Topology And Dynamics Of Chaos: In Celebration Of Robert Gilmore's 70th Birthday : In Celebration of Robert Gilmore's 70th Birthday

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 chaos, and knot theory, respectively.Very few books cover the topological approach for investigating nonlinear dynamical systems. The present book will provide not only some historical — not necessarily widely known — contributions (about the different types of chaos introduced by Rössler and not just the “Rössler attractor”; Gumowski and Mira's contributions in electronics; Poincaré's heritage in nonlinear dynamics) but also some recent applications in laser dynamics, biology, etc. 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 Rassler, Ren(r) Lozi And Joan Birman, Have Made A Significant Impact On Chaos Theory, Discrete Chaos, And Knot Theory, Respectively. Very Few Books Cover The Topological Approach For Investigating Nonlinear Dynamical Systems. The Present Book Will Provide Not Only Some Historical Oco Not Necessarily Widely Known Oco Contributions (about The Different Types Of Chaos Introduced By Rassler And Not Just The Rassler Attractor; Gumowski And Mira's Contributions In Electronics; Poincar(r)'s Heritage In Nonlinear Dynamics) But Also Some Recent Applications In Laser Dynamics, Biology, 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 Rossler, Rene Lozi and Joan Birman, have made a significant impact on chaos theory, discrete chaos, and knot theory, respectively. Very few books cover the topological approach for investigating nonlinear dynamical systems. The present book will provide not only some historical - not necessarily widely known - contributions (about the different types of chaos introduced by Rossler and not just the "Rossler attractor "; Gumowski and Mira's contributions in electronics; Poincare's heritage in nonlinear dynamics) but also some recent applications in laser dynamics, biology, etc Introduction to topological analysis / Christophe Letellier and Robert Gilmore The peregrinations of Poincaré / R. Abraham A Toulouse research group in the "prehistoric" times of chaotic dynamics / Christian Mira Can we trust in numerical computations of chaotic solutions of dynamical systems? / René Lozi Chaos hierarchy - a review, thirty years later / Otto E. R̈össler and Christophe Letellier The mathematics of Lorenz knots / Joan S. Birman A braided view of a knotty story / Mario Natiello and Hernán Solari How topology came to chaos / Robert Gilmore Reflections from the fourth dimension / Marc Lefranc The symmetry of chaos / Christophe Letellier The shape of ocean color / Nicholas Tufillaro Low Dimensional dynamics in biological motor patterns / Gabriel B. Mindlin Minimal smooth chaotic flows / Jean-Marc Malasoma The chaotic marriage of physics and financial economics / Claire Gilmore Introduction of the sphere map with application to spin-torque nano-oscillators / Keith Gilmore and Robert Gilmore Robert Gilmore, a portrait / Hernán G. Solari. 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 5. Chaos hierarchy - A review, thirty years later Otto E. Rossler & Christophe Letellier1. Introduction; 2. A Short Autobiography; 3. Three Main Influences; 3.1. The multivibrator by Andronov, Khaikin and Vitt; 3.2. The Lorenz paper; 3.3. The Li-Yorke theorem; 4. My Earliest Paper on Chaos; 4.1. Phase space and chaotic attractor; 4.2. First-return map to a Poincare section; 4.3. Qualitative properties of the expected dynamics; 4.4. The equations and their chaotic solution; 4.5. Topological analysis; 5. A Short Walk through Various Topologically Inequivalent Classes of Chaos; 6. Conclusion Preface; Contents; 1. Introduction to topological analysis Christophe Letellier & Robert Gilmore; 1. Strange versus Chaotic; 2. Topological Analysis: the Program; 3. Topological Analysis: an Explicit Case; 4. Classification; 5. Hopes for the Future; References; Emergence of a Chaos Theory; 2. The peregrinations of Poincare R. Abraham; 1. Introduction; 2. The Origin; 2.1. Jules Henri Poincare; 3. Westward Journey; 3.1. George David Birkhoff; 3.2. Solomon Lefschetz; 3.3. Stephen Smale; 4. Eastward Journey; 4.1. Sophie Kovalevsky; 4.2. Aleksandr Andronov; 4.3. Leonid Pavlovich Shilnikov 6. Study of One-Dimensional Non-Invertible Maps (from 1972)7. Applications; 8. Two Exhibitions of Chaotic Images (1973 and 1975); 9. Conclusion; Acknowledgement; References; 4. Can we trust in numerical computations of chaotic solutions of dynamical systems? Rene Lozi; 1. Introduction; 2. Continuous and Discrete Chaotic Dissipative Dynamical Systems: a Paradigm for Possibly Flawed Computations; 2.1. Some classes of dynamical system; 2.2. Poincare map: a bridge between continuous and discrete dynamical system; 3. Collapsing Effects; 3.1. Undesirable chaotic transient 4.4. Chihiro Hayashi4.5. Yoshisuke Ueda; 5. Conclusion; Acknowledgments; References; 3. A Toulouse research group in the "prehistoric" times of chaotic dynamics Christian Mira; 1. Introduction. Birth of the Group. Approach of Dynamic Problems; 2. Basin Boundaries of Two-dimensional Noninvertible Maps (1963- 1975); 2.1. Genesis of the results; 2.2. Complex organization of basins (1968-1973); 3. Chaotic Attractors of Two-Dimensional Noninvertible Maps (1968-1975); 4. Normal Forms for Resonant Bifurcations (1969-1974); 5. Two-Dimensional Conservative Maps (1970-1975) 3.2. Enigmatic computations for the logistic map (1838)3.3. Collapsing orbit of the symmetric tent map; 3.4. Statistical properties; 4. Shadowing and Parameter-shifted Shadowing Property of Mappings of the Plane; 4.1. Henon map (1976) found by mistake; 4.2. Lozi map (1977) a tractable model; 4.3. Shadowing, parameter-shifted shadowing and orbit-shifted shadowing properties; 5. Continuous Models; 5.1. Lorenz attractor (1963); 5.2. Geometric Lorenz attractor; 5.3. Lorenz map; 5.4. Rossler attractor (1976); 5.5. Chua attractor (1983); 6. Conclusion; References