📂Dynamics

Riddled Basin

Definition ¹

In a dynamical system, assume there are $n$ basins for $R_{1} , \cdots , R_{n}$ different attractors. A set $\mathcal{R}$ is termed a riddled basin if for all $\mathbf{x} \in \mathcal{R}$ and for all $\varepsilon > 0$, the open ball $B \left( \mathbf{x} ; \varepsilon \right)$ is not disjoint from all $R_{1} , \cdots , R_{n}$: $$ \forall \mathbf{x} \in \mathcal{R} , \varepsilon > 0 , k = 1 , \cdots n : B \left( \mathbf{x} ; \varepsilon \right) \cap R_{k} \ne \emptyset \implies \mathcal{R} \text{ is a riddled basin} $$

Example ²

In simple terms, a riddled basin is a basin with a complex and intertwined structure for multiple attractors, such that no matter how small a neighborhood is chosen, it includes parts of all the basins. By definition, even the smallest subset of a riddled basin will continuously exhibit these properties, revealing a connection with fractals.

$$ f : z \mapsto z^{2} - \left( 1 + i a \right) \bar{z} $$ Consider the example of a dynamical system defined by a map on the complex plane as described above. Here, the line $N_{1} = \left\{ z = x + iy : y = a/2 \right\}$ remains invariant under $f$ regardless of $a$, that is, $$ \begin{align*} & f \left( x + i a / 2 \right) \\ =& \left( x + i a / 2 \right)^{2} - \left( 1 + i a \right) \left( x - i {\frac{ a }{ 2 }} \right) \\ =& x^{2} - {\frac{ 3 a^{2} }{ 4 }} - x + i {\frac{ a }{ 2 }} \end{align*} $$ and hence under $f \left( N_{1} \right) \subset N_{1}$. The line rotated around the origin by ${\frac{ 4 \pi }{ 3 }}$ is denoted as $N_{2}$, and another rotation results in $N_{3}$, which are said to indeed be the three attractors under $f$.

When the basins for the three attractors are visualized in red, green, and blue respectively, they appear as shown above. A riddled basin has overlaps with all three basins regardless of how small a circle is chosen anywhere. From the perspective of each color, no matter where you look, the surface appears riddled with holes, justifying the aptness of the term.