Summary: Critical points and stability.
Suppose J is an open interval in R and
f : J R
is such that whenever a, b J and a < b there is a nonnegative real number L such that
|f(x1) - f(x2)| L|x1 - x2| whenever a x1 x2 b.
The theory we have developed guarantees the following.
Theorem. Suppose a, b J and a < b. There is > 0 such that if a x0 b and t0 in R then there is a
solution x of (ODE) defined on (t0 - , t0 + ) which satisfies x(t0) = x0.
Definition. We say x0 is critical point for ODE if x0 J and
f(x0) = 0.
We say the critical point x0 is stable if for each > 0 there > 0 such that (x0 - , x0 + ) J and such
that if |x1 - x0| < and if x is the unique maximal solution of ODE such that x(0) = x1 then [0, ) is a
subset of the domain of x and
|x(t) - x0| < .
We say the critical point x0 is unstable if x0 is not stable.
Theorem. Suppose x0 is a critical point for (ODE), x1 J and x : I J is the unique maximal solution