Chaos Lab

Discrete map · 1D · dim 1

Sine map

Topologically equivalent to the logistic in the chaotic regime; the smoothness changes critical exponents in transient analysis. Common in studies of period-doubling universality.

Bifurcation diagram of x ↦ a·sin(π x), a ∈ [0.7, 1]. Same universality class as logistic.

Equations

x_{n+1} = a · sin(π x_n)

At a glance

Parametersa ∈ (0, 1]
Chaotic fora close to 1 (a > a∞ ≈ 0.86)
Lyapunov exponentvaries with a
HistoryFeigenbaum's universality calculations used this map as a smoothness check.

Try it

Open the interactive playground at /tools/cobweb.

See also

Quick quiz

Test yourself on sine

8 multiple-choice questions. Pick an answer for each, then submit to see explanations.

  1. Q1.The sine map x ↦ a·sin(πx) becomes chaotic above:

  2. Q2.Sine and logistic share the same:

  3. Q3.At a = 1 the sine map maps [0,1] onto:

  4. Q4.The maximum of x ↦ a sin(π x) on [0, 1] is at:

  5. Q5.Compared to logistic at r = 4, the full sine map at a = 1 is:

  6. Q6.The sine map's critical exponent (Feigenbaum α) is:

  7. Q7.Iterating x ↦ sin(πx) from x_0 = 0.5 gives:

  8. Q8.The sine map invariant density (a = 1) is:

0 of 8 answered