Attractor

An attractor $A$ is a subset of a Phase Space $\Omega$ such that:

• $a\in A\Rightarrow U_{t}a\in A$, $\forall t\in {\mathbb{R}}^{+}$
• The Basin of attraction $B_A:=\{b\in \Omega :U_{t}a\in A\forall t\in {\mathbb{R}}_{\ge T_b}\exists T_b\in {\mathbb{R}}^{+}\}$ is a Neighbourhood of $A$.
• $\forall C\subset A$, $C$ is not an attractor

In the above

• $U_t$ describes the time evolution function dynamics of the system.