In the mathematics of
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
, the concept of Lyapunov dimension was suggested by
Kaplan and Yorke for estimating the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
of
attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
s.
Further the concept has been developed and rigorously justified in a number of papers, and nowadays various different approaches to the definition of Lyapunov dimension are used. Remark that the attractors with noninteger Hausdorff dimension are called
strange attractors
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
.
Since the direct
numerical computation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
of the Hausdorff dimension of attractors is often a problem of high numerical complexity, estimations via the Lyapunov dimension became widely spread.
The Lyapunov dimension was named
after the Russian mathematician
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 ...
because of the close connection with the
Lyapunov exponents
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 ...
.
Definitions
Consider a
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 ...
, where
is the shift operator along the solutions:
,
of
ODE
An ode (from grc, ᾠδή, ōdḗ) is a type of lyric poetry. Odes are elaborately structured poems praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structured in three majo ...
,
,
or difference equation
,
,
with continuously differentiable vector-function
.
Then
is the
fundamental matrix of solutions of linearized system
and denote by
,
singular values In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self- ...
with respect to their
algebraic multiplicity
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
,
ordered by decreasing for any
and
.
Definition via finite-time Lyapunov dimension
The concept of finite-time Lyapunov dimension and related definition of the Lyapunov dimension, developed in the works by
N. Kuznetsov,
is convenient for the numerical experiments where only finite time can be observed.
Consider an analog of the
Kaplan–Yorke formula for the finite-time Lyapunov exponents:
:
:
with respect to the ordered set of ''finite-time Lyapunov exponents''
at the point
.
The ''finite-time Lyapunov dimension'' of dynamical system with respect
to
invariant set
In mathematics, an invariant is a property of a mathematical object (or a Class (set theory), class of mathematical objects) which remains unchanged after Operation (mathematics), operations or Transformation (function), transformations of a ce ...
is defined as follows
:
In this approach the use of the analog of Kaplan–Yorke formula
is rigorously justified by the Douady–Oesterlè theorem,
which proves that for any fixed
the ''finite-time Lyapunov dimension'' for a closed bounded invariant set
is an upper estimate of the Hausdorff dimension:
:
Looking for best such estimation
, ''the Lyapunov dimension'' is defined as follows:
[
:
The possibilities of changing the order of the time limit and the supremum over set is discussed, e.g., in.]
Note that the above defined Lyapunov dimension is invariant under Lipschitz diffeomorphisms
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ma ...
.
Exact Lyapunov dimension
Let the Jacobian matrix at one of the equilibria have simple real eigenvalues:
,
then
:
If the supremum of local Lyapunov dimensions on the global attractor, which involves all equilibria, is achieved at an equilibrium point, then this allows one to get analytical formula of the exact Lyapunov dimension of the global attractor (see corresponding Eden’s conjecture In the mathematics of dynamical systems, Eden's conjecture states that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.
The validity ...
).
Definition via statistical physics approach and ergodicity
Following the statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...
approach and assuming the ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
the Lyapunov dimension of attractor is estimated[ by
limit value of the local Lyapunov dimension
of a ''typical'' trajectory, which belongs to the attractor.
In this case
and .
From a practical point of view, the rigorous use of ergodic Oseledec theorem,
verification that the considered trajectory is a ''typical'' trajectory,
and the use of corresponding Kaplan–Yorke formula is a challenging task
(see, e.g. discussions in]).
The exact limit values of finite-time Lyapunov exponents,
if they exist and are the same for all ,
are called the ''absolute'' ones[ and used in the Kaplan–Yorke formula.
Examples of the rigorous use of the ergodic theory for the computation of the Lyapunov exponents and dimension can be found in.]
References
{{Reflist
Dynamical systems