Logistic Map Explorer

The Logistic Map

The logistic map \(f(x)=rx(1-x)\) is the canonical example of how a simple one-dimensional recurrence can produce extraordinarily complex behavior as a single parameter \(r\) varies. Originally introduced by Robert May (1976) as a model of population dynamics, it has become the standard laboratory for period-doubling cascades and deterministic chaos.

Regime Summary

Fixed Points and Stability

Fixed points satisfy \(f(x^*)=x^*\). Stability is determined by \(|f'(x^*)|\):

• \(x^*=0\): \(f'(0)=r\). Stable iff \(r<1\), unstable for \(r>1\).
• \(x^*=\tfrac{r-1}{r}\): \(f'(x^*)=2-r\). Stable iff \(|2-r|<1\), i.e., \(1<r<3\).
• At \(r=3\): \(|f'(x^*)|=1\) — period-doubling bifurcation point.

The Tent Map

The tent map \(f_\mu(x)=\mu\min(x,1-x)\) is piecewise linear with slope \(\pm\mu\) almost everywhere. It is topologically conjugate to the logistic map at \(\mu=2,\,r=4\) via \(x=\sin^2(\pi\theta/2)\). Key facts:

• Fixed points: \(x^*=0\) and \(x^*=\tfrac{\mu}{\mu+1}\) (unstable for \(\mu>1\)).
• \(\mu < 1\): every orbit converges to \(x^*=0\).
• \(\mu = 1\): every \(x\le\tfrac{1}{2}\) is a fixed point.
• \(\mu = 2\): chaotic on \([0,1]\), Lyapunov exponent \(\lambda=\ln\mu=\ln 2\).

The Sine Map

The sine map \(f(x)=a\sin(\pi x)\) on \([0,1]\) is a smooth unimodal map sharing the same qualitative bifurcation structure as the logistic map. Both are conjugate to the same universality class governed by the Feigenbaum constant \(\delta\approx4.6692\), which controls the ratio of successive bifurcation intervals.

• \(a\in(0,1)\): stable fixed point near 0.
• \(a\approx0.7278\): first period-doubling bifurcation.
• \(a\approx0.8326\): period-4.
• \(a=1\): chaotic — largest Lyapunov exponent \(\lambda>0\).

The Feigenbaum Constant

All unimodal maps with a quadratic maximum share the same ratio of successive bifurcation intervals:

This universality means that the logistic map and the sine map undergo period-doubling at different parameter values, but the ratios of the intervals are identical — a hallmark of universality in chaotic systems.

Lyapunov Exponent

\(\lambda<0\): attracting orbit. \(\lambda=0\): bifurcation. \(\lambda>0\): chaos.

How to Use This Tool

• Use the map selector to switch between the logistic map, tent map, and sine map.
• Use the quick-preset buttons to jump to interesting parameter regimes.
• Drag the parameter slider and watch the cobweb update in real time.
• Scroll to zoom, drag to pan, double-click to reset view.
• Click the canvas to set a new initial condition \(x_0\).
• Use Play/Step/All to animate the iteration step by step.