This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems.

1 Linear Systems.- 1.1 Uncoupled Linear Systems.- 1.2 Diagonalization.- 1.3 Exponentials of Operators.- 1.4 The Fundamental Theorem for Linear Systems.- 1.5 Linear Systems in R2.- 1.6 Complex Eigenvalues.- 1.7 Multiple Eigenvalues.- 1.8 Jordan Forms.- 1.9 Stability Theory.- 1.10 Nonhomogeneous Linear Systems.- 2 Nonlinear Systems: Local Theory.- 2.1 Some Preliminary Concepts and Definitions.- 2.2 The Fundamental Existence-Uniqueness Theorem.- 2.3 Dependence on Initial Conditions and Parameters.- 2.4 The Maximal Interval of Existence.- 2.5 The Flow Defined by a Differential Equation.- 2.6 Linearization.- 2.7 The Stable Manifold Theorem.- 2.8 The Hartman-Grobman Theorem.- 2.9 Stability and Liapunov Functions.- 2.10 Saddles, Nodes, Foci and Centers.- 2.11 Nonhyperbolic Critical Points in R2.- 2.12 Center Manifold Theory.- 2.13 Normal Form Theory.- 2.14 Gradient and Hamiltonian Systems.- 3 Nonlinear Systems: Global Theory.- 3.1 Dynamical Systems and Global Existence Theorems.- 3.2 Limit Sets and Attractors.- 3.3 Periodic Orbits, Limit Cycles and Separatrix Cycles.- 3.4 The Poincaré Map.- 3.5 The Stable Manifold Theorem for Periodic Orbits.- 3.6 Hamiltonian Systems with Two Degrees of Freedom.- 3.7 The Poincaré-Bendixson Theory in R2.- 3.8 Lienard Systems.- 3.9 Bendixson's Criteria.- 3.10 The Poincaré Sphere and the Behavior at Infinity.- 3.11 Global Phase Portraits and Separatrix Configurations.- 3.12 Index Theory.- 4 Nonlinear Systems: Bifurcation Theory.- 4.1 Structural Stability and Peixoto's Theorem.- 4.2 Bifurcations at Nonhyperbolic Equilibrium Points.- 4.3 Higher Codimension Bifurcations at Nonhyperbolic Equilibrium Points.- 4.4 Hopf Bifurcations and Bifurcations of Limit Cycles from a Multiple Focus.- 4.5 Bifurcations at Nonhyperbolic Periodic Orbits.- 4.6One-Parameter Families of Rotated Vector Fields.- 4.7 The Global Behavior of One-Parameter Families of Periodic Orbits.- 4.8 Homoclinic Bifurcations.- 4.9 Melnikov's Method.- 4.10 Global Bifurcations of Systems in R2.- 4.11 Second and Higher Order Melnikov Theory.- 4.12 The Takens-Bogdanov Bifurcation.- 4.13 Coppel's Problem for Bounded Quadratic Systems.- References.