Definition of The Lyapunov Spectrum
For a dynamical system with evolution equation in an n–dimensional phase space, the spectrum of Lyapunov exponents
in general, depends on the starting point . (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. Note: Hamiltonian systems do not have attractors, so this particular discussion does not apply to them.) The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the Jacobian matrix
The matrix describes how a small change at the point propagates to the final point . The limit
defines a matrix (the conditions for the existence of the limit are given by the Oseledec theorem). If are the eigenvalues of, then the Lyapunov exponents are defined by
The set of Lyapunov exponents will be the same for almost all starting points of an ergodic component of the dynamical system.
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