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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 the Lyapunov vectors , point in the directions in which an infinitesimal perturbation will grow asymptotically, exponentially at an average rate given by the Lyapunov exponents . * When expanded in terms of Lyapunov vectors a perturbation asymptotically aligns with the Lyapunov vector in that expansion corresponding to the largest Lyapunov exponent as this direction outgrows all others. Therefore almost all perturbations align asymptotically with the Lyapunov vector corresponding to the largest Lyapunov exponent in the system.〔 * In some cases Lyapunov vectors may not exist.〔 * Lyapunov vectors are not necessarily orthogonal. * Lyapunov vectors are not identical with the local principal expanding and contracting directions, i.e. the eigenvectors of the Jacobian. While the latter require only local knowledge of the system, the Lyapunov vectors are influenced by all Jacobians along a trajectory. * The Lyapunov vectors for a periodic orbit are the Floquet vectors of this orbit. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lyapunov vector」の詳細全文を読む スポンサード リンク
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