Lyapunov vector
Encyclopedia
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 .
map the system with state vector 
at time 
to the state 
at time 
. The linearization of this map, i.e. the Jacobian matrix 
describes the change of an infinitesimal perturbation 
. That is
Starting with an identity matrix
the iterations
where
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. The diagonal entries 
of 
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 
columns of 
will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In particular, the first column of 
will point in the direction of the Lyapunov vector with the largest Lyapunov exponent if 
is large enough. When iterated backward in time a random vector contained in the space spanned by the first 
collums of 
will almost surely, asymptotically align with the Lyapunov vector corresponding to the 
th largest Lyapunov exponent, if 
and 
are sufficiently large. Defining 
we find 
. Choosing the first 
entries of 
randomly and the other entries zero, and iterating this vector back in time, the vector 
aligns almost surely with the Lyapunov vector 
corresponding to the 
th largest Lyapunov exponent if 
and 
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.
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
theory, Lyapunov vectors, named after Aleksandr Lyapunov
Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov was a Russian mathematician, mechanician and physicist. His surname is sometimes romanized as Ljapunov, Liapunov or Ljapunow....
, describe characteristic expanding and contracting directions of a dynamical system. They have been used in predictability analysis and as initial perturbations for ensemble forecasting
Ensemble forecasting
Ensemble forecasting is a numerical prediction method that is used to attempt to generate a representative sample of the possible future states of a dynamical system...
in numerical weather prediction
Numerical weather prediction
Numerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic...
. 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 exponentsLyapunov exponentIn mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
. 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.
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 map the system with state vector at time to the state at time . The linearization of this map, i.e. the Jacobian matrix describes the change of an infinitesimal perturbation . That isStarting with an identity matrix
the iterationswhere
is given by the Gram-Schmidt QR decompositionQR decomposition
In linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...
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. The diagonal entries of are local growth factors in the directions of the Lyapunov vectors. The Lyapunov exponents are given by the average growth ratesand 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 columns of will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In particular, the first column of will point in the direction of the Lyapunov vector with the largest Lyapunov exponent if is large enough. When iterated backward in time a random vector contained in the space spanned by the first collums of will almost surely, asymptotically align with the Lyapunov vector corresponding to the th largest Lyapunov exponent, if and are sufficiently large. Defining we find . Choosing the first entries of randomly and the other entries zero, and iterating this vector back in time, the vector aligns almost surely with the Lyapunov vector corresponding to the th largest Lyapunov exponent if and 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.