Bifurcation originally meant "splitting into two parts. " Namely, a system under- goes a bifurcation when there is a qualitative change in the behavior of the sys- tem. Bifurcation in the context of dynamical systems, where the time evolution of systems are involved, has been the subject of research for many scientists and engineers for the past hundred years simply because bifurcations are interesting. A very good way of understanding bifurcations would be to see them first and study theories second. Another way would be to first comprehend the basic concepts and theories and then see what they look like. In any event, it is best to both observe experiments and understand the theories of bifurcations. This book attempts to provide a general audience with both avenues toward understanding bifurcations. Specifically, (1) A variety of concrete experimental results obtained from electronic circuits are given in Chapter 1. All the circuits are very simple, which is crucial in any experiment. The circuits, however, should not be too simple, otherwise nothing interesting can happen. Albert Einstein once said "as simple as pos- sible, but no more" .
One of the major reasons for the circuits discussed being simple is due to their piecewise-linear characteristics. Namely, the voltage- current relationships are composed of several line segments which are easy to build. Piecewise-linearity also simplifies rigorous analysis in a drastic man- ner. (2) The piecewise-linearity of the circuits has far reaching consequences.
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Schweitzer Klassifikation
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.
