Differential Equations and Dynamical Systems

by Perko, Lawrence

• ISBN13: 9780387951164
• ISBN10: 0387951164
• Edition: 3rd
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Springer Verlag

Summary

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

Series Preface	vii
Preface to the Third Edition	ix
Linear Systems	1	(64)
Uncoupled Linear Systems	1	(5)
Diagonalization	6	(4)
Exponentials of Operators	10	(6)
The Fundamental Theorem for Linear Systems	16	(4)
Linear Systems in R2	20	(8)
Complex Eigenvalues	28	(4)
Multiple Eigenvalues	32	(1)
Jordan Forms	32	(19)
Stability Theory	51	(9)
Nonhomogeneous Linear Systems	60	(5)
Nonlinear Systems: Local Theory	65	(116)
Some Preliminary Concepts and Definitions	65	(5)
The Fundamental Existence-Uniqueness Theorem	70	(9)
Dependence on Initial Conditions and Parameters	79	(8)
The Maximal Interval of Existence	87	(8)
The Flow Defined by a Differential Equation	95	(6)
Linearization	101	(4)
The Stable Manifold Theorem	105	(14)
The Hartman-Grobman Theorem	119	(10)
Stability and Liapunov Functions	129	(7)
Saddles, Nodes, Foci and Centers	136	(11)
Nonhyperbolic Critical Points in R2	147	(7)
Center Manifold Theory	154	(9)
Normal Form Theory	163	(8)
Gradient and Hamiltonian Systems	171	(10)
Nonlinear Systems: Global Theory	181	(134)
Dynamical Systems and Global Existence Theorems	182	(9)
Limit Sets and Attractors	191	(11)
Periodic Orbits, Limit Cycles and Separatrix Cycles	202	(9)
The Poincare Map	211	(9)
The Stable Manifold Theorem for Periodic Orbits	220	(14)
Hamiltonian Systems with Two Degrees of Freedom	234	(10)
The Poincare-Bendixson Theory in R2	244	(9)
Lienard Systems	253	(11)
Bendixson's Criteria	264	(3)
The Poincare Sphere and the Behavior at Infinity	267	(26)
Global Phase Portraits and Separatrix Configurations	293	(5)
Index Theory	298	(17)
Nonlinear Systems: Bifurcation Theory	315	(224)
Structural Stability and Peixoto's Theorem	316	(18)
Bifurcations at Nonhyperbolic Equilibrium Points	334	(9)
Higher Codimension Bifurcations at Nonhyperbolic Equilibrium Points	343	(6)
Hopf Bifurcations and Bifurcations of Limit Cycles from a Multiple Focus	349	(13)
Bifurcations at Nonhyperbolic Periodic Orbits	362	(21)
One-Parameter Families of Rotated Vector Fields	383	(12)
The Global Behavior of One-Parameter Families of Periodic Orbits	395	(6)
Homoclinic Bifurcations	401	(14)
Melnikov's Method	415	(16)
Global Bifurcations of Systems in R2	431	(21)
Second and Higher Order Melnikov Theory	452	(14)
Francoise's Algorithm for Higher Order Melnikov Functions	466	(11)
The Takens-Bogdanov Bifurcation	477	(10)
Coppel's Problem for Bounded Quadratic Systems	487	(41)
Finite Codimension Bifurcations in the Class of Bounded Quadratic Systems	528	(11)
References	539	(2)
Additional References	541	(6)
Index	547