Double Pendulum

The double pendulum demonstrates sensitive dependence on initial conditions. Here two identical pendulums may be released by the user at what appear to be the same point, but as the release points of the pendulums are slightly different, the behavior of the pendulums will soon diverge noticeably.

One way we study this mathematically is through the idea of a Lyapunov exponent. The basic idea is to imagine the length of a vector between two initial states. If a system is well-behaved then this length should either stay relatively constant (e.g., a stable system) or shrink to 0 as the two states approach an attracting equilibrium state. For chaotic systems with sensitive dependence on initial conditions this length can grow. Indeed, one signature of chaos is so-called "exponential divergence of initial states", whereby the length of the vector between two close initial states grows exponentially. Roughly speaking, if d is the length between the initial states, then say d = d₀ e ^{r t} for some r > 0. This constant r is known as a Lyapunov exponent. One should consider this calculation in state space and over large sets of initial conditions to get an accurate estimate.

The image above illustrates the idea. The top graphic shows the magnitude of the difference of two initial state vectors and the bottom graphic shows the log of the magnitude. The positive slope indicates the trajectories are growing apart and the slope of this line corresponds to the Lyapunov exponent (for this sample trial run). Since the trajectories live in a bounded region of state space they can not grow apart forever and thus the line eventually flattens out. For chaotic systems with strange attractors this can be seen by the fact that the states can eventually only be as far apart as the width of the attractor.