Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines traditional teaching on ordinary differential equations with an introduction to the more modern theory of dynamical systems, placing this theory in the context of applications to physics, biology, chemistry, and engineering. Beginning with linear systems, including matrix algebra, the focus then shifts to foundational material on non-linear differential equations, drawing heavily on the contraction mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts - flow, chaos, invariant manifolds, bifurcation, etc. An appendix provides simple codes written in Maple®, Mathematica®, and MATLAB® software to give students practice with computation applied to dynamical systems problems. For senior undergraduates and first-year graduate students in pure and applied mathematics, engineering, and the physical sciences. Readers should be comfortable with differential equations and linear algebra and have had some exposure to advanced calculus.

• Combines a traditional theoretical development of ordinary differential equations with a modern dynamical systems viewpoint and an emphasis on applications • Provides a broad perspective on the development of invariant manifolds, bifurcation theory, chaos and geometric Hamiltonian dynamics • Exercises and examples include applications to biological, electronic, mechanical, fluid, plasma and chemical dynamics

### Contents

Preface; List of figures; List of tables; 1. Introduction; 2. Linear systems; 3. Existence and uniqueness; 4. Dynamical systems; 5. Invariant manifolds; 6. The phase plane; 7. Chaotic dynamics; 8. Bifurcation theory; 9. Hamiltonian dynamics; A. Mathematical software; Bibliography; Index.