Lecture plan/topics
The plan will be updated throughout the semester. J.S.=Jordan-Smith, B.R.=Birkhoff-Rota, HO=Hanche-Olsen.
Because I am reorganizing the course, the details in this plan will often be given after the lectures.
| Week | Topic | Sections | Homework | Remarks |
|---|---|---|---|---|
| 2 | Introduction - example: the pendulum Linear systems of ODEs | J-S chp. 1 J-S chp. 2.4 | Read yourself: J-S chp. 1 |
|
| 3 | Linear systems of ODEs - 2x2 autonomous systems - solution technique, fundamental matrix, stability | J-S chp. 2.4 J.S. 8.1 Definition 8.2 2x2 version of J.S 8.5 - 8.8 | 1 | |
| 4 | Linear systems of ODEs - 2x2 systems: equilibrium points, phase portaits - general nxn systems: existence, uniqueness, fundamental matrix | J-S chp. 2.1, 2.5-2.6 J.S. chp. 8.5-8.6 HO chp. 4 | 2 | Read yourself: J-S chp. 2.6 |
| 5 | 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 | |
| 6 | Linear systems of ODEs - matrix exponential - stability Nonlinear systems of ODEs - on well-posedness, Lipschitz | J.S. chp. 8.7, 8.8, 8.10, 10.9 Parts of HO chp. 3 B.R. 6.1 | 4 | |
| 7 | 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 |
| 8 | 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 |
| 9 | Phase diagrams and linearization - Hartman-Grobman theorem - classification of hyperbolic equilibrium points - indications of proof for 2x2 systems | HO chp. 2 HO p. 23-24 J-S chp. 2.1, 2.3, 3.6 | 7 | Read yourself: HO p. 23-24 |
| 10 | Stability of equilibrium points - nearly linear systems - linearized stability - Liapunov method | HO chp. 3 J-S chp. 8.9, 8.11 J-S 10.6, 10.8, 10.10 | 8 | |
| 11 | Liapunov method - instability Invariant domains and domains of attraction - Lasalles invariance principle | HO chp. 3 J-S 10.7, 10.8 copy of lecture notes | 9 | |
| 12 | Hamiltonian systems Index theory | J-S 2.8, 3.1 | 10 | |
| 13 | Periodic solutions, negative criterions Poincare sequences omega limit sets Poincare-Bendixon's theorem | J-S 3.4, 3.6 J-S 13.1 J-S 11.1 HO chp. 6 | 11 | |
| 14 | Easter | |||
| 15 | Poincare-Bendixon's theorem: Proof, applications Lienard equations: Cycles, center, limit cycles | J-S 11.1 - 11.3 HO chp. 6 | Read yourself: Proofs of 11.2 - 11.3 |
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| 16 | Exam problems | 12 | ||