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Scale Invariance in Nonlinear Dynamical Systems（非线性动力系统中的标度不变性）（英文版）

样章
作者：
Edson Denis Leonel 著
定价：
199.00 元
版面字数：
487.00千字
开本：
特殊
装帧形式：
精装
页数：
348
最新
印次时间：
2026年07月
ISBN：
978-7-04-067613-6
物料号：
67613-00
出版时间：
2026-08-05
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
动力系统
暂无
  • 目录
    • 前辅文
      • 1 Initial Discussion
        • 1.1 Objectives
          • 1.2 Initial Concepts
            • 1.3 Summary
            • 2 The Concept of Attractor
              • 2.1 Objectives
                • 2.2 Initial Concepts
                  • 2.3 The Damped Oscillator
                    • 2.3.1 Overdamping
                      • 2.3.2 Critical Damping
                        • 2.3.3 Underdamped Case
                        • 2.4 Van der Pol Oscillator
                          • 2.5 Chaotic Attractor
                            • 2.5.1 The Lorenz System
                              • 2.5.2 Duffing Equation
                              • 2.6 Strange Nonchaotic Attractor
                                • 2.7 Concept of Attractor
                                  • 2.8 Summary
                                    • 2.9 Proposed Exercises
                                    • 3 Stability of Fixed Points
                                      • 3.1 Objectives
                                        • 3.2 First-Order Linear Differential Equation
                                          • 3.3 Linear Systems
                                            • 3.4 Nonlinear Systems
                                              • 3.4.1 Example 1
                                                • 3.4.2 Example 2
                                                  • 3.4.3 Example 3
                                                  • 3.5 Summary
                                                    • 3.6 Proposed Exercises
                                                    • 4 Some Local Bifurcations
                                                      • 4.1 Objectives
                                                        • 4.2 Local Bifurcations
                                                          • 4.3 Saddle-Node Bifurcation
                                                            • 4.3.1 Example of Saddle-Node Bifurcation
                                                            • 4.4 Transcritical Bifurcation
                                                              • 4.4.1 Example of Transcritical Bifurcation
                                                              • 4.5 Supercritical Pitchfork Bifurcation
                                                                • 4.5.1 Example of Supercritical Pitchfork Bifurcation
                                                                • 4.6 Subcritical Pitchfork Bifurcation
                                                                  • 4.7 Normal Forms
                                                                    • 4.8 Summary
                                                                      • 4.9 Proposed Exercises
                                                                      • 5 Scaling Analysis in Local Bifurcations
                                                                        • 5.1 Objectives
                                                                          • 5.2 Convergence to Fixed Points
                                                                            • 5.3 Convergence to the Fixed Point: A Phenomenological Description
                                                                              • 5.3.1 Scaling Hypotheses
                                                                              • 5.4 Scaling Analysis in the Saddle-Node Bifurcation
                                                                                • 5.5 Scaling Analysis in the Transcritical Bifurcation
                                                                                  • 5.6 Scaling Analysis in the Supercritical Pitchfork Bifurcation
                                                                                    • 5.7 Summary
                                                                                      • 5.8 Proposed Exercises
                                                                                      • 6 One-Dimensional Discrete Maps
                                                                                        • 6.1 Objectives
                                                                                          • 6.2 Introduction
                                                                                            • 6.3 The Concept of Stability
                                                                                              • 6.3.1 Asymptotically Stable Fixed Point
                                                                                                • 6.3.2 Neutral Stability
                                                                                                  • 6.3.3 Unstable Fixed Point
                                                                                                  • 6.4 Applications of Fixed-Point Calculation in the Logistic Map
                                                                                                    • 6.5 Bifurcations
                                                                                                      • 6.5.1 Transcritical Bifurcation
                                                                                                        • 6.5.2 Period-Doubling Bifurcation
                                                                                                          • 6.5.3 Tangent Bifurcation
                                                                                                          • 6.6 Summary
                                                                                                            • 6.7 Proposed Exercises
                                                                                                            • 7 Some Dynamical and Statistical Properties of the Logistic Map
                                                                                                              • 7.1 Objectives
                                                                                                                • 7.2 Convergence to the Steady State
                                                                                                                  • 7.2.1 Transcritical Bifurcation
                                                                                                                    • 7.2.2 Period-Doubling Bifurcation
                                                                                                                      • 7.2.3 Route to Chaos via Period Doubling
                                                                                                                        • 7.2.4 Tangent Bifurcation
                                                                                                                        • 7.3 Lyapunov Exponents
                                                                                                                          • 7.4 Summary
                                                                                                                            • 7.5 Proposed Exercises
                                                                                                                            • 8 The Logistic-Like Map
                                                                                                                              • 8.1 Objectives
                                                                                                                                • 8.2 The Mapping Equation
                                                                                                                                  • 8.3 Transcritical Bifurcation
                                                                                                                                    • 8.3.1 Analytical Determination of the Exponents α, β, z, and δ
                                                                                                                                      • 8.3.2 Critical Exponents in the Period-Doubling Bifurcation
                                                                                                                                      • 8.4 Extension of Results to Other Maps
                                                                                                                                        • 8.4.1 Hassell Map
                                                                                                                                          • 8.4.2 Maynard Map
                                                                                                                                          • 8.5 Summary
                                                                                                                                            • 8.6 Proposed Exercises
                                                                                                                                            • 9 Introduction to Two-Dimensional Discrete Maps
                                                                                                                                              • 9.1 Objectives
                                                                                                                                                • 9.2 Linear Maps
                                                                                                                                                  • 9.3 Nonlinear Maps
                                                                                                                                                    • 9.4 Applications of Two-Dimensional Maps
                                                                                                                                                      • 9.4.1 Hénon Map
                                                                                                                                                        • 9.4.2 Lyapunov Exponents
                                                                                                                                                          • 9.4.3 Ikeda Map
                                                                                                                                                          • 9.5 Summary
                                                                                                                                                            • 9.6 Proposed Exercises
                                                                                                                                                            • 10 The Fermi Accelerator Model: Non-dissipative Version
                                                                                                                                                              • 10.1 Objectives
                                                                                                                                                                • 10.2 The Fermi-Ulam Model
                                                                                                                                                                  • 10.2.1 Jacobian Matrix for Indirect Collisions
                                                                                                                                                                    • 10.2.2 Jacobian Matrix for Multiple Collisions
                                                                                                                                                                      • 10.2.3 Fixed Points
                                                                                                                                                                        • 10.2.4 Phase Space
                                                                                                                                                                          • 10.2.5 Measure Preservation in Phase Space
                                                                                                                                                                          • 10.3 Simplified Version of the Fermi-Ulam Model
                                                                                                                                                                            • 10.4 Scaling Properties of the Chaotic Sea
                                                                                                                                                                              • 10.5 Location of the First Invariant Spanning Curve
                                                                                                                                                                                • 10.6 The Growth Regime
