Lyapunov exponent

Summary

A Lyapunov exponent measures the exponential rate at which a solution of a differential equation grows or decays as time goes to infinity. For a linear system it is the upper limit of $\frac{1}{t} \ln \norm{x(t)}$ along a nonzero solution $x$. For a nonlinear system one applies it to the linearised equation along a trajectory, and it then measures how fast nearby trajectories separate. A negative exponent signals contraction, related to Lyapunov stability, and a positive one signals sensitive dependence on initial conditions, a mark of chaos. See the definition.