# Logistic Map and Lyapunov Exponent

LogisticMap is a C# Windows application that I developed for homework in my Statistical Mechanics class.  The application plots the spread between two points on a logistic map and calculates the Lyapunov exponent for the spreading.  LogisticMap is a solution to Exercise 5.9 in Dr. James Sethna’s Statistical Mechanics: Entropy, Order Parameters, and Complexity.

You can get the LogisticMap application and its source code on GitHub.

Here’s some background on the logistic map, chaos, and the Lyapunov exponent.  In the application, we are evaluating the following function:

f(x) = 4μx(1-x)

This function is called a logistic map because it takes a point between 0 and 1 and returns a different point that is also between 0 and 1.  It maps the unit interval (0,1) into itself.  We can think about the trajectory of an initial point, x0, on the map as being the successive results of plugging the previous result back into the function: f(x0), f(f(x0)), f(f(f(x0))), …

The trajectory depends on the value of the constant μ.  When μ = 0 obviously all trajectories will immediately converge to 0.  When μ = 0.5 all trajectories converge on 0.5 but not immediately. When μ = 0.9 the trajectories do not converge on one value, but instead wind up within a certain range of values (roughly between 0.3 and 0.6 and between 0.8 and 0.9).