Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition

ISBN-13: 9780824723378 / Angielski / Twarda / 2007 / 402 str.

Jane Cronin-Scanlon; Cronin-Scanlon Cronin-Scanlon

Opis

Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition provides a detailed account, based on the work of Lefschetz, of the Bendixson theory of solutions of two-dimensional autonomous systems in a neighborhood of a singular point. Updated and revised, this third edition presents a unified treatment of the perturbation problem for periodic solutions, covering nonautonomous equations and bifurcation problems. The text includes a self-contained account of topological degree and its use as well as a detailed description of how the averaging method can be used to examine the problem of periodic solutions. It also contains additional exercises, applications, and solutions.

Recenzje

… a classic treatment of many of the topics an instructor would want in such a course, with particular emphasis on those aspects of the qualitative theory that are important for applications to mathematical biology. … A nice feature of this edition is an extended and unified treatment of the perturbation problem for periodic solutions. … a solid graduate-level introduction to ordinary differential equations, especially for applications. …
—MAA Reviews, August 2010

Spis treści

Prefaces Introduction Existence Theorems What This Chapter Is About Existence Theorem by Successive Approximations Differentiability Theorem Existence Theorem for Equation with a Parameter Existence Theorem Proved by Using a Contraction Mapping Existence Theorem without Uniqueness Extension Theorems Examples Linear Systems Existence Theorems for Linear Systems Homogeneous Linear Equations: General Theory Homogeneous Linear Equations with Constant Coefficients Homogeneous Linear Equations with Periodic Coefficients: Floquet Theory Inhomogeneous Linear Equations Periodic Solutions of Linear Systems with Periodic Coefficients Sturm–Liouville Theory Autonomous Systems Introduction General Properties of Solutions of Autonomous Systems Orbits near an Equilibrium Point: The Two-Dimensional Case Stability of an Equilibrium Point Orbits near an Equilibrium Point of a Nonlinear System The Poincaré–Bendixson Theorem Application of the Poincaré–Bendixson Theorem Stability Introduction Definition of Stability Examples Stability of Solutions of Linear Systems Stability of Solutions of Nonlinear Systems Some Stability Theory for Autonomous Nonlinear Systems Some Further Remarks Concerning Stability The Lyapunov Second Method Definition of Lyapunov Function Theorems of the Lyapunov Second Method Applications of the Second Method Periodic Solutions Periodic Solutions for Autonomous Systems Stability of the Periodic Solutions Sell’s Theorem Periodic Solutions for Nonautonomous Systems Perturbation Theory: The Poincaré Method Introduction The Case in which the Unperturbed Equation Is Nonautonomous and Has an Isolated Periodic Solution The Case in which the Unperturbed Equation Has a Family of Periodic Solutions: The Malkin–Roseau Theory The Case in which the Unperturbed Equation Is Autonomous Perturbation Theory: Autonomous Systems and Bifurcation Problems Introduction Using the Averaging Method: An Introduction Introduction Periodic Solutions Almost Periodic Solutions Appendix Ascoli’s Theorem Principle of Contraction Mappings The Weierstrass Preparation Theorem Topological Degree References Index Exercises appear at the end of each chapter.

O autorze

Cronin, Jane