Differential equations and dynamical systems

✍ Scribed by Lawrence Perko

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 570
• Edition: 3ed.
• Category: Library

✦ Synopsis

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 Poincare 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. In addition to minor corrections and updates throughout, this new edition includes materials on higher order Melnikov theory and the bifurcation of limit cycles for planar systems of differential equations, including new sections on Francoise's algorithm for higher order Melnikov functions and on the finite codimension bifurcations that occur in the class of bounded quadratic systems.

✦ Table of Contents

Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Series Preface......Page 8
Preface to the Third Edition......Page 10
1.1 Uncoupled Linear Systems......Page 14
1.2 Diagonalization......Page 19
1.3 Exponentials of Operators......Page 23
1.4 The Fundamental Theorem for Linear Systems......Page 29
1.5 Linear Systems in R^2......Page 33
1.6 Complex Eigenvalues......Page 41
1.7 Multiple Eigenvalues......Page 45
1.8 Jordan Forms......Page 52
1.9 Stability Theory......Page 64
1.10 Nonhomogeneous Linear Systems......Page 73
2.1 Some Preliminary Concepts and Definitions......Page 78
2.2 The Fundamental Existence-Uniqueness Theorem......Page 83
2.3 Dependence on Initial Conditions and Parameters......Page 92
2.4 The Maximal Interval of Existence......Page 100
2.5 The Flow Defined by a Differential Equation......Page 108
2.6 Linearization......Page 114
2.7 The Stable Manifold Theorem......Page 118
2.8 The Hartman-Grobman Theorem......Page 132
2.9 Stability and Liapunov Functions......Page 142
2.10 Saddles, Nodes, Foci and Centers......Page 149
2.11 Nonhyperbolic Critical Points in R^2......Page 160
2.12 Center Manifold Theory......Page 167
2.13 Normal Form Theory......Page 176
2.14 Gradient and Hamiltonian Systems......Page 184
3 Nonlinear Systems: Global Theory......Page 194
3.1 Dynamical Systems and Global Existence Theorems......Page 195
3.2 Limit Sets and Attractors......Page 204
3.3 Periodic Orbits, Limit Cycles and Separatrix Cycles......Page 215
3.4 The Poincare Map......Page 224
3.5 The Stable Manifold Theorem for Periodic Orbits......Page 233
3.6 Hamiltonian Systems with Two Degrees of Freedom......Page 247
3.7 The Poincare-Bendixson Theory in R^2......Page 257
3.8 Lienard Systems......Page 266
3.9 Bendixson's Criteria......Page 277
3.10 The Poincare Sphere and the Behavior at Infinity......Page 280
3.11 Global Phase Portraits and Separatrix Configurations......Page 306
3.12 Index Theory......Page 311
4 Nonlinear Systems: Bifurcation Theory......Page 328
4.1 Structural Stability and Pcixoto's Theorem......Page 329
4.2 Bifurcations at Nonhyperbolic Equilibrium Points......Page 347
4.3 Higher Codimension Bifurcations at Nonhyperbolic Equilibrium Points......Page 356
4.4 Hopf Bifurcations and Bifurcations of Limit Cycles from a Multiple Focus......Page 362
4.5 Bifurcations at Nonhyperbolic Periodic Orbits......Page 375
4.6 One-Parameter Families of Rotated Vector Fields......Page 396
4.7 The Global Behavior of One-Parameter Families of Periodic Orbits......Page 408
4.8 Homoclinic Bifurcations......Page 414
4.9 Melnikov's Method......Page 428
4.10 Global Bifurcations of Systems in R^2......Page 444
4.11 Second and Higher Order Melnikov Theory......Page 465
4.12 Frangoise's Algorithm for Higher Order Melnikov Functions......Page 479
4.13 The Takens-Bogdanov Bifurcation......Page 490
4.14 Coppel's Problem for Bounded Quadratic Systems......Page 500
4.15 Finite Codimension Bifurcations in the Class of Bounded Quadratic Systems......Page 541
References......Page 554
Additional References......Page 557
Index......Page 563
Back Cover......Page 570