I want to understand the global attractor of the dynamical system generated by the following ODE $$\dot{x} = x-x^3.$$ It has two stable fixed points at $x=1$ and $x=-1$ separated by an unstable fixed point at $x=0$. Let $\phi_t$ be the associated flow map. $A\subset\mathbb{R}$ is a global attractor if we have
- Invariance: $\phi_t(A)=A$ for all $t>0$
- Attracting compact sets: $\text{dist}(\phi_t(B),A)\rightarrow 0$. for all compact $B\subset \mathbb{R}$ (dist is the Hausdorff semi-metric)
It's stated in various places that the global attractor is the interval $A=[-1,1]$. I agree that this satisfies the definition but doesn't the set $A=\{-1,0,1\}$ also satisfy it? What am I missing here?
I got it. The reason is that $[-1,1]$ attracts $\{-1,0,1\}$ but not vice versa. It's basically just because the Hausdorff semi-metric is not symmetric. i.e. $$\text{dist}(\{-1,0,1\}, [-1,1]\} = 0$$ but $$\text{dist}([-1,1], \{-1,0,1\}\} = \frac{1}{2}$$