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AuthorPerko, Lawrence. author
TitleDifferential Equations and Dynamical Systems [electronic resource] / by Lawrence Perko
ImprintNew York, NY : Springer US, 1991
Connect tohttp://dx.doi.org/10.1007/978-1-4684-0392-3
Descript XII, 407 p. online resource

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 contains materials on higher order Melnikov functions 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


CONTENT

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 Theorem -- 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 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 Bendixsonโ{128}{153}s 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 Piexotoโ{128}{153}s Theorem -- 4.2 Bifurcations at Nonhyperbolic Equilibrium Points -- 4.3 Hopf Bifurcations and Bifurcations of Limit Cycles from a Multiple Focus -- 4.4 Bifurcations at Nonhyperbolic Periodic Orbits -- 4.5 One-Parameter Families of Rotated Vector Fields -- 4.6 The Global Behavior of One-Parameter Families of Periodic Orbits -- 4.7 Homoclinic Bifurcations -- 4.8 Melnikovโ{128}{153}s Method -- References


Mathematics Mathematical analysis Analysis (Mathematics) Mathematics Analysis



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