Chaos edges of z -logistic maps: Connection between the relaxation and sensitivity entropic indices
Makaleler > Tam Makale · 2006
Yazarlar: Uğur TIRNAKLI, C TSALLIS
Physical Review E, SCI Expanded İndekslerine Giren Dergi, Uluslararası, Wos, 2006, cilt 73, ss 037201-037201
DOI: 10.1103/physreve.73.037201
Özet
Chaos thresholds of the $z$-logistic maps ${x}_{t+1}=1\ensuremath{-}a\ensuremath{\mid}{x}_{t}{\ensuremath{\mid}}^{z}$ $(z>1;\phantom{\rule{0.3em}{0ex}}t=0,1,2,\dots{})$ are numerically analyzed at accumulation points of cycles 2, 3, and 5 (three different cycles 5). We verify that the nonextensive $q$-generalization of a Pesin-like identity is preserved through averaging over the entire phase space. More precisely, we computationally verify ${\mathrm{lim}}_{t\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{S}_{{q}_{sen}^{av}}⟩(t)∕t={\mathrm{lim}}_{t\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{\mathrm{ln}}_{{q}_{sen}^{av}}\phantom{\rule{0.2em}{0ex}}\ensuremath{\xi}⟩(t)∕t\ensuremath{\equiv}{\ensuremath{\lambda}}_{{q}_{sen}^{av}}^{av}$, where the entropy ${S}_{q}\ensuremath{\equiv}(1\ensuremath{-}{\ensuremath{\sum}}_{i}{p}_{i}^{q})∕(q\ensuremath{-}1)$ $({S}_{1}=\ensuremath{-}{\ensuremath{\sum}}_{i}{p}_{i}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{p}_{i})$, the sensitivity to the initial conditions $\ensuremath{\xi}\ensuremath{\equiv}{\mathrm{lim}}_{\mathrm{\ensuremath{\Delta}}x(0)\ensuremath{\rightarrow}0}\mathrm{\ensuremath{\Delta}}x(t)∕\mathrm{\ensuremath{\Delta}}x(0)$, and ${\mathrm{ln}}_{q}x\ensuremath{\equiv}({x}^{1\ensuremath{-}q}\ensuremath{-}1)∕(1\ensuremath{-}q)$ $({\mathrm{ln}}_{1}x=\mathrm{ln}\phantom{\rule{0.2em}{0ex}}x)$. The entropic index ${q}_{sen}^{av}<1$, and the coefficient ${\ensuremath{\lambda}}_{{q}_{sen}^{av}}^{av}>0$ depend on both $z$ and the cycle. We also study the relaxation that occurs if we start with an ensemble of initial conditions homogeneously occupying the entire phase space. The associated Lebesgue measure asymptotically decreases as $1∕{t}^{1∕({q}_{rel}\ensuremath{-}1)}\phantom{\rule{0.3em}{0ex}}({q}_{rel}>1)$. These results (i) illustrate the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely, ${q}_{rel}\ensuremath{-}1\ensuremath{\simeq}{A}_{n}(1\ensuremath{-}{q}_{sen}^{av}{)}^{{\ensuremath{\alpha}}_{n}}$, where the positive numbers $({A}_{n},{\ensuremath{\alpha}}_{n})$ depend on the cycle; (ii) exhibit an unexpected scaling, namely, ${q}_{sen}^{av}(\mathrm{cycle}\phantom{\rule{0.3em}{0ex}}n)={B}_{n}{q}_{sen}^{av}(\mathrm{cycle}\phantom{\rule{0.3em}{0ex}}2)+{ϵ}_{n}$.