In applied mathematics and
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
theory, Lyapunov vectors, named after
Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Liapo ...
, 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 is a method used in or within numerical weather prediction. Instead of making a single forecast of the most likely weather, a set (or ensemble) of forecasts is produced. This set of forecasts aims to give an indication of the ...
in
numerical weather prediction
Numerical weather prediction (NWP) 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 th ...
.
In modern practice they are often replaced by
bred vector
In applied mathematics, bred vectors are perturbations related to Lyapunov vectors, that capture fast-growing dynamical instabilities of the solution of a numerical model. They are used, for example, as initial perturbations for ensemble forec ...
s for this purpose.
Mathematical description
![LyapunovDiagram](https://upload.wikimedia.org/wikipedia/commons/0/04/LyapunovDiagram.svg)
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 exponent
In 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. Quantitatively, two trajectories in phase space with ini ...
s
.
* 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 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
columns 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.
References
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[{{cite journal , first=W. , last=Ott , first2=J. A. , last2=Yorke , title=When Lyapunov exponents fail to exist , journal=Phys. Rev. E , volume=78 , issue= 5, pages=056203 , year=2008 , doi=10.1103/PhysRevE.78.056203 , pmid=19113196 , bibcode=2008PhRvE..78e6203O ]
[{{cite book , first=Edward , last=Ott , year=2002 , title=Chaos in Dynamical Systems , edition=Second , publisher=Cambridge University Press ]
Functional analysis
Mathematical physics
Dynamical systems