= 0 is actually infinite! It’s not just quite an unstable fixed point - it’s infinitely unstable (that isn’t really a technical term...). It actually turns out that there are an infinite number of different possible behaviours starting at
We can make a more general statement:
Given a differential equation and initial condition
are continuous on an open interval in then there exists a unique solution on some interval (-τ,τ) about
. It says nothing about how long the solution will remain for, so after some time, something funky may happen, but if
is sufficiently smooth, then there will be a unique solution which will last for some time.
Doesn’t it seem strange that a solution may only exist for a finite time? Well, let’s look at another system which looks pretty well behaved.
Well, the first thing that we might notice about this is that there are no fixed points... that’s fine. The function is continuous and smooth for all
so apparently there will exist a unique solution at least for some time. It turns out that there is a solution to this differential equation which is given by:
is an integration constant and in fact with our initial condition,