Differential Equations and Dynamical Systems

✍ Scribed by Lawrence Perko

• Publisher: Springer Science & Business Media
• Year: 2013
• Tongue: English
• Leaves: 566
• Series: Texts in Applied Mathematics 7
• Edition: 3
• Category: Library

✦ Synopsis

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math ematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs.

✦ Table of Contents

Cover
Series Preface
Preface to the Third Edition
Contents
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 R^2
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 R^2
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 R^2
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 R^2
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
References
Additional References
Index