PART 3: PERIODIC SOLUTIONS OF  NONLINEAR SYSTEMS; TOPOLOGICAL METHODS

3/16  limit cycles in R^2: examples in polar coordinates, stable and unstable l
limit cycles
HW6: 8.2/8.3: 4,5(a), 9  8.4:2  (due 3/28)

3/19, 3/21, 3/23: SPRING BREAK

3/26: topological methods I- index  of a simple closed curve and at an isolated singularity

3/28: examples involving indices; comments on HW6

3/30: alpha- and omega-limit sets.  Transversal sections. Poincare'-Bendixson theorem

4/2: Poincare-Bendixson theorem: statment, examples

4/4: van der Pol-type systems: introduction

4/6: van der Pol systems: proof of Lienard's theorem, estimates of period and amplitude (handout)

4/9: Lienard's theorem (generalization); stability criterion for periodic slns; divergence criterion for
non-existence of periodic slns. Handout (problems)

4/11:  Exam 3-problems
 Exam 3-solutions

4/13: EASTER HOLIDAY

4/16: Area-preserving flows in R^2 and divergence-free vector fields (Liouville's theorem)
           (section 8.6) HW7: 5(a)(c),6,7 (due 4/23)

4/18: Conservative v.f's in R^2 are Hamiltonian.  Examples: potential motion on
the line, Duffing's equation