November 2, 2010
Summary -1
SUMMARY OF MATERIAL ON THE INDEX.
The following is a summary of the material on the
notion of index for non-zero vector elds along a curve
and for isolated critical points.
1
Paths, Lifts, and Homotopy
You should know the denit
December 6, 2004
17-1
17. Linear Homogeneous Systems with Constant Coecients Consider the system x = Ax (1)
where A is a constant nn matrix and x is an nvector in Rn. In the one-dimensional (scalar case), we found solutions using exponential functions, so
October 2, 2011
12-1
Linear Dierential Equations
Consider a dierential equation of the form
x = A(t)x + h(t)
(1)
where A(t) is a continuous real or complex n n matrix valued function and h(t) is a continuous n vectorvalued function. We assume that
A(t), h
18-1
Proofs of the Grobman-Hartman theorems Continued
We now develop the necessary results to prove the Main Step (A) above.
Lemma 1. Suppose H : V V is a bounded linear self-map of the
Banach space V with | H | < 1. Let I denote the identity map, Ix = x.
October 25, 2011
10a-1
Degree for maps of the circle
Let S 1 = cfw_z C : | z | = 1.
Let i = 1.
The standard covering projection from R onto S 1 is the map : R S 1
dened by
(t) = exp(2it)
Denition. An open interval in S 1 is the homeomorphic image by of
a