SUMMER 2001

1st. order linear equations (review)

2nd. order const. coeff. equations (review)

1st. order linear systems (const coeffs.)

fundamental solution, geometry of solutions

2X2 trace-determinant diagram

stability/structural stability/ generic behavior

higher dimensional systems- stable/unstable subspaces

standard forms for n x n matrices

conserved quantities for linear systems

non-homogeneous equations and systems

1st exam (6/13): up to here

EXAM
1-PROBLEMS

EXAM
1-solutions

Autonomous systems in R^n : integral curves of vector
fields

Autonomous equations: blowup in finite time, non- uniqueness,

qualitative analysis of solutions, vector fields in 1D

Lipschitz and locally Lipschitz functions and vector
fields

Picard's theorem: existence by Picard iteration (with
proof)

Gronwall's inequality and stability; criteria for infinite-time
existence

Geometric analysis of vector fields in R^2: linearization
at singular points,

Hartman's theorem. Example: the saddle-node bifurcation

Global behavior: saddle separatrices, basins of attraction,
invariant sets

Examples from math. ecology:Lotka-Volterra, predator-prey,
competing species

Conserved quantities

2nd exam (6/21): up to here

EXAM
2-PROBLEMS

EXAM
2-solutions

Examples
from mechanics: motion under a potential, Hamiltonian vector fields

Gradient vector fields. Stability and Liapunov functions

Comparison of Hamiltonian and gradient vector fields

Periodic solutions: examples of limit cycles

Future- and past- limit sets of a solution

Existence/nonexistence of periodic solutions: divergence
criterion, index

of a simple closed curve, singularity criterion,
index theorem.

The Poincare'-Bendixson theorem.

van der Pol's equation, Lienard's theorem. Example: Hopf
bifurcation

Stability criterion (average of div X along a periodic
solution)

Linear systems with time-dependent coefficients: global
existence/uniqueness

by Picard iteration (outline of proof)

Example: second-order equations with small nonlinearity
and periodic forcing

term- existence of periodic solutions in the non-resonance
case, stability criterion

(perturbation result)