# Math 436: Modeling with Dynamics and Control 1

## Contents

## Catalog Information

### Title

Differential and Integral Equations 1

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

The course runs through both Spring and Summer.

### Prerequisite

Math 322, Math 341, Math 346; concurrent with Math 437

### Description

Theory and applicatiions of dynamic systems and partial differential equations. Topics include dynamic systems; bifurcation theory; control theory; hyperbolic, parabolic, and elliptic partial differential equations; commonly-used algorithms.

## Desired Learning Outcomes

### Prerequisites

Math 322, Math 341, Math 346; concurrent with Math 437

### Minimal learning outcomes

Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will be able to recognize whether differential equation models apply in the context of a given application or not. They will be able to perform the relevant computations on small, simple problems.

- Dynamical Systems
- Matrix Exponentiation
- Linear Stability
- Variation of Constants Formula
- Lyapunov Matrices
- Nonlinear Stability
- Stable, Center, and Unstable Manifolds
- Poincare-Bendixson Theorem
- Chaotic Dynamics

- Bifurcation Theory
- Saddle-Node Bifurcation
- Trans-critical Bifurcation
- Pitchfork Bifurcation
- Hopf Bifurcation

- Control Theory
- State-Space Models and Realizations
- Observability and Controllability
- Transfer Functions and Minimal Realizations
- State Feedback

- Hyperbolic PDE
- Method of Characteristics
- Shock Waves and Rarefactions
- Wave Equation
- Fourier Transform
- Separation of Variables

- Parabolic PDE
- Heat Equation
- Maximum Principle
- Separation of Variables
- Convolution and the Gaussian Kernel

- Elliptic PDE
- Laplace equation
- Separation of Variables
- Green’s Function

### Textbooks

Possible textbooks for this course include (but are not limited to):