∞

Attractors

Undergraduate

Definition

An attractor is a state or set of states that trajectories converge to over time in a dynamical system. Types include point, periodic orbit, and strange attractors.

Formulas

lim_t → ∈fty d(x(t), A) = 0

Convergence to attractor A

\begincases \dotx = σ(y-x) \\ \doty = x(ρ - z) - y \\ \dotz = xy - β z \endcases

Lorenz equations (strange attractor)

Examples

Example 1

Explain difference between point, limit cycle, and strange attractors.

Example 2

What characterizes the Lorenz attractor?

History

Discovered by: Edward Lorenz (1963)

Discovered in weather model, Lorenz attractor became iconic in chaos theory.

Applications

Meteorology

Atmospheric models

Neuroscience

Brain activity patterns

Economics

Market dynamics