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

  • 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:

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