Lyapunov vector

In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system. They have been used in predictability analysis and as initial perturbations for ensemble forecasting in numerical weather prediction. In modern practice they are often replaced by bred vectors for this purpose.

Lyapunov vectors are defined along the trajectories of a dynamical system. If the system can be described by a d-dimensional state vector ${\displaystyle x\in \mathbb {R} ^{d}}$ the Lyapunov vectors ${\displaystyle v^{(k)}(x)}$, ${\displaystyle (k=1\dots d)}$ point in the directions in which an infinitesimal perturbation will grow asymptotically, exponentially at an average rate given by the Lyapunov exponents ${\displaystyle \lambda _{k}}$.

...
Wikipedia