Logistic Map Calculator: Chaos, Period Doubling and the Lyapunov Exponent

Iterate the chaos equation x(t+1) = k x (1 - x) and see whether the series settles, cycles or is chaotic.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the growth rate k, a starting value between 0 and 1 and the number of steps. The calculator gives the series, the value after the steps, the fixed point 1 - 1/k, and a graph of x against the step number.

For k under 3 the series settles to one value. From 3 to about 3.57 it cycles with a period that doubles, then it becomes chaotic. The Lyapunov exponent measures how fast two nearby starting values move apart: positive means chaos.

It is the simplest example of chaos: a short exact equation whose long-term behaviour cannot be predicted.

Formulas used

  • x_{t+1} = k\,x_t(1 - x_t)
  • x^* = 1 - \frac{1}{k}
  • \lambda = \lim \frac{1}{N}\sum\ln|k(1 - 2x_t)|

Worked example

With the default numbers in the calculator above:

  1. The equation: k = 3.90
  2. Value after 40 steps: 0.891241
  3. Fixed point (the value that stays the same): 0.74359
  4. Slope at the fixed point (the fixed point is stable if under 1): 1.90
  5. Lyapunov exponent: 0.4913

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Growth rate k (0 to 4) 2.5; Starting value x (between 0 and 1) 0.5; Number of steps to show 50.

  1. The equation: k = 2.50
  2. Value after 50 steps: 0.6
  3. Fixed point (the value that stays the same): 0.6
  4. Slope at the fixed point (the fixed point is stable if under 1): 0.50
  5. Lyapunov exponent: -0.6931

Common questions

What is chaos?

A system with exact rules whose long-term behaviour is unpredictable, because tiny differences in the starting value grow exponentially.

What does a positive Lyapunov exponent mean?

Nearby starting values separate exponentially, so the series is chaotic.

What is period doubling?

As k increases the series changes from one value to a cycle of 2, then 4, 8 and so on, until chaos.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: May (1976), Simple mathematical models with very complicated dynamics, Nature; Strogatz, Nonlinear Dynamics and Chaos.

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