Logistic Map & Bifurcation Calculator

What Is the Logistic Map? The logistic map is one of the simplest mathematical equations that produces chaotic behavior: x_{n+1} = r × x_n × (1 − x_n) Where x_n is a value between 0 and 1 (representing population fraction of maximum capacity), and r is the growth parameter (0 to 4). Despite its simplicity, the logistic map exhibits extraordinarily complex dynamics as r increases. It was popularized by biologist Robert May in his landmark 1976 paper, “Simple mathematical models with very complicated dynamics,” published in the journal Nature.

The Four Behavioral Regimes r < 1: population goes extinct, with x converging to 0. 1 < r < 3: stable fixed point, x converges to a single equilibrium value: x* = 1 − 1/r. 3 < r < 3.57: period-doubling bifurcations, the orbit cycles between 2, 4, 8, 16… fixed points. r ≥ 3.57: chaos. The orbit appears random and is extremely sensitive to initial conditions, except inside some “windows” of stability. The widest of those is the period-3 window near r = 3.83, which you can find on the diagram below.

Period-Doubling and Feigenbaum’s Constant As r increases toward 3.57, period-doubling bifurcations occur at increasingly close intervals. Mitchell Feigenbaum, working at the Los Alamos National Laboratory in the United States in 1975, discovered that the ratio of successive bifurcation intervals converges to the universal constant: δ = 4.669201609… (the Feigenbaum constant) This constant appears in all similar period-doubling routes to chaos, across physics, chemistry, biology and engineering, not just the logistic map. The bifurcation diagram of the logistic map contains the famous “Feigenbaum tree” shape.

Sensitive Dependence on Initial Conditions In the chaotic regime, two starting values that differ by 0.000001 produce completely different trajectories within 20 to 30 iterations. This is the “butterfly effect”, which traces to Edward Lorenz, a mathematician and meteorologist at MIT in the United States. He described the phenomenon in his 1963 paper “Deterministic Nonperiodic Flow”, calling it sensitive dependence on initial conditions after finding it in his weather simulations. The butterfly itself came later: his 1972 talk was titled “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”, and the name stuck from there. The logistic map demonstrates the same thing with a far simpler equation: deterministic but unpredictable.

Self-Similarity and Fractals The bifurcation diagram of the logistic map has fractal structure, looking much the same at every scale. Zooming into any window of periodic behavior reveals smaller copies of the entire diagram. At the accumulation point of the cascade, r∞ ≈ 3.5699, the attractor itself is a Cantor set with fractal dimension about 0.538. Note that this is the set of x-values the orbit visits, not the set of r-values that produce chaos. Those are two different things and they are easy to confuse: Jakobson proved in 1981 that the chaotic r-values occupy a set of positive measure, so they are far from a thin Cantor dust. Pick an r at random above 3.57 and you have a real chance of landing in chaos. The Mandelbrot set (the famous fractal) is mathematically related to the logistic map.

Applications of Chaos Theory Population ecology: models boom-bust cycles in animal populations. Weather forecasting: limits the predictability horizon to approximately 2 weeks. Engineering: chaos is used in secure communication (chaotic encryption) and random number generators. Medicine: heart arrhythmias and epileptic seizures show chaotic dynamics. Economics: some financial models use chaotic attractors for market dynamics.