Lyapunov dimension

In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. 
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. Since the direct numerical computation 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 because of the close connection with the Lyapunov exponents.
Definitions.
Consider a dynamical system 
formula_1, where formula_2 is the shift operator along the solutions:
formula_3,
of ODE formula_4, formula_5, 
or difference equation formula_6, formula_7,
with continuously differentiable vector-function formula_8.
Then formula_9 is the fundamental matrix of solutions of linearized system
and denote by formula_10,
singular values with respect to their algebraic multiplicity,
ordered by decreasing for any formula_11 and formula_12.
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"
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