Ordinary Differential Equations with ApplicationsSpringer Science & Business Media, 18. maí 2006 - 636 síður Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scienti?c disciplines and a resurgence of interest in the modern as well as the cl- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). Thedevelopmentofnewcoursesisanaturalconsequenceofahighlevelof excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Ma- ematical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden New York, New York L. Sirovich College Park, Maryland S.S. Antman Preface 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. |
Efni
Introduction to Ordinary Differential Equations | 1 |
Linear Systems and Stability of Nonlinear Systems | 145 |
Applications | 225 |
Hyperbolic Theory | 323 |
Continuation of Periodic Solutions | 367 |
Homoclinic Orbits Melnikovs Method and Chaos | 449 |
Averaging 511 | 510 |
Local Bifurcation | 545 |
| 603 | |
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apply approximation Banach space bifurcation center manifold compute constant continuous corresponding curve defined definition denote determine displacement function dynamical eigenvalues example Exercise fact finite fixed point Floquet flow formula fundamental matrix fundamental matrix solution given Hamiltonian system homoclinic Hopf hyperbolic implicit function theorem inequality initial condition initial value problem integral interval invariant manifold invariant set invertible Lemma let us note limit cycle linear system linear transformation Lyapunov Moreover motion nonlinear nonzero norm obtain open set ordinary differential equation origin oscillator parameter period annulus periodic orbit periodic solution perturbed phase portrait planar plane Poincaré map Poincaré section polynomial principal fundamental matrix proof Proposition prove resonance rest point result smooth function submanifold subset sufficiently small Suppose tangent theory tion unstable manifolds variables vector field zero