Ordinary Differential Equations with ApplicationsSpringer Science & Business Media, 8. apr. 2008 - 563 síður This book is based on a two-semester course in ordinary di?erential eq- tions that I have taught to graduate students for two decades at the U- versity of Missouri. The scope of the narrative evolved over time from an embryonic collection of supplementary notes, through many classroom tested revisions, to a treatment of the subject that is suitable for a year (or more) of graduate study. If it is true that students of di?erential equations giveaway their point of viewbythewaytheydenotethederivativewith respecttotheindependent variable, then the initiated reader can turn to Chapter 1, note that I write x ?,not x , and thus correctly deduce that this book is written with an eye toward dynamical systems. Indeed, this book contains a thorough int- duction to the basic properties of di?erential equations that are needed to approach the modern theory of (nonlinear) dynamical systems. However, this is not the whole story. The book is also a product of my desire to demonstrate to my students that di?erential equations is the least insular of mathematical subjects, that it is strongly connected to almost all areas of mathematics, and it is an essential element of applied mathematics. |
Efni
1 | |
Linear Systems and Stability | 127 |
Applications | 199 |
Hyperbolic Theory | 283 |
Continuation of Periodic Solutions 317 | 316 |
Homoclinic Orbits Melnikovs Method and Chaos | 391 |
Averaging | 434 |
Local Bifurcation | 483 |
530 | |
545 | |
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apply approximation averaging Banach space bifurcation boundary bounded called complete compute consider constant contained continuous coordinates corresponding curve defined definition denote depends derivative determine differential equation eigenvalues equivalent example Exercise existence fact Figure finite flow follows force formula function fundamental given Hamiltonian hyperbolic identity important initial value integral interval invariant invertible limit cycle linear linear system manifold matrix method Moreover motion multiple norm obtain operator origin oscillator parameter partial particular periodic orbit periodic solution perturbed phase portrait plane Poincaré map positive proof Proposition prove resonant respect rest point result simple smooth solution space stable subset Suppose tangent theorem theory tion transformation unique unperturbed unstable usual value problem variables vector field zero