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 has nothing say about the rate of convergence. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a measure that characterizes rate of separation of infinitesimaly close trajectories. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...

The theorem is also known as the multiplicative ergodic theorem.

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 A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...

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. A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...

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. The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a measure that characterizes rate of separation of infinitesimaly close trajectories. ...

The theorem

Let μ be an ergodic 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 A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...

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_iR_i+1 and i = 1, ..., m.

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 and its inverse exist then the values of the Lyapunov exponents do not change. The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a measure that characterizes rate of separation of infinitesimaly close trajectories. ... The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a measure that characterizes rate of separation of infinitesimaly close trajectories. ...

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.