                                                                                                                                                                                  • 10.7 Summary
                                                                                                                                                                                    • 10.8 Proposed Exercises
                                                                                                                                                                                    • 11 Dissipation in the Fermi Accelerator Model
                                                                                                                                                                                      • 11.1 Objectives
                                                                                                                                                                                        • 11.2 Dissipation via Inelastic Collisions
                                                                                                                                                                                          • 11.2.1 Jacobian Matrix for Multiple Collisions
                                                                                                                                                                                            • 11.2.2 Jacobian Matrix for Indirect Collisions
                                                                                                                                                                                              • 11.2.3 Phase Space
                                                                                                                                                                                                • 11.2.4 Fixed Points
                                                                                                                                                                                                  • 11.2.5 Construction of the Manifolds
                                                                                                                                                                                                    • 11.2.6 Determination of the Manifold Crossing and the Transient
                                                                                                                                                                                                      • 11.2.7 Determining the Exponent δ from the Eigenvalues of the Saddle Point
                                                                                                                                                                                                      • 11.3 Dissipation via Viscous Drag
                                                                                                                                                                                                        • 11.3.1 Drag Force F =−˜ηv
                                                                                                                                                                                                          • 11.3.2 Drag Force F =−˜ηv2
                                                                                                                                                                                                            • 11.3.3 Drag Force F =−˜ηvγ
                                                                                                                                                                                                            • 11.4 Summary
                                                                                                                                                                                                              • 11.5 Proposed Exercises
                                                                                                                                                                                                              • 12 Dynamical Properties of the Bouncer Model
                                                                                                                                                                                                                • 12.1 Objectives
                                                                                                                                                                                                                  • 12.2 The Model
                                                                                                                                                                                                                    • 12.3 Full Version of the Bouncer Model
                                                                                                                                                                                                                      • 12.3.1 Successive Collisions
                                                                                                                                                                                                                        • 12.3.2 Indirect Collisions
                                                                                                                                                                                                                          • 12.3.3 Jacobian Matrix
                                                                                                                                                                                                                            • 12.3.4 Phase Space
                                                                                                                                                                                                                            • 12.4 Simplified Version of the Bouncer Model
                                                                                                                                                                                                                              • 12.5 Numerical Investigation in the Simplified Version
                                                                                                                                                                                                                                • 12.6 Continuous-Time Approximation
                                                                                                                                                                                                                                  • 12.7 Summary
                                                                                                                                                                                                                                    • 12.8 Proposed Exercises
                                                                                                                                                                                                                                    • 13 Localization of Invariant Curves
                                                                                                                                                                                                                                      • 13.1 Objectives
                                                                                                                                                                                                                                        • 13.2 The Standard Map
                                                                                                                                                                                                                                          • 13.3 Localization of the Curves
                                                                                                                                                                                                                                            • 13.4 Rescaling in Phase Space
                                                                                                                                                                                                                                              • 13.5 Summary
                                                                                                                                                                                                                                                • 13.6 Proposed Exercises
                                                                                                                                                                                                                                                • 14 Chaotic Diffusion in Non-dissipative Maps
                                                                                                                                                                                                                                                  • 14.1 Objectives
                                                                                                                                                                                                                                                    • 14.2 A Family of Discrete Maps
                                                                                                                                                                                                                                                      • 14.3 Properties of the Chaotic Sea: A Phenomenological Description
                                                                                                                                                                                                                                                        • 14.4 A Semi-phenomenological Approach
                                                                                                                                                                                                                                                          • 14.5 Obtaining the Probability via the Diffusion Equation
                                                                                                                                                                                                                                                            • 14.6 Summary
                                                                                                                                                                                                                                                              • 14.7 Proposed Exercises
                                                                                                                                                                                                                                                              • 15 Introduction to Billiard Dynamics
                                                                                                                                                                                                                                                                • 15.1 Objectives
                                                                                                                                                                                                                                                                  • 15.2 The Billiard
                                                                                                                                                                                                                                                                    • 15.2.1 Circular Billiard
                                                                                                                                                                                                                                                                      • 15.2.2 Elliptical Billiard
                                                                                                                                                                                                                                                                        • 15.2.3 Ovoid Billiard
                                                                                                                                                                                                                                                                        • 15.3 Summary
