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2. A critical point xo of an autonomous system of ordinary differential equations is called

attracting if there is a d > 0 so that for any solution x(t),

|x(to) xo <8 for some to ER⇒ lim x(t) = xo.

-

t→∞0

Consider the one-dimensional ordinary differential equation x = f(x), where f: R → R

is continuously differentiable.

(i) Show that, for any solution x(t), if f(x(to)) = 0 for some to, then f(x(t))

t€ R.

= 0 for all

(ii) Show also that, for any solution x(t), if f(x(to)) > 0 for some to, then f(x(t)) > 0 for

all t€ R; show that the same holds when > is replaced with <.

(iii) Use (i) and (ii) to show that if a critical point xo is attracting then it must also be stable.