Lyapunov Exponents for the Logistic Map

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This Demonstration plots the orbit diagram of the logistic map

x

k+1

=4λ

x

k

(1-

x

k

)

and the corresponding Lyapunov exponents for different ranges of the parameter

λ

.

The Lyapunov exponent is a parameter characterizing the behavior of a dynamical system. It gives the average rate of exponential divergence from nearby initial conditions. The Lyapunov exponent of the logistic map is given by

1

n

∑

k=1

log[4λ(1-2

x

k

)]

.If the Lyapunov exponent is positive, then the system is chaotic; if it is negative, the system will converge to a periodic state; and if it is zero, there is a bifurcation.

By dragging the locator to the left or right or clicking the plot, you can scroll through the whole range of

λ

-values (0.70–1.0), generate the bifurcation diagram, and plot the Lyapunov exponent over that range.

You can zoom to the position of the locator by using the zoom sliders. To replot the graphs at higher zoom scales, use the "detail" button to increase the number of

λ

values and the number of iterations to 5000.