Bifurcations

1 Bifurcations Observed from Electronic Circuits.- 1.1 Introduction.- 1.2 The Double Scroll Circuit.- 1.2.1 Circuit and its Dynamics.- 1.2.2 Implementation.- 1.2.3 Experiments.- A Hopf Bifurcation.- B Period-Doubling Bifurcations of the Periodic Orbit.- C Chaotic Attractor (Rossler's Spiral-type).- D Saddle-Node Bifurcations of the Periodic Orbit and Periodic Window.- E Interior Crisis (The Double Scroll).- F Near Heteroclinicity.- G Boundary Crisis.- H Sounds.- 1.2.4 Confirmations.- A Hopf Bifurcation.- B Period-Doubling Bifurcation.- C Chaotic Attractor (Rossler's Spiral-type).- D Saddle-Node Bifurcation and Periodic Window.- E Interior Crisis (The Double Scroll).- F Near Heteroclinicity.- G Boundary Crisis.- 1.2.5 Summary.- 1.3 Structure of the Double Scroll.- 1.3.1 Geometric Structure.- 1.3.2 Lyapunov Exponents and Lyapunov Dimension.- A Lyapunov Exponents.- B Computations.- C Explicit Formula.- D Lyapunov Dimension.- E Time Waveforms and Power Spectra.- 1.4 The Double Scroll Circuit is Chaotic in the Sense of Shil'nikov.- 1.4.1 Statement.- 1.4.2 The Class L.- 1.4.3 Equivalence and Conjugacy Classes of L.- 1.4.4 Subset LDS.- A Half-Return Map ?0.- B Half-Return Map ?1.- C The Map ?.- D Poincare Map ?.- 1.4.5 Completion of the Proof.- 1.5 Homoclinic Linkage.- 1.5.1 Introduction.- 1.5.2 Bifurcation Equations.- A Normal Form.- B Return Time Coordinates.- C Periodic Orbits.- D Bifurcation Conditions for Periodic Orbits.- E Homoclinic Orbits Passing Through O.- F Homoclinic Orbits Passing Through P+.- G Heteroclinic Orbits.- 1.5.3 Global Bifurcations.- A Homoclinic/Heteroclinic Bifurcation Sets.- B Homoclinic Linkage.- C Global Bifurcations of Periodic Windows.- 1.6 The Torus Breakdown Circuit.- 1.6.1 Introduction.- 1.6.2 Observations of Torus Breakdown.- A The Circuit and its Dynamics.- B Experiments.- C Period-Adding Sequence.- D Sounds.- 1.6.3 Analysis.- A Divergence Zero Boundary.- B Trajectories on the Torus.- C The Folded Torus and the Double Scroll.- 1.7 The Hyperchaotic Circuit.- 1.7.1 Introduction.- 1.7.2 Experiment.- A Observation.- B Sounds.- 1.7.3 Confirmation.- 1.8 The Neon Bulb Circuit.- 1.8.1 Introduction.- 1.8.2 Experiment.- A Observation.- B Sounds.- 1.8.3 Arnold Tongues.- 1.8.4 Rotation Numbers.- 1.9 The R-L-Diode Circuit.- 1.9.1 Experiment 1.- 1.9.2 Analysis 1.- A The Dynamics.- B Two-Dimensional Map Model.- C The Bifurcation Scenario.- 1.9.3 Experiment 2.- 1.9.4 Analysis 2.- 2 Bifurcations of Continuous Piecewise-Linear Vector Fields.- 2.1 Introduction.- 2.2 Definition and Standard Forms of Continuous Piecewise-Linear Maps.- 2.2.1 Definition of Piecewise-Linear Maps.- 2.2.2 Standard Forms of CPL Maps with the Boundary Set in General Position.- 2.2.3 Standard Forms of CPL Functions.- 2.2.4 Examples of CPL functions.- 2.3 Normal Forms of Piecewise-Linear Vector Fields.- 2.3.1 Notations.- 2.3.2 Normal Forms of Linear Vector Fields with a Boundary.- 2.3.3 Normal Forms of Degenerate Affine Vector Fields with a Boundary.- 2.3.4 Normal Forms of Two-Region Piecewise-Linear Vector Fields.- 2.3.5 Normal Forms of Proper Two-Region Piecewise-Linear Vector Fields.