Measuring Chaos in a Double Pendulum

A double pendulum exhibits chaotic behavior given proper initial conditions. This chaotic behavior is characterized by measuring the correlation of a pendulum’s track with itself over time as well as calculating the Lyapunov exponents. A chaotic path’s correlation with itself (or autocorrelation) should decrease exponentially with the lag. A Lyapunov exponent characterizes the separation of infinitesimally close points in phase space over time; exponential growth of this separation typically implies chaotic behavior. A simulation with and without introduced error, as well as a physical pendulum tracked using a high-speed camera are both used to derive auto-correlations and Lyapunov exponents. Expected results are to observe these indicators of chaotic motion in both the simulated and physical double pendulum.