Zaslavsky Map

FractalPark implements z(n+1) = exp(0.55 i) * (z(n) + 0.28 sin(z(n))) + c. Sine is a 0.28-weighted perturbation of the existing orbit, then the fixed complex phase rotates the full perturbed result by 0.55 radians (about 31.5°), and c shifts it.

In 1978, George M. Zaslavsky introduced the original map to model a dissipative kicked rotor: a rotating body repeatedly kicked while friction drains energy. That classical map is two-dimensional, using real action-angle variables. FractalPark recasts the idea in one complex variable for escape-time images; it is not the exact recurrence from the 1978 paper.

Complex sine can grow without bound along the imaginary axis, so this is not simple polynomial escape. An orbit can circle for a while, then fly apart. That hesitation leaves swirling arms and concentric rings, especially after a kaleidoscopic spatial transform.

George M. Zaslavsky (1929–2008) was a Soviet‑born mathematical physicist who made foundational contributions to the theory of dynamical chaos. He introduced the map that now bears his name in a 1978 paper titled "The Simplest Case of a Strange Attractor," published in Physics Letters A. The map was derived as a stroboscopic description of a dissipative kicked rotor — a rotating system that receives periodic impulses and loses energy through friction.

Zaslavsky later moved to the United States, joining New York University's physics department and the Courant Institute of Mathematical Sciences. He authored several books on Hamiltonian chaos and fractional dynamics. The Scholarpedia article on the Zaslavsky map, written by Zaslavsky himself in 2007, provides an authoritative technical reference.

Classical Zaslavsky maps operate on two real variables (action and angle). FractalPark's complex escape-time variant uses z(n+1) = exp(0.55 i) * (z(n) + 0.28 sin(z(n))) + c; it is a creative adaptation rather than a direct transcription of the original dissipative system.

Kaleidoscopic transforms make the swirling arms and mandala-like rings especially clear. The canonical image sets concentric arcs and radiating spokes around a central region where the perturbed orbit can remain bounded.

Warm oranges and golds sit against deep black in the canonical rendering. Constant rotation curves the arms; the sine perturbation puts fine ripples along the ring edges.