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Intro; Preface; Chapter Outline; Contents; 1 Origins of Discontinuity; 1.1 Discontinuities and Dynamics; 1.2 Discontinuities and Determinism; 1.3 Discontinuities in Approximations; 1.4 Discontinuities in Physics and Other Disciplines; 1.4.1 In Mechanics: Collisions and Contact Forces; 1.4.2 In Optics: Illuminating a Victorian Discontinuity; 1.4.3 In Sound: Wavefronts and Shocks; 1.4.4 In Graphs: Sigmoid Transition Functions; 1.5 `Can't We Just Smooth It?' (And Other Fudges); 1.6 Discontinuities and Simulation; 1.7 Discontinuity in Dynamics: A Brief History; 1.8 Looking Forward
2 One Switch in the Plane: A Primer2.1 The Elements of Piecewise-Smooth Dynamics; 2.2 The Value of sign(0): An Experiment; 2.3 Types of Dynamics: Sliding and Crossing; 2.4 The Switching Layer and Hidden Dynamics; 2.4.1 A Note on Modelling Basic Oscillators; 2.5 Local Singularities and Bifurcations; 2.5.1 Equilibria and Local Stability; 2.5.2 Tangencies and Their Bifurcations; 2.5.3 Equilibria, Sliding Equilibria, and Their Bifurcations; 2.6 Global Bifurcations and Tangencies; 2.7 Determinacy-Breaking: A First Glimpse; 2.8 Counting Limit Cycles; 2.9 Looking Forward
3 The Vector Field: Multipliers and Combinations3.1 Piecewise-Smooth Vector Fields; 3.2 The Discontinuity Surface; 3.3 Constituent Fields and Indexing; 3.4 The Switching Layer; 3.5 Inclusions and Existence of a Flow; 3.6 Looking Forward; 4 The Flow: Types of Solution; 4.1 Indexing; 4.2 Types of Trajectory; 4.3 Determinacy Breaking Events; 4.4 Equivalence and Stability; 4.5 Prototypes; 4.6 Flow Derivatives; 4.7 Looking Forward; 5 The Vector Field Canopy; 5.1 The Vector Field Canopy; 5.1.1 The Canopy for One Switch; 5.1.2 The Canopy for Two Switches; 5.1.3 The Canopy for m Switches
5.2 Deriving the Canopy5.2.1 Joint Expansions and Matching; 5.2.2 Series of Signs; 5.2.3 Uniqueness of the Multilinear Term; 5.3 Looking Forward; 6 Tangencies: The Shape of the Discontinuity Surface; 6.1 Flow Tangencies; 6.2 Fold (d=2,k=2); 6.3 Two-Fold (d=3,k=3); 6.4 Cusp (d=3,k=2); 6.5 Swallowtail (d=4,k=2); 6.6 Umbilic: Lips and Beaks (d=4,k=2); 6.7 Fold-Cusp (d=4,k=3); 6.8 Many-fold Singularities, Cusp-Cusps, and So On; 6.9 Proofs of Leading-Order Expressions for the Fold and Two-Fold; 6.10 A Note on Alternative Classifications; 6.11 Looking Forward; 7 Layer Analysis
7.1 The Sliding Manifold7.1.1 A Special Case: 'Higher-Order' Sliding Modes; 7.2 The Sliding Region's Attractivity; 7.3 Singularities of the Sliding Manifold M; 7.4 End Points of the Sliding Region; 7.5 Multiplicity and Attractivity of Sliding Modes; 7.5.1 One Switch, Multiple Sliding Modes; 7.5.2 Multiplicity of Sliding Modes at Intersections; 7.5.3 Classification of Sliding Modes/Equilibria; 7.6 Layer Variables; 7.7 Equilibria and Sliding Equilibria; 7.8 Invariant Subspaces; 7.9 Bifurcations of Equilibria and Sliding Equilibria; 7.10 A Saddlenode/Persistence Criterion; 7.11 Looking Forward
2 One Switch in the Plane: A Primer2.1 The Elements of Piecewise-Smooth Dynamics; 2.2 The Value of sign(0): An Experiment; 2.3 Types of Dynamics: Sliding and Crossing; 2.4 The Switching Layer and Hidden Dynamics; 2.4.1 A Note on Modelling Basic Oscillators; 2.5 Local Singularities and Bifurcations; 2.5.1 Equilibria and Local Stability; 2.5.2 Tangencies and Their Bifurcations; 2.5.3 Equilibria, Sliding Equilibria, and Their Bifurcations; 2.6 Global Bifurcations and Tangencies; 2.7 Determinacy-Breaking: A First Glimpse; 2.8 Counting Limit Cycles; 2.9 Looking Forward
3 The Vector Field: Multipliers and Combinations3.1 Piecewise-Smooth Vector Fields; 3.2 The Discontinuity Surface; 3.3 Constituent Fields and Indexing; 3.4 The Switching Layer; 3.5 Inclusions and Existence of a Flow; 3.6 Looking Forward; 4 The Flow: Types of Solution; 4.1 Indexing; 4.2 Types of Trajectory; 4.3 Determinacy Breaking Events; 4.4 Equivalence and Stability; 4.5 Prototypes; 4.6 Flow Derivatives; 4.7 Looking Forward; 5 The Vector Field Canopy; 5.1 The Vector Field Canopy; 5.1.1 The Canopy for One Switch; 5.1.2 The Canopy for Two Switches; 5.1.3 The Canopy for m Switches
5.2 Deriving the Canopy5.2.1 Joint Expansions and Matching; 5.2.2 Series of Signs; 5.2.3 Uniqueness of the Multilinear Term; 5.3 Looking Forward; 6 Tangencies: The Shape of the Discontinuity Surface; 6.1 Flow Tangencies; 6.2 Fold (d=2,k=2); 6.3 Two-Fold (d=3,k=3); 6.4 Cusp (d=3,k=2); 6.5 Swallowtail (d=4,k=2); 6.6 Umbilic: Lips and Beaks (d=4,k=2); 6.7 Fold-Cusp (d=4,k=3); 6.8 Many-fold Singularities, Cusp-Cusps, and So On; 6.9 Proofs of Leading-Order Expressions for the Fold and Two-Fold; 6.10 A Note on Alternative Classifications; 6.11 Looking Forward; 7 Layer Analysis
7.1 The Sliding Manifold7.1.1 A Special Case: 'Higher-Order' Sliding Modes; 7.2 The Sliding Region's Attractivity; 7.3 Singularities of the Sliding Manifold M; 7.4 End Points of the Sliding Region; 7.5 Multiplicity and Attractivity of Sliding Modes; 7.5.1 One Switch, Multiple Sliding Modes; 7.5.2 Multiplicity of Sliding Modes at Intersections; 7.5.3 Classification of Sliding Modes/Equilibria; 7.6 Layer Variables; 7.7 Equilibria and Sliding Equilibria; 7.8 Invariant Subspaces; 7.9 Bifurcations of Equilibria and Sliding Equilibria; 7.10 A Saddlenode/Persistence Criterion; 7.11 Looking Forward