Author | Perko, Lawrence. author |
---|---|
Title | Differential Equations and Dynamical Systems [electronic resource] / by Lawrence Perko |
Imprint | New York, NY : Springer US, 1996 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0249-0 |
Descript | XIV, 519 p. online resource |
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 Poincarรฉ Map -- 3.5 The Stable Manifold Theorem for Periodic Orbits -- 3.6 Hamiltonian Systems with Two Degrees of Freedom -- 3.7 The Poincarรฉ-Bendixson Theory in R2 -- 3.8 Lienard Systems -- 3.9 Bendixsonโs Criteria -- 3.10 The Poincarรฉ 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 Peixotoโs 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 Melnikovโs 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 Coppelโs Problem for Bounded Quadratic Systems -- References