Lyapunov exponent ($\lambda ={\lim }_{t\to \infty }{t}^{-1}\log \Vert D{\Phi }_{t}v\Vert$)

A Lyapunov exponent measures the asymptotic exponential rate of an infinitesimal tangent displacement under a dynamical system, when the displayed limit exists for a nonzero tangent vector $v$. A positive exponent represents exponential separation along that direction. In a smooth velocity field, the tangent displacement obeys $\dot{\mathbf{l}}=(\nabla \mathbf{u})\mathbf{l}$. Finite-separation growth eventually leaves this linear regime; moments of random separation need not grow at the same rate as typical separation.