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.