Lyapunov exponent
Significance of the Lyapunov spectrum

The Lyapunov spectrum can be used to give an estimate of the rate of entropy production,of the fractal dimension, and of the Hausdorff dimension of the considered dynamical system. In particular from the knowledge of the Lyapunov spectrum it is possible to obtain the so-called Lyapunov dimension (or Kaplan–Yorke dimension) D K Y {\displaystyle D_{KY}}, which is defined as follows: D K Y = k + ∑ i = 1 k λ i | λ k + 1 | {\displaystyle D_{KY}=k+\sum _{i=1}^{k}{\frac {\lambda _{i}}{|\lambda _{k+1}|}}} where k {\displaystyle k} is the maximum integer such that the sum of the k {\displaystyle k} largest exponents is still non-negative. D K Y {\displaystyle D_{KY}} represents an upper bound for the information dimension of the system. Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov–Sinai entropy accordingly to Pesin's theorem. Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, which is based on the direct Lyapunov method with special Lyapunov-like functions. The Lyapunov exponents of bounded trajectory and the Lyapunov dimension of attractor are invariant under diffeomorphism of the phase space.

The multiplicative inverse of the largest Lyapunov exponent is sometimes referred in literature as Lyapunov time, and defines the characteristic e -folding time. For chaotic orbits, the Lyapunov time will be finite, whereas for regular orbits it will be infinite.
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