                                                                                                                                                                                                                                                                          • 15.4 Proposed Exercises
                                                                                                                                                                                                                                                                          • 16 Time-Dependent Billiards
                                                                                                                                                                                                                                                                            • 16.1 Objectives
                                                                                                                                                                                                                                                                              • 16.2 The Billiard
                                                                                                                                                                                                                                                                                • 16.2.1 LRA Conjecture
                                                                                                                                                                                                                                                                                • 16.3 Time-Dependent Elliptical Billiard
                                                                                                                                                                                                                                                                                  • 16.4 Oval Billiard
                                                                                                                                                                                                                                                                                    • 16.5 Summary
                                                                                                                                                                                                                                                                                      • 16.6 Proposed Exercises
                                                                                                                                                                                                                                                                                      • 17 Suppression of Fermi Acceleration in the Oval Billiard
                                                                                                                                                                                                                                                                                        • 17.1 Objectives
                                                                                                                                                                                                                                                                                          • 17.2 The Model and the Mapping
                                                                                                                                                                                                                                                                                            • 17.3 Results for the Case Fα−V
                                                                                                                                                                                                                                                                                              • 17.4 Results for the Case Fα−V2
                                                                                                                                                                                                                                                                                                • 17.5 Results for the Case Fα−Vδ
                                                                                                                                                                                                                                                                                                  • 17.6 Summary
                                                                                                                                                                                                                                                                                                    • 17.7 Proposed Exercises
                                                                                                                                                                                                                                                                                                    • 18 A Thermodynamic Model for Time-Dependent Billiards
                                                                                                                                                                                                                                                                                                      • 18.1 Objectives
                                                                                                                                                                                                                                                                                                        • 18.2 Motivation
                                                                                                                                                                                                                                                                                                          • 18.3 Heat Transfer
                                                                                                                                                                                                                                                                                                            • 18.4 Billiard Formalism
                                                                                                                                                                                                                                                                                                              • 18.4.1 Steady State
                                                                                                                                                                                                                                                                                                                • 18.4.2 Dynamical Regime
                                                                                                                                                                                                                                                                                                                  • 18.4.3 Numerical Simulations
                                                                                                                                                                                                                                                                                                                    • 18.4.4 Velocity Average Over n
                                                                                                                                                                                                                                                                                                                      • 18.4.5 Critical Exponents
                                                                                                                                                                                                                                                                                                                        • 18.4.6 Velocity Distribution
                                                                                                                                                                                                                                                                                                                        • 18.5 Connection Between the Two Formalisms
                                                                                                                                                                                                                                                                                                                          • 18.6 Summary
                                                                                                                                                                                                                                                                                                                            • 18.7 Proposed Exercises
                                                                                                                                                                                                                                                                                                                            • Appendix A: Euler Relations
                                                                                                                                                                                                                                                                                                                              • Appendix B: Numerical Integration Methods
                                                                                                                                                                                                                                                                                                                                • Appendix C: Expressions for the Coefficients j in the Dynamical Approach
                                                                                                                                                                                                                                                                                                                                  • Appendix D: Change of Reference Frame
                                                                                                                                                                                                                                                                                                                                    • Appendix E: Solution of the Diffusion Equation
                                                                                                                                                                                                                                                                                                                                      • Appendix F: Heat Flux Equation
                                                                                                                                                                                                                                                                                                                                        • Appendix G: Connection Between t and n in the Time-Dependent Ovoid Billiard
                                                                                                                                                                                                                                                                                                                                          • Appendix H: Solution of the Integral for the Relation Between n and t in the Time-Dependent Ovoid Billiard
                                                                                                                                                                                                                                                                                                                                            • References
                                                                                                                                                                                                                                                                                                                                              • Index

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