- 2.4 Multiregion Systems and Chaotic Attractors.- 2.4.1 Attractors in Three-Dimensional Three-Region System.- 2.4.2 The Piecewise-Linear Lorenz Attractor.- 2.4.3 The Piecewise-Linear Duffing Attractor.- 2.5 Bifurcation Equations of Piecewise-Linear Vector Fields.- 2.5.1 Normal Forms of Three-Dimensional Two-Region Systems.- 2.5.2 The Tangent Map of Poincares Spiral-type).- D Saddle-Node Bifurcations of the Periodic Orbit and Periodic Window.- E Interior Crisis (The Double Scroll).- F Near Heteroclinicity.- G Boundary Crisis.- H Sounds.- 1.2.4 Confirmations.- A Hopf Bifurcation.- B Period-Doubling Bifurcation.- C Chaotic Attractor (Rossler's Spiral-type).- D Saddle-Node Bifurcation and Periodic Window.- E Interior Crisis (The Double Scroll).- F Near Heteroclinicity.- G Boundary Crisis.- 1.2.5 Summary.- 1.3 Structure of the Double Scroll.- 1.3.1 Geometric Structure.- 1.3.2 Lyapunov Exponents and Lyapunov Dimension.- A Lyapunov Exponents.- B Computations.- C Explicit Formula.- D Lyapunov Dimension.- E Time Waveforms and Power Spectra.- 1.4 The Double Scroll Circuit is Chaotic in the Sense of Shil'nikov.- 1.4.1 Statement.- 1.4.2 The Class L.- 1.4.3 Equivalence and Conjugacy Classes of L.- 1.4.4 Subset LDS.- A Half-Return Map ?0.- B Half-Return Map ?1.- C The Map ?.- D Poincare Map ?.- 1.4.5 Completion of the Proof.- 1.5 Homoclinic Linkage.- 1.5.1 Introduction.- 1.5.2 Bifurcation Equations.- A Normal Form.- B Return Time Coordinates.- C Periodic Orbits.- D Bifurcation Conditions for Periodic Orbits.- E Homoclinic Orbits Passing Through O.- F Homoclinic Orbits Passing Through P+.- G Heteroclinic Orbits.- 1.5.3 Global Bifurcations.- A Homoclinic/Heteroclinic Bifurcation Sets.- B Homoclinic Linkage.- C Global Bifurcations of Periodic Windows.- 1.6 The Torus Breakdown Circuit.- 1.6.1 Introduction.- 1.6.2 Observations of Torus Breakdown.- A The Circuit and its Dynamics.- B Experiments.- C Period-Adding Sequence.- D Sounds.- 1.6.3 Analysis.- A Divergence Zero Boundary.- B Trajectories on the Torus.- C The Folded Torus and the Double Scroll.- 1.7 The Hyperchaotic Circuit.- 1.7.1 Introduction.- 1.7.2 Experiment.- A Observation.- B Sounds.- 1.7.3 Confirmation.- 1.8 The Neon Bulb Circuit.- 1.8.1 Introduction.- 1.8.2 Experiment.- A Observation.- B Sounds.- 1.8.3 Arnold Tongues.- 1.8.4 Rotation Numbers.- 1.9 The R-L-Diode Circuit.- 1.9.1 Experiment 1.- 1.9.2 Analysis 1.- A The Dynamics.- B Two-Dimensional Map Model.- C The Bifurcation Scenario.- 1.9.3 Experiment 2.- 1.9.4 Analysis 2.- 2 Bifurcations of Continuous Piecewise-Linear Vector Fields.- 2.1 Introduction.- 2.2 Definition and Standard Forms of Continuous Piecewise-Linear Maps.- 2.2.1 Definition of Piecewise-Linear Maps.- 2.2.2 Standard Forms of CPL Maps with the Boundary Set in General Position.- 2.2.3 Standard Forms of CPL Functions.- 2.2.4 Examples of CPL functions.- 2.3 Normal Forms of Piecewise-Linear Vector Fields.- 2.3.1 Notations.- 2.3.2 Normal Forms of Linear Vector Fields with a Boundary.- 2.3.3 Normal Forms of Degenerate Affine Vector Fields with a Boundary.- 2.3.4 Normal Forms of Two-Region Piecewise-Linear Vector Fields.