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Preface. Perspectives and Problems of Nonlinear System Theory. A Nonlinear World. The Central Questions of System Theory. Classes of Nonlinear Processes. Linear versus Nonlinear System Theory. Notes and References. Mathematical Tools of Nonlinear System Theory. Introduction. Algebraic Concepts. Some Ideas from Differential Geometry. Concepts from Algebraic Geometry. Grassmann Manifolds and the Classification Problem. Problems and Exercises. Notes and References. A Modern View of Linear System Theory. Introduction. Mathematical Description of a Linear System. Some System-Theoretic Consequences. Transfer Functions. Realization of Transfer Functions. The Construction of Canonical Realizations. Partial Realizations. Pole-Shifting and Stability. Some Geometric Aspects of Linear Systems. Feedback, the McMillan Degree, and Kronecker Indices. Problems and Exercises. Notes and References. Reachability and Controllability. Introduction. Basic Definitions and Problem Statement. Solution Manifolds for Dynamical Systems. Some General Results. Linear-Analytic Systems. The Role of the State Manifold M. Bilinear Systems. Polynomial Systems. Discrete-Time Systems. Input Construction. Control Canonical Form and System Invariants. Problems and Exercises. Notes and References. Observability, Realization, and Estimation. Measurements and State Determination. Some Basic Definitions. Basic Observability Results. Polynomial Systems. Realization Theory. Nonlinear Filtering and Estimation. Problems and Exercises. Notes and References. Stability Theory: Singularities, Bifurcations, and Catastrophes. Concepts of Stability. Lyapunov Stability. The Center Manifold Theorem. Families of Vector Fields and Normal Forms. Elementary Bifurcations. Singularity Theory and the Elementary Catastrophes. Applications of Catastrophe Theory. Chaos and Strange Attractors. Control Systems and Feedback. Problems and Exercises. Notes and References. Index