1
$\begingroup$

Fix an ODE system $\dot{x} = f(x)$ where $f : \mathbb{R}^n \to \mathbb{R}^n$ is locally Lipschitz. In the case where $x^\ast$ is a global attractor of $f$, it obviously holds true that $x^\ast$ is the unique solution of the system of equations $0 = f(x)$.

I was asking myself whether there are examples that actually show that the converse is not true in general. That is, does anyone know a Lipschitz function $g : \mathbb{R}^n \to \mathbb{R}^n$ where $g(y) = 0$ admits a unique solution $y^\ast$ such that $y^\ast$ is NOT a global attractor of the ODE system $\dot{y} = g(y)$?

$\endgroup$
0
$\begingroup$

Of course. The simplest is for $n=1$: $$\dot{x} = x$$

If you want an example where the unique equilibrium is a local attractor but not a global attractor, you can try e.g.

$$\eqalign{ \dot{x} &= y + (x^2 + y^2 - 1)\; x \cr \dot{y} &= -x + (x^2 + y^2 - 1)\; y\cr}$$

where the basin of attraction of $(0,0)$ is the open unit disk.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.