# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 207A: Methods of Applied Mathematics

**Approved:**2015-10-01, Joseph Biello and John Hunter

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

**Prerequisites:**

**Suggested Schedule:**

### Part I: One-Dimensional Systems

Chapter 1: Scalar Autonomous Equations - 3 lectures

- existence and uniqueness
- flows, phase lines, and equilibria

Chapter 2: Bifurcations of Equilibria - 3 lectures

- saddle-node, pitchfork, and transcritical

Chapter 3: Scalar Maps - 3 lectures

- visualization of iterates of scalar maps
- fixed points and stability
- period doubling bifurcation
- logistic map and chaotic behavior

### Part III: Two-Dimensional Systems

Chapter 7 - Planar Autonomous Systems - 6 lectures

- phase plane
- examples from mechanics and ecology
- conservative systems
- gradient systems
- periodic orbits and limit cycles
- bifurcations of equilibria
- bifurcation diagrams

Chapter 8 - Linear Systems - 2 lectures

- matrix exponential
- eigenvectors and eigenvalues
- classification of 2-d linear systems
- phase plane

Chapter 9 - Near Equilibria - 3 lectures

- - linearization at equilibria

- classification

- stability and Lyapunov functions

- stable and unstable manifolds of hyperbolic equilibria

Chapter 10 - Center Manifolds - 1 lecture

Chapter 11 - Hopf Bifurcation - 2 lectures

Chapter 12 - Periodic Orbits - 3 lectures

- Poincare-Bendixson theorem
- Poincare maps
- stability of periodic orbits
- homoclinic bifurcations

Chapter 15 - Planar Maps - 2 lectures

- fixed points
- linearization
- stability and bifurcations

Total 28 Lectures

**Assessment:**