This Is What Chaos Equals | Feigenbaum's Constant

本站添加: 2026-05-05

Feigenbaum's constant (approximately 4.6692016091029) is a universal mathematical constant discovered by physicist Mitchell Feigenbaum in 1975 that describes the ratio between consecutive bifurcation points in period-doubling cascades. This constant appears identically across all smooth one-dimensional dynamical systems with a quadratic maximum, regardless of their specific equations, because all such functions are locally quadratic near their peak (as explained by Taylor's theorem). The constant represents the fixed point of a renormalization operator that zooms into the bifurcation diagram, revealing self-similar copies of the entire branching structure at each scale. Feigenbaum's constant governs the route from order to chaos in systems like the logistic map, sign map, and Gaussian feedback loop, and also appears in the Mandelbrot set, where the ratio of diameters of period-doubling discs converges to this same value.