, the main field of Robert Gilmore's works in dynamical systems. They include a review on the topological analysis of chaotic dynamics, works done in the past as well as the very latest issues. Most of the contributors who published during the 90's, including the very well-known scientists Otto Rössler, René Lozi and Joan Birman, have made a significant impact on chaos theory, discrete ch

Attractors (Mathematics) --- Chaotic behavior in systems --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Differentiable dynamical systems

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This accessible research monograph investigates how 'finite-dimensional' sets can be embedded into finite-dimensional Euclidean spaces. The first part brings together a number of abstract embedding results, and provides a unified treatment of four definitions of dimension that arise in disparate fields: Lebesgue covering dimension (from classical 'dimension theory'), Hausdorff dimension (from geometric measure theory), upper box-counting dimension (from dynamical systems), and Assouad dimension (from the theory of metric spaces). These abstract embedding results are applied in the second part of the book to the finite-dimensional global attractors that arise in certain infinite-dimensional dynamical systems, deducing practical consequences from the existence of such attractors: a version of the Takens time-delay embedding theorem valid in spatially extended systems, and a result on parametrisation by point values. This book will appeal to all researchers with an interest in dimension theory, particularly those working in dynamical systems.

Dimension theory (Topology) --- Attractors (Mathematics) --- Topological imbeddings. --- Embeddings, Topological --- Imbeddings, Topological --- Topological embeddings --- Embeddings (Mathematics) --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Differentiable dynamical systems --- Topology

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This book is the first monograph devoted exclusively to strange nonchaotic attractors (SNA), recently discovered objects with a special kind of dynamical behavior between order and chaos in dissipative nonlinear systems under quasiperiodic driving. A historical review of the discovery and study of SNA, mathematical and physically-motivated examples, and a review of known experimental studies of SNA are presented. The main focus is on the theoretical analysis of strange nonchaotic behavior by means of different tools of nonlinear dynamics and statistical physics (bifurcation analysis, Lyapunov

Attractors (Mathematics) --- Chaotic behavior in systems. --- Differentiable dynamical systems. --- Nonlinear systems. --- Systems, Nonlinear --- System theory --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics)

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This unique book's subject is meanders (connected, oriented, non-self-intersecting planar curves intersecting the horizontal line transversely) in the context of dynamical systems. By interpreting the transverse intersection points as vertices and the arches arising from these curves as directed edges, meanders are introduced from the graphtheoretical perspective. Supplementing the rigorous results, mathematical methods, constructions, and examples of meanders with a large number of insightful figures, issues such as connectivity and the number of connected components of meanders are studied in detail with the aid of collapse and multiple collapse, forks, and chambers. Moreover, the author introduces a large class of Morse meanders by utilizing the right and left one-shift maps, and presents connections to Sturm global attractors, seaweed and Frobenius Lie algebras, and the classical Yang-Baxter equation. Contents Seaweed Meanders Meanders Morse Meanders and Sturm Global Attractors Right and Left One-Shifts Connection Graphs of Type I, II, III and IV Meanders and the Temperley-Lieb Algebra Representations of Seaweed Lie Algebras CYBE and Seaweed Meanders

Curves, Algebraic. --- Attractors (Mathematics) --- Lie algebras. --- Yang-Baxter equation. --- Baxter-Yang equation --- Factorization equation --- Star-triangle relation --- Triangle equation --- Mathematical physics --- Quantum field theory --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Lie groups --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Differentiable dynamical systems --- Algebraic curves --- Algebraic varieties

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Group theory --- Differential equations --- Attractors (Mathematics) --- Lyapunov exponents. --- Stokes equations. --- Attracteurs (Mathématiques) --- Liapounov, Exposants de --- Equations de Stokes --- 51 <082.1> --- Mathematics--Series --- Attracteurs (Mathématiques) --- Navier-Stokes, Équations de. --- Liapounov, Exposants de. --- Attracteurs (mathématiques) --- Lyapunov exponents --- Stokes equations --- Stokes differential equations --- Stokes's differential equations --- Stokes's equations --- Differential equations, Partial --- Liapunov exponents --- Lyapunov characteristic exponents --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Differentiable dynamical systems