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)
