Click on diagram to explore behavior at any r value
Cobweb diagram showing iteration at current r
Period Doubling and the Road to Chaos
Click on diagram to explore behavior at any r value
Cobweb diagram showing iteration at current r
Australian ecologist Robert May showed how this simple population model exhibits extraordinarily complex behavior. A population that grows proportionally but is limited by resources can oscillate, bifurcate, and become chaotic—proving that simple rules don't mean simple outcomes!
Mitchell Feigenbaum discovered that the ratio between successive bifurcation intervals approaches 4.669201609... This universal constant appears in ANY system with period-doubling—as fundamental as π for circles! It connects this simple map to deep mathematical structure.
Mathematician Li and Yorke proved (1975) that if a system has a period-3 cycle, it must have cycles of EVERY period. At r ≈ 3.83, period-3 appears in a "window of order" within chaos. This theorem: "Period three implies chaos" is a foundational result!
Remarkably, the bifurcation diagram is embedded in the Mandelbrot set! The real axis of the Mandelbrot set, when properly mapped, reproduces this exact structure. Chaos theory and complex dynamics are deeply connected through these visual patterns.