What is the logistic map?

The logistic map is the recurrence relation x_{n+1} = rx_n(1-x_n), where x ? (0,1) represents a normalised population and r ? [0,4] is the growth rate. Despite being a one-line equation, it exhibits incredibly rich behaviour ranging from stable fixed points to period-doubling cascades and full chaos.

What is the Feigenbaum constant?

The Feigenbaum constant d 4.6692 is the universal ratio at which successive period-doubling bifurcations occur: the distance between bifurcation points shrinks by a factor of d each time. Remarkably, d appears in every unimodal map, making it a universal constant of chaos.

What does the Lyapunov exponent tell us?

The Lyapunov exponent ? measures the average rate of divergence of nearby trajectories. ? 0 indicates chaos: tiny differences in initial conditions grow exponentially, leading to sensitive dependence.

What is a cobweb diagram?

A cobweb diagram graphically shows iteration of a function. Starting at x_0, draw a vertical line to the curve f(x), then a horizontal line to the diagonal y=x this gives x_1. Repeat to trace the orbit. Spiralling cobwebs indicate stable fixed points; diverging zigzags signal chaos.

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🧪 Logistic Map and Bifurcation Diagram - Chaos Theory and Feigenbaum Constant

Explore the logistic map. Visualise the bifurcation diagram, cobweb diagram, and time-series. Discover the Feigenbaum constant, Lyapunov exponents, and the period-doubling route to chaos in interactive real time.

Chaos & Dynamics2DModerate60 FPS

⚙ Under the hood

This simulation demonstrates the Logistic Map, a fundamental model in chaos theory. Users can observe how changes to the map's parameter lead to complex behavior, including period doubling and the emergence of chaotic dynamics, illustrating concepts like the Feigenbaum constant.

Logistic MapBifurcation DiagramChaos Theory

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