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:
formula_13
formula_14
with respect to the ordered set of "finite-time Lyapunov exponents"
formula_15 at the point formula_11.
The "finite-time Lyapunov dimension" of dynamical system with respect
to invariant set formula_17
is defined as follows
formula_18
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 formula_19
the "finite-time Lyapunov dimension" for a closed bounded invariant set formula_17
is an upper estimate of the Hausdorff dimension:
formula_21
Looking for best such estimation 
formula_22, "the Lyapunov dimension" is defined as follows:
formula_23
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.
Exact Lyapunov dimension.
Let the Jacobian matrix formula_24 at one of the equilibria have simple real eigenvalues:
formula_25,
then 
formula_26
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).
Definition via statistical physics approach and ergodicity.
Following the statistical physics approach and assuming the ergodicity
the Lyapunov dimension of attractor is estimated by 
limit value of the local Lyapunov dimension formula_27 
of a "typical" trajectory, which belongs to the attractor.
In this case formula_28 
and formula_29.
From a practical point of view, the rigorous use of ergodic Oseledec theorem,
verification that the considered trajectory formula_30 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 formula_31,
are called the "absolute" ones formula_32 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.