Logistic Map Calculator (Bifurcations and Chaos)

Logistic map calculator x_{n+1} = r x_n (1-x_n): trajectories, fixed points, Feigenbaum bifurcations delta=4.6692, Lyapunov exponent and chaotic regime.

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The logistic map

The logistic map, studied by Robert May in 1976 in population dynamics context, is one of the most famous nonlinear recurrence equations in mathematics. Despite its apparent simplicity:

x_{n+1} = r * x_n * (1 - x_n)

displays the astonishing complexity of 'deterministic chaos': from a single parameter r emerges the whole zoo of dynamical behaviors, from stable fixed points to infinite bifurcations and chaos.

Parameter r and bifurcation diagram

As r varies in [0, 4], dynamics goes through several phases:

  • 0 < r < 1: x* = 0 is unique attractor, all trajectories go to 0 (extinction).
  • 1 < r < 3: x* = 1 - 1/r stable, non-trivial attractor.
  • r = 3: period-doubling bifurcation. Fixed point becomes unstable, period-2 cycle emerges.
  • r ~ 3.4495: period-4 bifurcation.
  • r ~ 3.5440: period-8 bifurcation.
  • r_infinity ~ 3.5699456...: Feigenbaum accumulation point, chaos threshold.
  • 3.5699 < r <= 4: chaos with periodic windows (e.g., period 3 at r ~ 3.828).
  • r = 4: complete chaos, statistically equivalent to random binary sequence.

Feigenbaum constant

Mitchell Feigenbaum (1975) discovered a remarkable result: ratios between consecutive bifurcation intervals converge to universal constant:

delta = lim (r_n - r_{n-1}) / (r_{n+1} - r_n) = 4.6692016091029906718...

UNIVERSAL: holds for all unimodal maps (single internal maximum) with period-doubling bifurcations, not just logistic. Produced 'universal theory of chaos', showing road to chaos follows precise mathematical laws independent of specific system.

Lyapunov exponent

Quantitative measure of sensitivity to initial conditions, chaos indicator:

L = lim_{N -> infinity} (1/N) * Sigma_{n=0}^{N-1} log|f'(x_n)|

with f'(x) = r * (1 - 2x).

  • L < 0: regular (stable) attractor, small perturbations decay.
  • L = 0: edge of chaos (bifurcation).
  • L > 0: chaotic regime, small perturbations grow exponentially (butterfly effect).

For r = 4 (complete chaos): L = log(2) ~ 0.693.

Five modes

  1. Trajectory: explicit iteration x_0, x_1, x_2, ... for given r and x_0.
  2. Fixed points: solutions of x* = f(x*) and stability via f'(x*).
  3. Period: automatic detection of cycles of length 1, 2, 4, 8, ...
  4. Lyapunov exponent: numerical L estimate for regime characterization.
  5. Feigenbaum: universal constant and bifurcations discussion.

Sensitivity to initial conditions

Lorenz 'butterfly effect' (1961) in 1D: for r > 3.5699, two initial conditions x_0 and x_0 + delta (with small delta) produce trajectories diverging exponentially: |x_n - x'_n| ~ delta * e^(L*n). Impossible to predict long-term behavior without infinite precision initial conditions. This 'predictive horizon' is limited by initial condition measurement capacity.

Period-3 window

Surprising phenomenon: even in chaotic region r > 3.5699, 'periodicity windows' exist. Most famous is 'period-3 window' centered at r ~ 3.828. Sharkovskii-Li-Yorke theorem: if iteration has period-3 cycle, then it has cycles of EVERY period. Foundation of famous Li-Yorke 'Period Three Implies Chaos' theorem (1975).

Applications

  • Biology: population dynamics (May 1976 model), epidemics, genetics.
  • Economics: nonlinear market models, business cycles.
  • Meteorology: weather prediction (infinite-dimensional chaotic atmosphere).
  • Physics: turbulence, quantum chaotic systems, neural networks.
  • Engineering: nonlinear system control, electronic oscillators.
  • Cryptography: pseudo-random number generators based on chaotic maps.

