Homework due dates and the date for the midterm will be announced in class. Exact content of the exams will also be announced in class. The final exam is not comprehensive. There is no set schedule for when homework will be assigned. Some assignments will be longer than others. As a result, the HW grade above will be the percentage obtained.

Topics Covered:

• Dynamics on R
  • fixed points, stability, linear stability
  • Existence, Uniqueness, Picard Iteration, Taylor series methods
  • Blowup, Comparison arguments
• Elementary Bifurcations on R
  • Saddle Node, Transcritical, Pitchfork bifurcations
  • Normal Forms
  • Stability Diagrams
• Dynamics on S^1
  • Phase portraits, periodic orbits
  • Bifurcations
  • Asymptotic Period estimates and bottlenecks
  • Compactification of dynamics on R onto S^1
• Linear Planar Systems
  • Phase portraits, fixed points
  • Fundamental matrix solutions
  • Stability in from Tr(Df) and det(A)
  • linear manifolds E^s, E^c and E^u
• Planar Systems Introduction
  • Nullclines, Flow, Hyperbolic fixed points
  • Homeomorphisms and Topological Equivalence
  • Liapunov Stable, Attracting, isolated, asymptotically stable
  • Hartman-Grobman Theorem
• Special Structures in Planar Systems
  • Conservative Systems
  • Hamiltonian Systems
  • Reversible Systems
  • Theorems for Nonlinear Centers
• Index Theory
  • Definition and Calculation
  • Integral formulation
  • Key properties and Theorems
  • Proving nonexistence of Periodic orbits
• Periodic Orbits in Planar Systems
  • Conversion to Polar coordinates
  • Limit Cycles and Stability Definitions

Homework Assignments

• Homework 1 in PDF format Due Sept 10. Autonomous systems, fixed points, stability, linear stability
• Homework 2 in PDF format Due Sept 22. Existence, Uniqueness, Picard Iteration,Taylor series methods, Blowup, Comparison arguments
• Homework 3 in PDF format Due Oct 4. Saddle Node and Transcritical bifurcations, normal forms
• Homework 4 in PDF format Due Oct 11. Pitchforks, isolated and nonhyperbolic fixed points.
• Homework 5 in PDF format Due Oct 18. Stability diagrams, simple dynamics on S^1
• Homework 6 in PDF format Due Oct 29. Periodic orbits, asymptotic period approximations, bifurcations and nonhyperbolic fixed points for dynamics on S^1
• Homework 7 in PDF format Due Nov. 8. Fundamental matrix solutions, Stability in linear systems and E^s, E^c and E^u in linear systems.
• Homework 8 in PDF format Due Nov. 19. Nullclines, Phase portraits, Homeomorphisms and Topological Equivalence.
• Homework 9 in PDF format Due Dec. 1. Conservative, Hamiltonian and Reversible systems.
• Homework 10 in PDF format Not due ever!!!! Sample index problems for the final exam.

Supplementary Notes (regularly being upgraded)

• click here for the current version (81 pages in a PDF file). Last updated Oct 29,2004.