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:

• Planar Ordinary Differential Systems
  • Liapunov Functions
  • Gradient systems
  • Dulac's Criteria
  • Omega limit sets, invariance and trapping regions
  • Poincare-Bendixson Theorem
  • Relaxation Oscillators
  • Multiple Scales solutions of weakly nonlinear systems
  • Poinare Return Maps
• Bifurcations in Planar ODE's
  • Fixed point location and the Implicit Function Theorem
  • Saddle-Node, Saddle-Saddle, Pitchfork bifurcations
  • Hopf bifurcations
  • Global bifurcations of periodic orbits
  • Saddle-Node on Invariant Circle bifurcation
• One Dimensional Maps
  • Fixed points, orbits, stability definitions
  • Cobwebbing
  • Linear Stability and hyperbolicity
  • Logistic map period doubling cascade
  • Periodic orbits and their stability
  • Itineraries, Transition graphs, Liapunov exponents
  • Chaos
  • Conjugate maps
  • Binary shift maps
• Planar Maps
  • Linear Maps and their solutions
  • Area preserving maps
  • Saddles, Flip saddles, Spirals, Pure Rotation
  • Linearization, Stability and hyperbolicity
  • Poinare-Andronov-Hopf Bifurcations

Homework Assignments

• Homework 1 Index Theory
• Homework 2 Gradients systems, Liapunov Functions and Dulac's Criteria.
• Homework 3 Poincare-Bendixson Theorem
• Homework 4 Relaxation Oscillators, Multiple Scales Methods and Fixed Point Bifurcations
• Homework 5 Hopf bifurcations and bifurcations involving periodic orbits.
• Homework 6 Poincare Return Maps, Fixed points of 1-D maps, periodic nonautonomous differential equations
• Homework 7 Bifurcations in the quadratic map, period doubling, and the existence of other periodic orbits
• Homework 8 Itineraries, Transition graphs, Conjugacy, Eventual periodicity
• Homework 9 Shift maps, Sharkovskii's Theorem, Inverse of planar maps

Notes (regularly being upgraded)

• Download A complete set of supplementary notes from M454 (Dynamics I).
• Download A complete set of supplementary notes for M455 (Dynamics II).
  • Liapunov Functions
  • Limit sets, Positive invariance
  • Poincare-Bendixson Theorem
  • Bifurcation Theory for fixed points (in progress)
  • Hopf Theory

Computer Lab(s)

• LAB 1: Fixed Point, Hopf and Homoclinic Bifurcations
• LAB 2: A catalog of systems (in progress)
• LAB 3: Examples of Chaotic Systems

Misc. Code

• Matlab script used to generate cobweb diagrams.
• Maple code to help determine if a planar map is area preserving