Oseledec theorem

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In mathematics, the Oseledec theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. The theorem states conditions for the existence of the defining limits and describes the Lyapunov exponents look. It says nothing about the rate of convergence.

The theorem is also known as the multiplicative ergodic theorem.

1 Cocycles
2 The theorem
3 Transformation invariance
4 Multiplicative vs. additive ergodic theorems
5 References

[edit] Cocycles

The theorem has a broader application than just Lyapunov exponents: it holds for arbitrary cocycles; see the following definition and examples.

A cocycle of an autonomous dynamical system is a map C : X×T → R^n×n satisfying

$C(x,0)=I_n {\rm~for~all~} x\in X$

$C(x,t+s)=C(x(t),s)\,C(x,t) {\rm~for~all~} x\in X {\rm~and~} t,s\in T$

where X and T (with T=Z or T=R) are the phase space and the time range, respectively, of the dynamical system, and I_n is the n-dimensional unit matrix. The dimension n of the matrices C is not related to the phase space X.

The Matrix J^t (see Lyapunov exponent) is the most famous example of a cocycle; here the dimension n of C is the same as that of X. A one dimensional example is the determinant det C(x, t) for any cocycle C.

[edit] The theorem

Let μ be an invariant measure on X and C a cocycle of the dynamical system such that ||C(x,t)|| and ||C(x,t)⁻¹|| are L¹-integrable (i.e. such that C(x, t)⁻¹ exists if T = Z). Then for μ-almost all x and each non-zero vector u∈Rⁿ the limit

$\lambda=\lim_{t\to\infty}{1\over t} \log{||C(x,t)u|| \over ||u||}$

exists and assumes, depending on u but not on x, up to n different values.

Further, if λ₁ > ... > λ_m are the different limits then there are subspaces Rⁿ = R₁ ⊃ ... ⊃ R_m ⊃ R_m+1 = {0} such that the limit is λ_i for v ∈ R_i\R_i+1 and i = 1, ..., m.

[edit] Transformation invariance

The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate transformations. Suppose that g : X → X is a one-to-one map such that $\partial g/\partial x$ and its inverse exist then the values of the Lyapunov exponents do not change.

[edit] Multiplicative vs. additive ergodic theorems

Verbally, ergodicity means that time and space averages are equal:

$\lim_{t\to\infty}{1\over t} \int_0^t f(x(s))ds = \int_X f(x)\mu(dx)$

Space average (right hand side) is the accumulation of f(x) values weighted by μ(dx). Since addition is commutative, the accumulation of the f(x)μ(dx) values may be done in arbitrary order. In contrast, the time average (left hand side) insists on a prescribed ordering of the f(x(s)) values along the trajectory.

Since matrix multiplication is, in general, not commutative, accumulation of multiplied cocycle values (and limits thereof) according to C(x(t₀),t_k) = C(x(t_k-1),t_k-t_k-1) ... C(x(t₀),t₁-t₀) makes sense only for a prescribed ordering. Thus, the time average may exist (and the theorem states that it actually exists), but there is no space average counterpart. In other words, the theorem is different from other ergodic theorems (such as G. D. Birkhoff's and J. von Neumann's) in that it guarantees the existence of the time average, but does nothing claim about the space average.

[edit] References

V. I. Oseledets, "Multiplicative ergodic theorem: Characteristic Lyapunov exponents of dynamical systems", Trudy MMO 19 (1968), 179-210. (in Russian).
D. Ruelle, "Ergodic theory of differentiable dynamic systems", IHES Publ. Math. 50 (1979), 27-58.

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Categories: Dynamical systems | Ergodic theory