Nonlinear Systems

For a first-year graduate-level course on nonlinear systems. It may also be used for self-study or reference by engineers and applied mathematicians. The text is written to build the level of mathematical sophistication from chapter to chapter. It has been reorganized into four parts: Basic analysis, Analysis of feedback systems, Advanced analysis, and Nonlinear feedback control.All chapters conclude with Exercises. 1. Introduction. Nonlinear Models and Nonlinear Phenomena. Examples. 2. Second-Order Systems. Qualitative Behavior of Linear Systems. Multiple Equilibria. Qualitative Behavior Near Equilibrium Points. Limit Cycles. Numerical Construction of Phase Portraits. Existence of Periodic Orbits. Bifurcation. Systems. 3. Fundamental Properties. Existence and Uniqueness. Continuos Dependence on Initial Conditions and Parameters. Differentiability of solutions and Sensitivity Equations. Comparison Principle. 4. Lyapunov Stability. Autonomous Systems. The Invariance Principle. Linear Systems and Linearization. Comparison Functions. Nonautonomous Systems. Linear Time-Varying Systems and Linearization. Converse Theorems. Boundedness and Ultimate Boundedness. Input-to-State Stability. 5. Input-Output Stability. L Stability. L Stability of State Models. L2 Gain. Feedback Systems: The Small-Gain Theorem. 6. Passivity. Memoryless Functions. State Models. Positive Real Transfer Functions. L2 and Lyapunov Stability. Feedback Systems: Passivity Theorems. 7. Frequency-Domain Analysis of Feedback Systems. Absolute Stability. The Describing Function Method. 8. Advanced Stability Analysis. The Center Manifold Theorem. Region of Attraction. Invariance-like Theorems. Stability of Periodic Solutions. 9. Stability of Perturbed Systems. Vanishing Pertubation. Nonvanishing Pertubation. Comparison Method. Continuity of Solutions on the Infinite Level. Interconnected Systems. Slowly Varying Systems. 10. Perturbation Theory and Averaging. The Perturbation Method. Perturbation on the Infinite Level. Periodic Perturbation of Autonomous Systems. Averaging. Weekly Nonlinear Second-Order Oscillators. General Averaging. 11. Singular Perturbations. The Standard Singular Perturbation Model. Time-Scale Properties of the Standard Model. Singular Perturbation on the Infinite Interval. Slow and Fast Manifolds. Stability Analysis. 12. Feedback Control. Control Problems. Stabilization via Linearization. Integral Control. Integral Control via Linearization. Gain Scheduling. 13. Feedback Linearization. Motivation. Input-Output Linearization. Full-State Linearization. State Feedback Control. Index.