Example of unstable attractor

Question

While defining the notion of assymptotically stable solution of an ODE (stable + atractor), my notes warn that there are unstable attractors and accompany this with the following diagram:

What is a simple example of an unstable attractor.

Answer 1

An unstable attractor $A$ essentially is an invariant set with a basin of attraction containing a positive measure set and for all neighborhoods $U(A)$, a positive Lebesgue measure of points leave that neighborhood $U(A)$. A simple example is provided by a dynamical system defined via the differential equation

$$\frac{d x}{d t} = x^2 +I $$, where $x,I \in \mathbb{R}$.

At $I>0$ there is no fixed point, at $I<0$ there are two fixed points, and at $I=0$ there is one "half-stable" fixed point at $x=0$, such that trajectories starting from initial conditions $x(0)<0$ converge towards it and trajectories starting from initial conditions $x(0)>0$ diverge from it. Thus the basin of attraction contains a positive measure set $[-1,0)$ and at the same time, every neighborhood intersects with the positive real line, so a positive measure subset leaves the neighborhood.

Here not the full measure of points of a neighborhood leave it. Such a setting is impossible for sufficiently smooth right hand sides of ordinary (non-delayed) differential equations, but it may exist for maps, as illustrated in the UNSTABLE ATTRACTOR reference provided by Baymax.

The original work on attractors that may be unstable is by John Milnor, On the Concept of attractors, Communications in Mathematical Physics 1985, https://doi.org/10.1007/BF01212280