# Double Pendulum

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 = d_{0} 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.