- 2.3.5 Normal Forms of Proper Two-Region Piecewise-Linear Vector Fields.- 2.4 Multiregion Systems and Chaotic Attractors.- 2.4.1 Attractors in Three-Dimensional Three-Region System.- 2.4.2 The Piecewise-Linear Lorenz Attractor.- 2.4.3 The Piecewise-Linear Duffing Attractor.- 2.5 Bifurcation Equations of Piecewise-Linear Vector Fields.- 2.5.1 Normal Forms of Three-Dimensional Two-Region Systems.- 2.5.2 The Tangent Map of Poincare Full Return Maps.- 2.5.3 The Return Time Coordinates.- 2.5.4 Bifurcation Equations of Three-Dimensional Two-Region Systems.- A Homoclinic Bifurcations.- B Heteroclinic Bifurcations.- 2.5.5 Bifurcation Equations of Periodic Orbits.- 2.6 Bifurcation Sets.- 2.6.1 Homoclinic/Heteroclinic Bifurcation Sets.- A Bifurcation Sets for Principal Homoclinic Orbits.- B Subsidiary Homoclinic Bifurcation Sets and Heteroclinic Bifurcation Sets.- 2.6.2 Bifurcation Sets for Periodic Orbits.- A Saddle-Node Bifurcation Sets.- B Period-Doubling Bifurcation Sets.- C Windows.- 2.6.3 Computing Bifurcation Sets.- 3 Fundamental Concepts in Bifurcations.- 3.1 Introduction.- 3.2 Fundamental Notions for Dynamical Systems.- 3.2.1 Definitions and Examples of Dynamical Systems.- 3.2.2 Orbits and Invariant Sets in Dynamical Systems.- 3.2.3 Linearization at Equilibrium Points and the Theorem of Hartman-Grobman.- 3.2.4 Stable and Unstable Manifolds.- 3.2.5 Topological Equivalence and Structural Stability.- 3.2.6 Bifurcation.- 3.2.7 Framework for the Bifurcation Theory.- 3.3 Local Bifurcations around Equilibrium Points in Vector Fields.- 3.3.1 Center Manifolds.- 3.3.2 Normal Forms.- 3.3.3 Codimension One Bifurcations.- A Saddle-Node Bifurcation.- B Hopf Bifurcation.- 3.3.4 Bogdanov-Takens Bifurcation.- 3.3.5 Symmetry and Bifurcations.- 3.3.6 Other Degenerate Singularities.- 3.4 Dynamics and Bifurcations for Discrete Dynamical Systems.- 3.4.1 Discrete Dynamical Systems.- 3.4.2 Basic Theorems and Structural Stability.- 3.4.3 Elementary Bifurcations.- A Saddle-Node Bifurcation.- B Period-Doubling Bifurcation.- C Hopf Bifurcation.- 3.4.4 One-Dimensional Mapping (1).- A Elementary Bifurcations for Quadratic Family.- B The Case of ? < ?2.- C The Case of ? = ?2.- 3.4.5 One-Dimensional Mapping (2).- 3.4.6 Horseshoe.- A Topological Horseshoe.- B Hyperbolicity.- C Transverse Homoclinic Points and Horseshoes.- 3.4.7 Further Developments.- A One-Dimensional Quadratic Family.- B Lozi Map.- C Henon Map.- D Homoclinic Tangency.- 3.5 Bifurcations of Homoclinic and Heteroclinic Orbits in Vector Fields.- 3.5.1 Persistence of Homoclinic/Heteroclinic Orbits and the Melnikov Integral.- 3.5.2 Shil'nikov Theorem.- 3.5.3 Gluing Bifurcations for Heteroclinic Orbits and Exponential Expansion.- 3.5.4 T-points and Gluing Bifurcations with Different Saddle-Indices.- 3.5.5 Homoclinic Doubling Bifurcation.- A Motivation.- B Homoclinic Doubling Bifurcation Theorems.- C Proof of the Homoclinic Doubling Bifurcation Theorems.- D Further Development.- 3.5.6 Bifurcation Generating Geometric Lorenz Attractors from Homoclinic Orbits.- 3.5.7 Local Bifurcations and Global Bifurcations.- References.- Credits.