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Über dieses Buch

This text is a rigorous treatment of the basic qualitative theory of ordinary differential equations, at the beginning graduate level. Designed as a flexible one-semester course but offering enough material for two semesters, A Short Course covers core topics such as initial value problems, linear differential equations, Lyapunov stability, dynamical systems and the Poincaré—Bendixson theorem, and bifurcation theory, and second-order topics including oscillation theory, boundary value problems, and Sturm—Liouville problems. The presentation is clear and easy-to-understand, with figures and copious examples illustrating the meaning of and motivation behind definitions, hypotheses, and general theorems. A thoughtfully conceived selection of exercises together with answers and hints reinforce the reader's understanding of the material. Prerequisites are limited to advanced calculus and the elementary theory of differential equations and linear algebra, making the text suitable for senior undergraduates as well.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Initial Value Problems

Abstract
This chapter discusses the basic problems for solutions of initial value problems: existence and uniqueness, continuation, and dependence on parameters and initial conditions.
Qingkai Kong

Chapter 2. Linear Differential Equations

Abstract
For linear differential equations, the structures of solution spaces are characterized and solution formulas are derived in terms of fundamental matrix solutions. The calculations of fundamental matrix solutions are studied for the constant coefficient case and the Floquet theory is presented for the periodic coefficient case.
Qingkai Kong

Chapter 3. Lyapunov Stability Theory

Abstract
Basic concepts for the Lyapunov stability are introduced. Conditions are obtained for the stability of linear equations with constant, periodic, and general variable coefficients. Linearization and Lyapunov functions are used to deal with nonlinear stability problems.
Qingkai Kong

Chapter 4. Dynamical Systems and Planar Autonomous Equations

Abstract
Basic concepts for dynamical systems are introduced. The Poincaré–Bendixson theorem is proved and used to study the existence and orbital stability of periodic solutions for planar equations. Invariant manifolds for n-dimensional nonlinear equations are investigated.
Qingkai Kong

Chapter 5. Introduction to Bifurcation Theory

Abstract
One-dimensional bifurcations are discussed for scalar equations and planar systems. Results on Hopf bifurcations for planar systems are derived using the Lyapunov function method and the Friedrich method.
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Chapter 6. Second-Order Linear Equations

Abstract
Several topics for second-order linear equations including the Sturm theorems, oscillation, boundary value problems, and Sturm-Liouville problems are studied.
Qingkai Kong

Backmatter

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