Editing Lyapunov exponent (section)

==Definition of the Lyapunov spectrum==
[[File:LyapunovDiagram.svg|class=skin-invert-image|alt=Lyapunov exponent|thumb|236x236px|The leading Lyapunov vector.]]
For a dynamical system with evolution equation <math>\dot{x}_i = f_i(x)</math> in an ''n''–dimensional phase space, the spectrum of Lyapunov exponents
<math display="block"> \{ \lambda_1, \lambda_2, \ldots , \lambda_n \} \,, </math>
in general, depends on the starting point <math>x_0</math>. However, we will usually be interested in the [[attractor]] (or attractors) of a dynamical system, and there will normally be one set of exponents associated with each attractor. The choice of starting point may determine which attractor the system ends up on, if there is more than one. (For Hamiltonian systems, which do not have attractors, this is not a concern.) The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the [[Jacobian matrix]]
<math display="block"> J_{ij}(t) = \left. \frac{ d f_i(x) }{dx_j} \right|_{x(t)} </math>
this Jacobian defines the evolution of the tangent vectors, given by the matrix <math>Y</math>, via the equation
<math display="block"> \dot{Y} = J Y </math>
with the initial condition <math>Y_{ij}(0) = \delta_{ij}</math>. The matrix <math>Y</math> describes how a small change at the point <math>x(0)</math> propagates to the final point <math>x(t)</math>. The limit
<math display="block"> \Lambda = \lim_{t \rightarrow \infty} \frac{1}{2t} \log (Y(t) Y^T(t)) </math>
defines a matrix <math>\Lambda</math> (the conditions for the existence of the limit are given by the [[Oseledets theorem]]). The Lyapunov exponents <math>\lambda_i</math> are defined by the eigenvalues of <math>\Lambda</math>.

The set of Lyapunov exponents will be the same for almost all starting points of an [[Dynamical system#Ergodic systems|ergodic]] component of the dynamical system.