Critical points and stability. Suppose J is an open interval in R and 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. (ODE) dx dt = f(x). 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 Collections: Mathematics