Date 
Lecture topics 
Practicum notes 
17 August 2009 
Planar ODEs: Solutions of planar autonomous ODE systems. Orbits and phase portraits. Equilibria and cycles. Homo and heteroclinic orbits to equilibria. Classification of equilibria, cycles, and homoclinic orbits. Poincaré return maps. PoincareBendixson Theorem. Dulac criteria. Planar Hamiltonian systems and their dissipative perturbations. Equivalence of planar ODEs and their structural stability. 

18 August 2009 
practicum 1 

19 August 2009 
Oneparameter
bifurcations of planar ODEs: Bifurcations and their codimension. Fold (saddlenode) and AndronovHopf bifurcations of equilibria and their normal forms. Fold bifurcation of cycles and the normal form for its Poincaré return map. Saddle homoclinic and heteroclinic bifurcations. Bifurcation of a homoclninc orbit to a saddlenode. 
practicum 2 
21 August 2009 
Twoparameter
bifurcations of planar ODEs: Curves of fold and AndronovHopf bifurcations in the parameter plane. Local codim 2 bifurcations (cusp, BogdanovTakens, and Bautin) and their normal forms. Some global codim 2 bifurcations (triple cycle, neutral saddle homoclinic orbit, noncentral homoclininc orbit to a saddlenode, saddle heteroclinic cycle). 
practicum 3 
24 August 2009 
Some bifurcations of
ndimensional ODEs: Equilibria, cycles, invariant tori, and chaotic invariant sets of ndimensional ODEs. Centermanifold reduction for bifurcations of equilibria and cycles. Codim 1 bifurcations of equilibria (fold and AndronovHopf) in ndimensional systems. Normal form coefficients. Remarks on multidimensional codim 2 equilibrium bifurcations (foldHopf and double Hopf). Codim 1 bifurcations of cycles (fold, perioddoubling, and NeimarkSacker) and the normal forms for their Poincaré return maps. Codim 1 bifurcations of saddle homoclinic orbits. Shilnikov's Theorems. Bifurcations of homoclinic orbits to the saddlenode and saddlesaddle equilibria. 

25 August 2009 
practicum 4 