Differential equations and dynamical systems / Lawrence Perko

Includes index

Includes bibliographical references (p. [539]-545 and index.

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 -- 4.6 One-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 Françoise's algorithm for higher order Melnikov functions -- 4.13 The Takens-Bogdanov bifurcation -- 4.14 Coppel's problem for bounded quadratic systems -- 4.15 Finite codimension bifurcations in the class of bounded quadratic systems.

This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems. Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text. All the material necessary for a clear understanding of the qualitative behavior of dynamical systems is contained in this textbook, including an outline of the proof and examples illustrating the proof of the Hartman-Grobman theorem, the use of the Poincaré map in the theory of limit cycles, the theory of rotated vector fields and its use in the study of limit cycles and homoclinic loops, and a description of the behavior and termination of one-parameter families of limit cycles.