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 Poincare-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.
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.
The G3 max-norm of a is | a lomax : max{| ti l, • * * * | tn |}. The basic fact about norms is the equivalence of norms: Proposition 1 Let N : R* → R be any norm. There exist constants A > 0, B > 0 such that (4) A || 3 | < N(x) < B | a ...
Olver, P. J. (1993). Applications of Lie Groups to Differential Equations. New York, SpringerVerlag. Olver, P. J., and C. Shakiban (2006). Applied Linear Algebra. Upper Saddle River, NJ, Pearson Prentice–Hall. Perko, L. (2000).
The subject material is presented from both the qualitative and the quantitative point of view, with many examples to illustrate the theory, enabling the reader to begin research after studying this book.
The book also presents elementary dynamical systems in a unique and flexible way that is suitable for all courses, regardless of length.
The authors are tops in the field of advanced mathematics, including Steve Smale who is a recipient of the Field's Medal for his work in dynamical systems. * Developed by award-winning researchers and authors * Provides a rigorous yet ...
Seminar on Differential Equations and Dynamical Systems: Part 1
This second edition of Noonburg's best-selling textbook includes two new chapters on partial differential equations, making the book usable for a two-semester sequence in differential equations.
Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations
This text discusses the qualitative properties of dynamical systems including both differential equations and maps.