2005/02/25 by Uǧur Tırnaklı, Ugur Tirnakli, Constantino Tsallis
Economics, Econometrics and Finance · Physics and Astronomy · #Chaos control and synchronization #Complex Systems and Time Series Analysis #Statistical Mechanics and Entropy #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.73.037201
published as Phys. Rev. E 73 (2006) 037201 · 5 pages, 5 figures
arxiv created 2005/02/25 · openalex publication_date 2006/03/15 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Chaos thresholds of the z-logistic maps xt+1=1\ensuremath-a\ensuremath|xt\ensuremath|z (z>1;\phantom\rule0.3em0ext=0,1,2,…) 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 lim_t\ensuremath→\ensuremath∞⟨S_qsenav⟩(t)∕t=lim_t\ensuremath→\ensuremath∞⟨ln_qsenav\phantom\rule0.2em0ex\ensuremathξ⟩(t)∕t\ensuremath≡\ensuremathλ_qsenavav, where the entropy Sq\ensuremath≡(1\ensuremath-\ensuremath∑ipiq)∕(q\ensuremath-1) (S1=\ensuremath-\ensuremath∑ipiln\phantom\rule0.2em0expi), the sensitivity to the initial conditions \ensuremathξ\ensuremath≡lim_\mathrm\ensuremathΔx(0)\ensuremath→0\mathrm\ensuremathΔx(t)∕\mathrm\ensuremathΔx(0), and lnqx\ensuremath≡(x^1\ensuremath-q\ensuremath-1)∕(1\ensuremath-q) (ln1x=ln\phantom\rule0.2em0exx). The entropic index qsenav<1, and the coefficient \ensuremathλ_qsenavav>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∕(qrel\ensuremath-1)\phantom\rule0.3em0ex(qrel>1). These results (i) illustrate the connection (conjectured by one of us) between sensitivity and relaxation entropic indices, namely, qrel\ensuremath-1\ensuremath≃An(1\ensuremath-qsenav)^\ensuremathαn, where the positive numbers (An,\ensuremathαn) depend on the cycle; (ii) exhibit an unexpected scaling, namely, qsenav(cycle\phantom\rule0.3em0exn)=Bnqsenav(cycle\phantom\rule0.3em0ex2)+ϵn.