Scaling laws in dynamical systems / / Edson Denis Leonel

• Record Nr.: UNISA996466731803316
• Autore: Leonel Edson Denis
• Pubbl/distr/stampa: Singapore : , : Springer, , [2021]; ©2021
• ISBN: 981-16-3544-7
• Descrizione fisica: 1 online resource (258 pages)
• Collana: Nonlinear physical science
• Disciplina: 530.1595
• Soggetti: Scaling laws (Statistical physics)
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Acknowledgements -- Contents -- List of Figures -- List of Tables -- 1 Introduction -- 1.1 Initial Concepts -- 1.2 Summary -- 2 One-Dimensional Mappings -- 2.1 Introduction -- 2.2 The Concept of Stability -- 2.2.1 Asymptotically Stable Fixed Point -- 2.2.2 Neutral Stability -- 2.2.3 Unstable Fixed Point -- 2.3 Fixed Points to the Logistic Map -- 2.4 Bifurcations -- 2.4.1 Transcritical Bifurcation -- 2.4.2 Period Doubling Bifurcation -- 2.4.3 Tangent Bifurcation -- 2.5 Summary -- 2.6 Exercises -- 3 Some Dynamical Properties for the Logistic Map -- 3.1 Convergence to the Stationary State -- 3.1.1 Transcritical Bifurcation -- 3.1.2 Period Doubling Bifurcation -- 3.1.3 Route to Chaos via Period Doubling -- 3.1.4 Tangent Bifurcation -- 3.2 Lyapunov Exponent -- 3.3 Summary -- 3.4 Exercises -- 4 The Logistic-Like Map -- 4.1 The Mapping -- 4.2 Transcritical Bifurcation -- 4.2.1 Analytical Approach to Obtain α, β, z and δ -- 4.2.2 Critical Exponents for the Period Doubling Bifurcation -- 4.3 Extensions to Other Mappings -- 4.3.1 Hassell Mapping -- 4.3.2 Maynard Mapping -- 4.4 Summary -- 4.5 Exercises -- 5 Introduction to Two Dimensional Mappings -- 5.1 Linear Mappings -- 5.2 Nonlinear Mappings -- 5.3 Applications of Two Dimensional Mappings -- 5.3.1 Hénon Map -- 5.3.2 Lyapunov Exponents -- 5.3.3 Ikeda Map -- 5.4 Summary -- 5.5 Exercises -- 6 A Fermi Accelerator Model -- 6.1 Fermi-Ulam Model -- 6.1.1 Jacobian Matrix for the Indirect Collisions -- 6.1.2 Jacobian Matrix for the Direct Collisions -- 6.1.3 Fixed Points -- 6.1.4 Phase Space -- 6.1.5 Phase Space Measure Preservation -- 6.2 A Simplified ; Version of the Fermi-Ulam Model -- 6.3 Scaling Properties for the Chaotic Sea -- 6.4 Localization of the First Invariant Spanning Curve -- 6.5 The Regime of Growth -- 6.6 Summary -- 6.7 Exercises -- 7 Dissipation in the Fermi-Ulam Model.; 7.1 Dissipation via Inelastic Collisions -- 7.1.1 Jacobian Matrix for the Direct Collisions -- 7.1.2 Jacobian Matrix for the Indirect Collisions -- 7.1.3 The Phase Space -- 7.1.4 Fixed Points -- 7.1.5 Construction of the Manifolds -- 7.1.6 Transient and Manifold Crossings Determination -- 7.1.7 Determining the Exponent δ from the Eigenvalues of the Saddle Point -- 7.2 Dissipation by Drag Force -- 7.2.1 Drag Force of the Type F=-tildeηv -- 7.2.2 Drag Force of the Type F=pmtildeηv2 -- 7.2.3 Drag Force of the Type F=-tildeηvγ -- 7.3 Summary -- 7.4 Exercises -- 8 Dynamical Properties for a Bouncer Model -- 8.1 The Model -- 8.2 Complete Version of the Bouncer Model -- 8.2.1 Successive Collisions -- 8.2.2 Indirect Collisions -- 8.2.3 Jacobian Matrix -- 8.2.4 The Phase Space -- 8.3 A Simplified Version of the Bouncer Model -- 8.4 Numerical Investigation on the Simplified Version -- 8.5 Approximation of Continuum Time -- 8.6 Summary -- 8.7 Exercises -- 9 Localization of Invariant Spanning Curves -- 9.1 The Standard Mapping -- 9.2 Localization of the Curves -- 9.3 Rescale in the Phase Space -- 9.4 Summary -- 9.5 Exercises -- 10 Chaotic Diffusion in Non-Dissipative Mappings -- 10.1 A Family of Discrete Mappings -- 10.2 Dynamical Properties for the Chaotic Sea: A Phenomenological Description -- 10.3 A Semi Phenomenological Approach -- 10.4 Determination of the Probability via the Solution of the Diffusion Equation -- 10.5 Summary -- 10.6 Exercises -- 11 Scaling on a Dissipative Standard Mapping -- 11.1 The Model -- 11.2 A Solution for the Diffusion Equation -- 11.3 Specific Limits -- 11.4 Summary -- 11.5 Exercises -- 12 Introduction to Billiard Dynamics -- 12.1 The Billiard -- 12.1.1 The Circle Billiard -- 12.1.2 The Elliptical Billiard -- 12.1.3 The Oval Billiard -- 12.2 Summary -- 12.3 Exercises -- 13 Time Dependent Billiards -- 13.1 The Billiard.; 13.1.1 The LRA Conjecture -- 13.2 The Time Dependent Elliptical Billiard -- 13.3 The Oval Billiard -- 13.4 Summary -- 13.5 Exercises -- 14 Suppression of Fermi Acceleration in the Oval Billiard -- 14.1 The Model and the Mapping -- 14.2 Results for the Case of Fpropto-V -- 14.3 Results for the Case of FproptopmV2 -- 14.4 Results for the Case of Fpropto-Vδ -- 14.5 Summary -- 14.6 Exercises -- 15 A Thermodynamic Model for Time Dependent Billiards -- 15.1 Motivation -- 15.2 Heat Transference -- 15.3 The Billiard Formalism -- 15.3.1 Stationary Estate -- 15.3.2 Dynamical Regime -- 15.3.3 Numerical Simulations -- 15.3.4 Average Velocity over n -- 15.3.5 Critical Exponents -- 15.3.6 Distribution of Velocities -- 15.4 Connection Between the Two Formalism -- 15.5 Summary -- 15.6 Exercises -- Appendix A Expressions for the Coefficients j -- Appendix B Change of Referential Frame -- B.1 Introduction -- B.2 Elastic Collisions -- B.3 Inelastic Collisions -- Appendix C Solution of the Diffusion Equation -- C.1 Introduction -- Appendix D Heat Flow Equation -- D.1 Introduction -- Appendix E Connection Between t and n in a Time Dependent Oval Billiard -- E.1 Introduction -- Appendix F Solution of the Integral to Obtain the Relation Between n and t in the Time Dependent Oval Billiard -- F.1 Introduction -- Appendix Bibliography.