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.

For a unit perturbation direction $e$ and Jacobian matrix $J$, the instantaneous logarithmic growth rate of its Euclidean norm is $e^{T}Se$, where $S=(J+J^T)/2$. Indeed $d|\eta {|}^2/dt=2{\eta }^{T}S\eta$ for $\dot{\eta }=J\eta$. The eigenvalues of the symmetric matrix $S$ give the principal rates and its orthonormal eigenvectors give the principal directions. These directions generally rotate as the trajectory evolves; their instantaneous rates need not equal asymptotic Lyapunov exponents.