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    Logistic Map

    Logistic map

    x_n+1 = r x_n (1 - x_n) (n = (0, 1, 2, ...))

    Input values

    parameter r | 4 initial condition x_0 | 0.1

    Iterates

    n | 0 | 1 | 2 | 3 | 4 x_n | 0.10000 | 0.36000 | 0.92160 | 0.28901 | 0.82194

    Iterates

    Closed form solution

    x_n = sin^2(2^(n - 1) cos^(-1)(1 - 2 x_0))≈sin^2(0.321751 2^n)

    Cobweb diagram

    (lines successively connect the first 50 iterates and the dashed line y = x)

    Bifurcation diagram

    (iterates 100 through 150 for each r)

    Zoomed in: (iterates 300 through 450 for each r)

    Possible limit cycles for this choice of parameter

    period | iterates | linear stability 1 | 0 | unstable 1 | 0.75 | unstable 2 | 0.345492, 0.904508 | unstable 3 | 0.116978, 0.413176, 0.969846 | unstable 3 | 0.188255, 0.61126, 0.950484 | unstable 4 | 0.0337639, 0.130496, 0.453866, 0.991487 | unstable 4 | 0.0432273, 0.165435, 0.552264, 0.989074 | unstable 4 | 0.277131, 0.801317, 0.636831, 0.925109 | unstable

    Invariant density

    (20000 iterates; bin width 0.005)

    Perturbed trajectories

    (initial perturbation 1×10^-6)

    Lyapunov exponent

    λ≈0.693

    Plot of the Lyapunov exponent vs. r

    Plot of the Lyapunov exponent vs. r

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