Lecture plan/topics
Tentative plan, it will be changed and updated throughout the semester. J.S.=JordanSmith, B.R.=BirkhoffRota, HO=HancheOlsen.
Week  Topic  Sections  Homework  Remarks 

3  Introduction  example: the pendulum Linear systems of ODEs  JS chp. 1 JS chp. 2.4  Read yourself: JS chp. 1 

4  Linear systems of ODEs  2x2 autonomous systems  solution technique, fundamental matrix, stability  JS chp. 2.4 J.S. 8.1 Definition 8.2 2x2 version of J.S 8.5  8.8  1  
5  Linear systems of ODEs  2x2 systems: equilibrium points, phase portaits  general nxn systems: existence, uniqueness, fundamental matrix  JS chp. 2.1, 2.52.6 J.S. chp. 8.58.6 HO chp. 4  2  Read yourself: JS chp. 2.6 
6  Linear systems of ODEs  inhomogeneous equations  constant coefficient equations  cannonical and Jordan forms  matrix exponential  J.S. chp. 8.6, 8.8, 10.9 Parts of HO chp. 2, 3, and 4  3  
7  Linear systems of ODEs  matrix exponential  stability Nonlinear systems of ODEs  on wellposedness, Lipschitz  J.S. chp. 8.7, 8.8, 8.10, 10.9 Parts of HO chp. 3 B.R. 6.1  4  
8  Nonlinear systems of ODEs  Lipschitz condition, uniquess  continuous dependence on data  applications: projectile, curves and curvature  B.R. 6.2  6.5  5  Read yourself: B.R. chp. 6.5 Material partially covered in HO chp. 1 
9  Nonlinear systems of ODEs  equivalent integral formulation  global and local existence  continuation of solutions  C1 dependence on initial data  B.R. 6.6  6.9, 6.10 (only theorem 9) 6.12, 6.13  6  Read yourself: B.R. page 201 Material partially covered in HO chp. 1 
10  Phase diagrams and linearization  HartmanGrobman theorem  classification of hyperbolic equilibrium points  indications of proof for 2x2 systems  HO chp. 2 HO p. 2324 JS chp. 2.1, 2.3, 3.6  7  Read yourself: HO p. 2324 
11  Stability of equilibrium points  nearly linear systems  linearized stability  Liapunov method  HO chp. 3 JS chp. 8.9, 8.11 JS 10.6, 10.8, 10.10  8  
12  Liapunov method  instability Invariant domains and domains of attraction  Lasalles invariance principle  HO chp. 3 JS 10.7, 10.8 copy of lecture notes  9  
13  Easter  
14  Hamiltonian systems Index theory  JS 2.8, 3.1  
15  Index theory Periodic solutions, negative criterions Poincare sequences  JS 3.1, 3.4, 3.6 JS 13.1  10  
16  Omega limit sets PoincareBendixson's theorem Lienard equations  JS 11.1 HO chp. 6  11  
17  Lienard equations: Cycles, center, limit cycles Exam problems  JS 11.2  11.3  12  Read yourself: Proofs of 11.2  11.3 
18  Exam problems 