Math 777 Homework 3
1. Consider a smooth system x = v (x). Let p be a nonequilibrium point:
v (p) = 0. Prove the Flow Box Theorem: there is a neighborhood B of p
(ow box) such that the ow t |B is smoothly conjugate to the linear ow
x (x1 + t, x2 , . . . ,
Math 777 Homework 2
1. Suppose a C 1 system x = v(x) satises |v(x)| c1 + c2 |x| for all x Rn .
Show that the ow t : Rn Rn is complete (dened for all t R).
Hint: let Q(x) = |x|2 + 1. Show that there is a constant c > 0 such
that |Q| cQ. This implies that a
Math 777 Homework 1
1. Consider the logistic equation x = ax(1 x), a > 0. For any x0 nd the
time interval I = I (x0 ) on which the solution x(t) with x(0) = x0 exists.
2. Prove that a one-dimensional discrete dynamical system f : I I , where
I R is a clos