What is a bifurcation diagram?

A bifurcation diagram plots the long-run behaviour (attractor) of a dynamical system against a control parameter. For the logistic map x_{n+1}=r·x_n(1−x_n), the diagram shows which values of x the system settles on for each r. At r≈3 the fixed point bifurcates into a period-2 cycle; subsequent bifurcations double the period until chaos emerges near r≈3.5699.

What is Feigenbaum's constant δ≈4.6692?

The Feigenbaum constant δ = lim (r_n − r_{n-1})/(r_{n+1} − r_n) ≈ 4.6692040890... gives the rate at which successive period-doubling bifurcations accelerate. Remarkably, this constant is universal — it appears in any one-dimensional map with a quadratic maximum, regardless of the specific functional form.

What is the period-3 window and why is it significant?

Around r≈3.831 the chaotic dynamics give way to a stable period-3 cycle. By Sharkovskii's theorem, the existence of a period-3 orbit implies orbits of all other periods also exist. Li and Yorke's 1975 paper 'Period Three Implies Chaos' used this to define mathematical chaos rigorously.

What does 'universality' mean in chaos theory?

Mitchell Feigenbaum showed in 1978 that the period-doubling route to chaos, and the constants δ≈4.6692 and α≈2.5029, appear in any smooth map with a single quadratic maximum — sin(πx), x(1−x²), etc. This universality is explained by renormalisation group theory applied to dynamical systems.

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🧪 Bifurcation Diagrams

Interactive bifurcation diagram of the logistic map showing period-doubling route to chaos, Feigenbaum's delta constant δ≈4.6692, period-3 window, and universality of chaos. Zoom into any region with click-drag.

Chaos & Dynamics2DModerate60 FPS

⚙ Under the hood

This simulation visualizes bifurcation diagrams, showcasing the transitions between different states of dynamical systems. It demonstrates the Feigenbaum constants and how they relate to chaotic behavior in these systems.

Bifurcation DiagramChaos Theory

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install