| Preface Acknowledgments Standard Notation 1 Linear vs. Nonlinear 1.1 Nonlinear Models 1.2 Complexity in Nonlinear Dynamics 1.2.1 Subtleties of Nonlinear Systems Analysis 1.2.2 Autonomous Systems and Equilibrium Points 1.3 Some Classical Examples 1.3.1 The Tunnel Diode Circuit 1.3.2 An Oscillating Circuit: Due to van der Po! 1.3.3 The Pendulum: Due to Newton 1.3.4 The Buckling Beam: Due to Euler 1.3.5 The Volterra-Lotka Predator-Prey Equations 1.4 Other Classics: Musical Instruments 1.4.1 Blowing of a Clarinet Reed: Due to Rayleigh 1.4.2 Bowing of a Violin String: Due to Rayleigh 1.5 Summary 1.6 Exercises 2 Planar Dynamical Systems 2.1 Introduction 2.2 Linearization About Equilibria of Second-Order Nonlinear Systems 2.2.1 Linear Systems in the Plane 2.2.2 Phase Portraits near Hyperbolic Equilibria 2.3 Closed Orbits of Planar Dynamical Systems 2.4 Counting Equilibria: Index Theory 2.5 Bifurcations 2.6 Bifurcation Study of Josephson Junction Equations 2.7 The Degenerate van der Pol Equation 2.8 Planar Discrete-Time Systems 2.8.1 Fixed Points and the Hartman-Grobman Theorem 2.8.2 Period N Points of Maps 2.8.3 Bifurcations of Maps 2.9 Summary 2.10 Exercises 3 Mathematical Background 3.1 Groups and Fields 3.2 Vector Spaces, Algebras, Norms, and Induced Norms 3.3 Contraction Mapping Theorems 3.3.1 Incremental Small Gain Theorem 3.4 Existence and Uniqueness Theorems for Ordinary Differential Equations 3.4.1 Dependence on Initial Conditions on Infinite Time Intervals 3.4.2 Circuit Simulation by Waveform Relaxation 3.5 Differential Equations with Discontinuities 3.6 Carleman Linearization 3.7 Degree Theory 3.8 Degree Theory and Solutions of Resistive Networks 3.9 Basics of Differential Topology 3.9.1 Smooth Manifolds and Smooth Maps 3.9.2 Tangent Spaces and Derivatives 3.9.3 Regular Values 3.9.4 Manifolds with Boundary 3.10 Summary 3.11 Exercises 4 Input-Output Analysis 4.1 Optimal Linear Approximants to Nonlinear Systems 4.1.1 Optimal Linear Approximations for Memoryless, Time-Invariant Nonlinearities 4.1.2 Optimal Linear Approximations for Dynamic Nonlinearities: Oscillations in Feedback Loops 4.1.3 Justification of the Describing Function 4.2 Input-Output Stability. 4.3 Applications of the Small Gain Theorems …… 5 Lyapunov Stability THeory 6 Applications of Lyapunov THeory 7 Dynamical Systems and Bifurcations 8 Basics of Differential Geometry 9 Linearization by State Feedback 10 Design Examples Using Linearization 11 Geometric Nonlinear Control 12 Exterior Differential Systems in Control 13 New Vistas: Multi-Agent Hybrid Systems References Index |
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