9780387951164

Differential Equations and Dynamical Systems

by
  • ISBN13:

    9780387951164

  • ISBN10:

    0387951164

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 1/1/2001
  • Publisher: Textstream
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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

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