Feigenbaum

Feigenbaum refers to a surname and, in the context of chaos theory, to two mathematical constants named after the American physicist Mitchell J. Feigenbaum. He identified universal scaling laws governing the onset of chaotic behavior through period-doubling bifurcations in dissipative dynamical systems. Feigenbaum’s work showed that many different systems share the same quantitative structure near the accumulation point of period doublings, a hallmark of universality in nonlinear dynamics.

The Feigenbaum constants are delta and alpha. Feigenbaum delta, δ, is approximately 4.669201609102990, and it is the limiting ratio of successive parameter intervals between period-doubling bifurcations in many one-dimensional maps. Feigenbaum alpha, α, is approximately 2.502907875095892, and it is the limiting scaling factor for the spatial structure of the attractor as the system undergoes successive bifurcations. These constants arise from the renormalization group approach to chaos and are observed across a broad class of systems, not tied to a single model.

The discovery and analysis of the Feigenbaum constants in the late 1970s helped establish a framework for understanding universality in nonlinear dynamics, illustrating how disparate systems exhibit identical scaling behavior at the transition to chaos. The results are most closely associated with the logistic map and other one-dimensional maps, where period-doubling bifurcations accumulate at a finite parameter value, giving rise to a universal cascade.