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.

Contents

[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 → Rn×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 In is the n-dimensional unit matrix. The dimension n of the matrices C is not related to the phase space X.

The Matrix Jt (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 > ... > λm are the different limits then there are subspaces Rn = R1 ⊃ ... ⊃ Rm ⊃ Rm+1 = {0} such that the limit is λi for v ∈ Ri\Ri+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] 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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