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]

Mathematical description

Depiction of the asymmetric growth of perturbations along an evolved trajectory.

Lyapunov vectors are defined along the trajectories of a dynamical system. If the system can be described by a d-dimensional state vector x\in\mathbb{R}^d the Lyapunov vectors v^{(k)}(x), (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 \lambda_k.

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.[5] Let x_{n+1}=M_{t_n\to t_{n+1}}(x_n) map the system with state vector x_n at time t_n to the state x_{n+1} at time t_{n+1}. The linearization of this map, i.e. the Jacobian matrix ~J_n describes the change of an infinitesimal perturbation h_n. That is


      M_{t_n\to t_{n+1}}(x_n + h_n) \approx M_{t_n\to t_{n+1}}(x_n) + J_n h_n = x_{n+1} + h_{n+1}


Starting with an identity matrix Q_0=\mathbb{I}~ the iterations


	Q_{n+1}R_{n+1} = J_n Q_n


where Q_{n+1}R_{n+1} is given by the Gram-Schmidt QR decomposition of J_n Q_n, will asymptotically converge to matrices that depend only on the points x_n of a trajectory but not on the initial choice of Q_0. The rows of the orthogonal matrices Q_n define a local orthogonal reference frame at each point and the first k rows span the same space as the Lyapunov vectors corresponding to the k largest Lyapunov exponents. The upper triangular matrices R_n describe the change of an infinitesimal perturbation from one local orthogonal frame to the next. The diagonal entries r^{(n)}_{kk} of R_n are local growth factors in the directions of the Lyapunov vectors. The Lyapunov exponents are given by the average growth rates


      \lambda_k = \lim_{m\to\infty}\frac{1}{t_{n+m}-t_n} \sum_{l=1}^m \log r^{(n+l)}_{kk}


and by virtue of stretching, rotating and Gram-Schmidt orthogonalization the Lyapunov exponents are ordered as \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_d. When iterated forward in time a random vector contained in the space spanned by the first k columns of Q_n will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In particular, the first column of Q_n will point in the direction of the Lyapunov vector with the largest Lyapunov exponent if n is large enough. When iterated backward in time a random vector contained in the space spanned by the first k columns of Q_{n+m} will almost surely, asymptotically align with the Lyapunov vector corresponding to the kth largest Lyapunov exponent, if n and m are sufficiently large. Defining c_n = Q_n^{T} h_n we find c_{n-1} = R_n^{-1} c_n. Choosing the first k entries of c_{n+m} randomly and the other entries zero, and iterating this vector back in time, the vector Q_n c_n aligns almost surely with the Lyapunov vector v^{(k)}(x_n) corresponding to the kth largest Lyapunov exponent if m and n 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.

References

  1. Kalnay, E. (2007), "Atmospheric Modeling, Data Assimilation and Predictability", Cambridge: Cambridge University Press
  2. Kalnay E, Corazza M, Cai M. "Are Bred Vectors the same as Lyapunov Vectors?", EGS XXVII General Assembly, (2002)
  3. Edward Ott (2002), "Chaos in Dynamical Systems", second edition, Cambridge University Press.
  4. W. Ott and J. A. Yorke, "When Lyapunov exponents fail to exist", Phys. Rev. E 78, 056203 (2008)
  5. F Ginelli, P Poggi, A Turchi, H Chaté, R Livi, and A Politi, "Characterizing Dynamics with Covariant Lyapunov Vectors", Phys. Rev. Lett. 99, 130601 (2007), arXiv
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