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.
Mathematical description.
Lyapunov vectors are defined along the trajectories of a dynamical system. If the system can be described by a d-dimensional state vector formula_1 the Lyapunov vectors formula_2, formula_3 point in the directions in which an infinitesimal perturbation will grow asymptotically, exponentially at an average rate given by the Lyapunov exponents formula_4. 
Numerical method.
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. Let formula_5 map the system with state vector formula_6 at time formula_7 to the state formula_8 at time formula_9. The linearization of this map, i.e. the Jacobian matrix formula_10 describes the change of an infinitesimal perturbation formula_11. That is
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Starting with an identity matrix formula_13 the iterations
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where formula_15 is given by the Gram-Schmidt QR decomposition of formula_16, will asymptotically converge to matrices that depend only on the points formula_6 of a trajectory but not on the initial choice of formula_18. The rows of the orthogonal matrices formula_19 define a local orthogonal reference frame at each point and the first formula_20 rows span the same space as the Lyapunov vectors corresponding to the formula_20 largest Lyapunov exponents. The upper triangular matrices formula_22 describe the change of an infinitesimal perturbation from one local orthogonal frame to the next. The diagonal entries formula_23 of formula_22 are local growth factors in the directions of the Lyapunov vectors. The Lyapunov exponents are given by the average growth rates
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and by virtue of stretching, rotating and Gram-Schmidt orthogonalization the Lyapunov exponents are ordered as formula_26. When iterated forward in time a random vector contained in the space spanned by the first formula_20 columns of formula_19 will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In particular, the first column of formula_19 will point in the direction of the Lyapunov vector with the largest Lyapunov exponent if formula_30 is large enough. When iterated backward in time a random vector contained in the space spanned by the first formula_20 columns of formula_32 will almost surely, asymptotically align with the Lyapunov vector corresponding to the formula_20th largest Lyapunov exponent, if formula_30 and formula_35 are sufficiently large. Defining formula_36 we find formula_37. Choosing the first formula_20 entries of formula_39 randomly and the other entries zero, and iterating this vector back in time, the vector formula_40 aligns almost surely with the Lyapunov vector formula_41 corresponding to the formula_20th largest Lyapunov exponent if formula_35 and formula_30 are sufficiently large. Since the iterations will exponentially blow up or shrink a vector it can be re-normalized at any iteration point without changing the direction.