Zaslavsky Map

The Zaslavsky Map bends an orbit with sine, turns the result by the fixed complex phase (e^{0.55i}), then adds c. A polynomial escape map gets its character from powers of z growing larger and larger. Here the plot comes from periodic bends meeting a steady turn.

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: sine supplies the nonlinear kick, the fixed complex multiplier supplies the turn, and c makes parameter-scanning escape-time images possible.

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). The complex variant used by FractalPark, with its sine perturbation and fixed complex multiplier, is a creative adaptation suitable for escape‑time rendering 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 sine perturbation keeps orbits bounded.

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