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 .

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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

Other

Lorenz Attractor Animation

Energy Method Demonstrations

Download Mathematica Notebook

Konstantinos Efstathiou
Konstantinos Efstathiou
Mathematics Dynamical Systems