Ordinary Differential Equations: A Dynamical Point Of View

Details about Ordinary Differential Equations: A Dynamical Point Of View:

Ordinary differential equations is a standard course in the undergraduate mathematics curriculum that usually comes after the first university calculus and linear algebra courses taken by a mathematics major. Such courses may also typically be attended by undergraduates from other areas of physical and social sciences, and engineering. The content of such a course has remained fairly static over time, despite the expansion of the topic into other disciplines as a result of the dynamical systems point of view.

This core undergraduate course updated from the dynamical systems perspective can easily be covered in one semester, with room for projects or more advanced topics tailored to the interests of the students.

Contents:

• Preface
• List of Figures
• List of Tables
• Getting Started: The Language of ODEs
• Special Structure and Solutions of ODEs
• Behavior Near Trajectories and Invariant Sets: Stability
• Behavior Near Trajectories: Linearization
• Behavior Near Equilibria: Linearization
• Stable and Unstable Manifolds of Hyperbolic Equilibria
• Lyapunov's Method and the LaSalle Invariance Principle
• Bifurcation of Equilibria, I
• Bifurcation of Equilibria, II
• Center Manifold Theory
• Jacobians, Inverses of Matrices, and Eigenvalues
• Integration of Some Basic Linear ODEs
• Solutions of Some Second Order ODEs Arising in Applications: Newton's Equations
• Finding Lyapunov Functions
• Center Manifolds Depending on Parameters
• Dynamics of Hamilton's Equations
• A Brief Introduction to the Characteristics of Chaos
• Bibliography
• Index

Readership: Undergraduate students in mathematics, physical science, social science, and engineering that use ordinary differential equations.

Key Features:

• Develops an undergraduate course in ordinary differential equations within the framework of dynamical systems theory
• Enables the student to understand the nature and significance of research in areas of science and engineering that use this point of view
• Provides students with the mathematical tools and an entry point for research in these areas