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Graduate Studies in Mathematics
2012; 356 pp; hardcover
List Price: US$64
Member Price: US$51.20
Order Code: GSM/140
Ordinary Differential Equations: Qualitative Theory - Luis Barreira and Claudia Valls
Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value Problems - Stuart P Hastings and J Bryce McLeod
This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students.
The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm-Liouville boundary value problems, including oscillation theory, are investigated.
The second part introduces the concept of a dynamical system. The Poincaré-Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems.
The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits.
The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.
Graduate students interested in ordinary differential equations and dynamical systems.
"It's easy to build all sorts of courses from this book -- a classical one-semester course with a brief introduction to dynamical systems, a one-semester dynamical systems course with just brief coverage of the existence and linear systems theory, or a rather nice two-semester course based on most (if not all) of the material."
-- MAA Reviews
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