October 26, 2011
9a-1
Dierential Equations on the two dimensional
torus
We will now study some simple dierential equations on a two-dimensional
torus. These will provide interesting minimal sets.
The 2torus T2 is the product S 1 S 1 of two circles.
Writin
September 6, 2011
3-1
3. General Properties of Dierential Equations
Let Rn+1 be the n + 1dimensional Euclidean space and let (t, x) denote
coordinates in Rn+1 with x Rn . Write x = dx .
dt
A rst order ordinary dierential equation in Rn is an expression of
September 27, 2011
6-1
We have proved that solutions to dierential equations depend continuously on parameters.
Now we wish to investigate the smooth dependence of solutions in systems
which depend smoothly on parameters.
Theorem. Suppose that f (t, x, )
October 16, 2010
9c-1
Index for planar C 2 vector elds
Let be an open set in Rn. We say that is connected
if it cannot be written as the disjoint union of two nonempty open subsets.
A curve or path in is a continuous map : [a, b]
where [a, b] is a close