# 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)
J. Hale and H. Kocak, Dynamics and Bifurcations (Parts I and III)
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

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
• 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:
The recommendation is one in-class midterm exam and a 2 hour, timed final exam during final exam week.