Lyapunov vector

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

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

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

${\displaystyle Q_{n+1}{R}_{n+1}=J_n{Q}_n}$

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

${\displaystyle {\lambda }_k=\underset{m\to \mathrm{\infty }}{lim}\frac{1}{t_{n+m}-t_n}\sum _{l=1}^m\log r_{kk}^{(n+l)}}$

and by virtue of stretching, rotating and Gram-Schmidt orthogonalization the Lyapunov exponents are ordered as ${\displaystyle {\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_d}$. When iterated forward in time a random vector contained in the space spanned by the first ${\displaystyle k}$ columns of ${\displaystyle Q_n}$ will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In particular, the first column of ${\displaystyle Q_n}$ will point in the direction of the Lyapunov vector with the largest Lyapunov exponent if ${\displaystyle n}$ is large enough. When iterated backward in time a random vector contained in the space spanned by the first ${\displaystyle k}$ columns of ${\displaystyle Q_{n+m}}$ will almost surely, asymptotically align with the Lyapunov vector corresponding to the ${\displaystyle k}$th largest Lyapunov exponent, if ${\displaystyle n}$ and ${\displaystyle m}$ are sufficiently large. Defining ${\displaystyle c_n=Q_n^T{h}_n}$ we find ${\displaystyle c_{n-1}=R_n^{-1}{c}_n}$. Choosing the first ${\displaystyle k}$ entries of ${\displaystyle c_{n+m}}$ randomly and the other entries zero, and iterating this vector back in time, the vector ${\displaystyle Q_n{c}_n}$ aligns almost surely with the Lyapunov vector ${\displaystyle v^{(k)}(x_n)}$ corresponding to the ${\displaystyle k}$th largest Lyapunov exponent if ${\displaystyle m}$ and ${\displaystyle n}$ 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.