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.

[1] In modern practice they are often replaced by bred vectors for this purpose.

[2] Lyapunov vectors are defined along the trajectories of a dynamical system.

If the system can be described by a d-dimensional state vector

point in the directions in which an infinitesimal perturbation will grow asymptotically, exponentially at an average rate given by the Lyapunov exponents

If the dynamical system is differentiable and the Lyapunov vectors exist, they can be found by forward and backward iterations of the linearized system along a trajectory.

map the system with state vector

The linearization of this map, i.e. the Jacobian matrix

describes the change of an infinitesimal perturbation

Starting with an identity matrix

is given by the Gram-Schmidt QR decomposition of

, will asymptotically converge to matrices that depend only on the points

of a trajectory but not on the initial choice of

The rows of the orthogonal matrices

define a local orthogonal reference frame at each point and the first

rows span the same space as the Lyapunov vectors corresponding to the

largest Lyapunov exponents.

The upper triangular matrices

describe the change of an infinitesimal perturbation from one local orthogonal frame to the next.

are local growth factors in the directions of the Lyapunov vectors.

The Lyapunov exponents are given by the average growth rates

and by virtue of stretching, rotating and Gram-Schmidt orthogonalization the Lyapunov exponents are ordered as

When iterated forward in time a random vector contained in the space spanned by the first

will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector.

When iterated backward in time a random vector contained in the space spanned by the first

will almost surely, asymptotically align with the Lyapunov vector corresponding to the

th largest Lyapunov exponent, if

aligns almost surely with the Lyapunov vector

Since the iterations will exponentially blow up or shrink a vector it can be re-normalized at any iteration point without changing the direction.