# Lecture plan/topics

Tentative plan, it will be changed and updated throughout the semester. J.S.=Jordan-Smith, B.R.=Birkhoff-Rota, HO=Hanche-Olsen.

Week Topic Sections Homework Remarks
3 Introduction
- example: the pendulum
Linear systems of ODEs
J-S chp. 1
J-S chp. 2.4
J-S chp. 1
4 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
5 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
J-S 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 well-posedness, 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
B.R. page 201
Material partially covered in HO chp. 1
10 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
HO p. 23-24
11 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
12 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
13 Easter
14 Hamiltonian systems
Index theory
J-S 2.8, 3.1
15 Index theory
Periodic solutions, negative criterions
Poincare sequences
J-S 3.1, 3.4, 3.6
J-S 13.1
10
16 Omega limit sets
Poincare-Bendixson's theorem
Lienard equations
J-S 11.1
HO chp. 6
11
17 Lienard equations: Cycles, center, limit cycles
Exam problems
J-S 11.2 - 11.3 12 Read yourself:
Proofs of 11.2 - 11.3
18 Exam problems