Tinkerbell Map
A 2D discrete dynamical system that iterates a pair of simple quadratic equations, producing a butterfly-shaped strange attractor. Adjust parameters to morph the attractor's geometry in real time.
About this lab
The Tinkerbell map was introduced as a simple example of a two-dimensional strange attractor governed by quadratic iteration. Starting from a single point near the origin, repeated application of the map traces out an intricate butterfly-like set that never repeats exactly yet remains bounded within a finite region of the plane.
Strange attractors arise in dynamical systems that are simultaneously dissipative (volumes in phase space contract over time) and chaotic (nearby trajectories diverge exponentially). The Tinkerbell attractor satisfies both conditions: the map contracts area in parts of the plane while its Lyapunov exponents confirm sensitive dependence on initial conditions. Changing parameters a, b, c, d continuously morphs the attractor's shape, and for some parameter values the orbit escapes to infinity rather than settling on the attractor.
This simulation plots millions of iterated points, each colored by iteration count to reveal the attractor's fine structure. The density of points reflects the natural invariant measure of the system — regions visited most often appear brightest.