Definition:Attractor

Definition

Let $S$ be a dynamical system in a space $X$.

Let $T$ be an iterated mapping in $S$:

$x_{n + 1} = \map T {x_n}$

An attractor is an invariant set $A$ in $X$ towards which nearby points $x$ converge, that is:

$T \sqbrk A = A$

$x_n = \map {T^n} x$ approaches $A$ as $n$ increases for points close to $A$.

Examples

Origin under Complex Square Function

Consider the complex function $f: \C \to \C$ defined as:

$\forall z \in \C: \map f z = z^2$

Then the origin $\tuple {0, 0}$ of the Argand plane is an attractor.

Also see

• Definition:Strange Attractor
• Definition:Hénon Attractor
• Definition:Lorenz Attractor

• Definition:Repellor

• Results about attractors can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): attractor
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): attractor
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chaos
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): attractor
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chaos
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): attractor
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): attractor