History of the logistic map

The equation underlying the logistic map goes back to the Belgian mathematician Pierre François Verhulst, who introduced it in 1838 as a model of population growth with limited resources, as an alternative to Malthus's unlimited exponential growth. For over a century it remained a simple demographic model. It was the biologist Robert May who revealed its surprising complexity in a famous 1976 paper in Nature, entitled "Simple mathematical models with very complicated dynamics", showing how an elementary equation could generate chaotic behaviour. In the same years Mitchell Feigenbaum discovered the universality of the bifurcations, that is that the same constant governs the transition to chaos across a whole class of systems. The logistic map thus became the most famous and didactic example of chaos theory.

The logistic map and deterministic chaos

The deepest lesson of the logistic map is the existence of deterministic chaos. The equation is completely deterministic: knowing the current value, the next is computed with absolute precision, without any randomness. Yet, for r beyond the Feigenbaum threshold, the long-term behaviour becomes unpredictable: two very close initial conditions diverge exponentially, the famous "butterfly effect". Determinism and predictability, which classical intuition regarded as equivalent, turn out to be distinct concepts. This discovery revolutionised 20th-century science, showing that simple systems can be intrinsically unpredictable not through ignorance of the laws, but through the amplification of errors. The logistic map is the minimal model that captures this deep idea, with applications from meteorology to ecology, from physics to economics.

Updated July 2026

Frequently Asked Questions

What is the logistic map?

It is a simple mathematical model defined by the iterative equation x_{n+1} = r·x_n·(1−x_n), where x represents a population normalised between 0 and 1 and r is a growth parameter. Despite its extreme simplicity, as r varies it shows a surprising range of behaviours, from stable equilibrium to periodic oscillations up to deterministic chaos. It is the most famous example of dynamical systems theory.

What is the bifurcation diagram?

It is the graph showing the long-term values of the logistic map as the parameter r varies. For small r you see a single stable point; at r=3 the line bifurcates into two (period 2), then four, eight, sixteen, in a cascade of ever closer period doublings, up to the chaos threshold at r≈3.5699. Beyond it, chaotic regions appear, interspersed with windows of periodic order.

What is the Feigenbaum constant?

It is the universal number δ ≈ 4.6692, discovered by Mitchell Feigenbaum in the 1970s, which describes the rate at which period doublings follow one another towards chaos. The ratio between consecutive intervals of r in the bifurcation cascade always converges to δ, independently of the details of the system. This universality is one of the deepest results of chaos theory: the same constant governs all unimodal maps.

What is the Lyapunov exponent?

It is a quantitative measure of chaos: L = lim (1/N)·Σ log|f'(x_n)|, the average of the logarithms of the derivatives along the trajectory. If L is negative, the system is regular and nearby trajectories converge; if L is positive, the system is chaotic and small perturbations grow exponentially. The value L=0 marks the edge of chaos, at the bifurcations. It is the standard tool for distinguishing order and chaos.

What is sensitivity to initial conditions?

It is the distinctive feature of chaos: two trajectories starting from almost identical initial values diverge exponentially over time, making long-term prediction impossible. It is the famous 'butterfly effect'. In the logistic map, for r in the chaotic regime, a microscopic error in the initial value is amplified until the two evolutions become completely different after a few dozen iterations.

What is the period-3 window?

It is an interval of values of r, centred around r≈3.828, in which chaos suddenly gives way to a stable cycle of period 3. It is notable for Sharkovskii's theorem and the famous 'period three implies chaos' result of Li and Yorke (1975): the presence of a period-3 cycle guarantees the existence of cycles of every period and of chaotic dynamics. It is one of the many periodic windows embedded in the chaotic regime.

What is deterministic chaos?

It is the unpredictable behaviour of a completely deterministic system, that is one without randomness. The logistic map is the archetypal example: the equation computes each value exactly from the previous one, without any random element, yet for r beyond the Feigenbaum threshold the evolution is unpredictable in the long term. Deterministic chaos showed that determinism and predictability are distinct concepts, revolutionising 20th-century science.

Who studied the logistic map?

The equation was introduced by Pierre François Verhulst in 1838 as a population growth model. Its chaotic nature was revealed by the biologist Robert May in a famous 1976 Nature paper. Mitchell Feigenbaum discovered the universality of the bifurcations in the same years. Today the logistic map is the didactic model par excellence of chaos theory, with applications in ecology, physics, economics and biology.

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