GUMOWSKI–MIRA_MAP

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SYSTEM_PARAMETERS

PRESETS

Island Structure

SHAPE_PARAMETERS

Mu (μ) 0.310

Alpha (a) 0.0080

Beta (b) 0.0500

x₀ 0.10

Single Orbit only

y₀ 0.00

Single Orbit only

Iterations 15000

Burn-in 500

RENDER_CONTROLS

Render Mode

Density (seeds) 300

Point Size 1.5

Color Mode

Opacity 0.60

g(x) = μ·x + 2(1−μ)·x² / (1 + x²)

x(n+1) = y + a·(1 − b·y²)·y + g(x)

y(n+1) = −x + g(x(n+1))

LIVE_RENDER // CANVAS

LIVE_RENDER

DATA_LOG: GUMOWSKI–MIRA_MAP

The Gumowski–Mira map is a two-dimensional discrete nonlinear system studied for its rich transitions between order and chaos. The parameter μ (mu) controls the shape of the Gumowski function g(x): negative values produce smooth invariant curves, values near zero create KAM-island chains where regular orbits coexist with chaotic layers, and larger positive values fill phase space with a chaotic sea. The damping coefficient a governs how far orbits spread, while b shapes the nonlinear term. Unlike continuous attractors such as Lorenz or Rössler, this map advances in discrete steps, making it a cousin of the Hénon, Ikeda, and Chaos Esthetique modules — but with a uniquely broad spectrum of visually distinct behaviors.