Math 415 (Fall 2014)

This is the archived site for the Fall 2014 section of “Math 415: Applied Dynamical Systems, Chaos and Modeling” as taught by Matthew D. Johnston at the University of Wisconsin – Madison.


  • Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry, And Engineering


Term Tests and Exams



  • Week 1 (Review, classification of differential equations, reduction to first-order differential equations)
  • Week 2 (One-dimensional flows, phase portraits, fixed points (linear stability), potentials)
  • Week 3 (Saddle-node bifurcation, transcritical bifurcation, pitchfork bifurcation)
  • Week 4 (Numerical solutions, Euler’s method, Runge-Kutta method)
  • Week 5 (Two-dimensional linear systems, eigenvalues and eigenvectors, sources, sinks, saddles)
  • Week 6 (Two-dimensional nonlinear systems, nullclines and fixed points, linearization)
  • Week 7 (Hartman-Grobman Theorem, applications)
  • Week 8 (Conservative systems (Note on centers), pendulum example, Hamiltonian systems)
  • Week 9 (Notions of stability, Lyapunov functions, damped pendulum)
  • Week 10 (Lyapunov Functions, limit cycles, Van der Pol Oscillator, Poincare-Bendixson Theorem)
  • Week 11 (Limit cycles, bifurcations in multiple dimensions, Hopf bifurcation
  • Week 12 (Numerical methods, higher-dimensional systems)
  • Week 13 (Chaotic behavior, Lorenz equations, omega-limit sets, strange attractors)
  • Week 14 (Recurrence relations, classifications and reduction, fixed points, cobweb diagrams, linear stability, periodicity)
  • Week 15 (Logistic map, chaos, course review)