# MATH 303 — ODE and Dynamical Systems (DKU)

## Overview

MATH 303 is a required course for DKU students in the major Applied Mathematics and Computational Science and elective for students in other majors.

## Course Description

Differential equations and dynamical systems are essential mathematical tools for modeling a large variety of phenomena. They are used in Physics, Chemistry, Biology, Ecology, Finance, Sociology, and beyond, to describe various processes. However, differential equations in most cases cannot be solved and we need to use qualitative methods in order to understand such processes. In this course we will do two things. First, we will see how to solve some first-order differential equations and linear differential equations. The fact that such differential equations can be solved makes them very special. Second, we will consider nonlinear differential equations. In general, these cannot be solved, and therefore we will need to qualitatively understand the behavior of the solutions, without actually finding them. To achieve this, we will combine our knowledge of the solutions of linear differential equations with other theoretical results from the theory of dynamical systems. This course uses results from single- and multi-variable Calculus and from Linear Algebra and it is a prime example of how such results can be applied in more complicated contexts with a wide range of applications in other scientific domains.

## Syllabus

The course syllabus is available on Github .

## Literature

R. K. Nagle, E. B. Saff, and A. D. Snider. Fundamentals of Differential Equations and Boundary Value Problems, 7th edition, Pearson (2017).

## Lecture Slides

Introduction

Fundamentals of Differential Equations

First-Order Differential Equations

Applications; Substitutions

Existence and Uniqueness Theorem

One-dimensional Dynamical Systems

One-dimensional Maps

Second-Order Homogeneous Linear Equations

Second-Order Non-Homogeneous Linear Equations

Higher-order Linear Differential Equations

Introduction to the Phase Plane

Examples and Applications

Dynamical Systems and Poincaré Maps

Linear Systems

Linear Systems with Constant Coefficients

Matrix Exponential

Planar Linear Systems

Almost Linear Systems

Energy Method

Lyapunov’s Method

Limit Cycles and Periodic Solutions

Bifurcations of Equilibria in One-dimensional Systems

Bifurcations in Planar Systems

Coda — Analysis of a System