Multifractality of the Feigenbaum attractor and fractional derivatives
Frisch, U. ; Khanin, K. ; Matsumoto, T.
arXiv, 0309068 / Harvested from arXiv
Text on arXiv
It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities $f(\alpha)$. This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions (Frisch and Matsumoto (J. Stat. Phys. 108:1181, 2002)). The relation between the thermodynamic approach (Vul, Sinai and Khanin (Russian Math. Surveys 39:1, 1984)) and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations.
Publié le : 2003-09-26
Classification:  Nonlinear Sciences - Chaotic Dynamics,  Mathematical Physics 
@article{0309068,
     author = {Frisch, U. and Khanin, K. and Matsumoto, T.},
     title = {Multifractality of the Feigenbaum attractor and fractional derivatives},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0309068}
}

Frisch, U.; Khanin, K.; Matsumoto, T. Multifractality of the Feigenbaum attractor and fractional derivatives. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0309068/