Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity
Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity
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DOI：
10.1103/physreve.56.245
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发表时间：
1997-07-01
影响因子：
2.4
通讯作者：
Tsallis, C
中科院分区：
文献类型：
--
作者：
Costa, UMS;Lyra, ML;Tsallis, C
研究了一类一维非线性类logisticlike映射x（t+1）= 1 -ax（t）z（z > 1; 0 < a小于或等于2; t = 0，1，2，.）对初始条件的幂律敏感性.我们的方法的主要成分是广义偏差律lim（Delta x（0）-->0）[Delta x（t）/Delta x（0）] = [1 +（1-q）lambda（q）t]（1/（l-q））（对于q = 1等于e（λ lt），对于大的t，对于q不等于1与t（1/（1-q））成比例; q是R的一个元素，是最近引入的非广延广义统计中出现的熵指数）。揭示了参数q与混沌吸引子的分形维数d（f）之间的关系：当d（f）从1（非分形的，遍历的，极限的）到0变化时，q呈现从1（Boltzmann-Gibbs的，广泛的，极限的）到无穷大的单调减小.
Power-law sensitivity to initial conditions, characterizing the behavior of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear one-dimensional logisticlike maps x(t+1)= 1 - a\x(t)\(z) (z > 1; 0 < a less than or equal to 2; t = 0,1,2,...). The main ingredient of our approach is the generalized deviation law lim (Delta x(0)-->0)[Delta x(t)/Delta x(0)] = [1 + (1-q)lambda(q)t](1/(l - q)) (equal to e(lambda lt) for q = 1, and proportional, for large t, to t(1/(1-q)) for q not equal 1; q is an element of R is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension d(f) of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to -infinity when d(f) varies from 1 (nonfractal, ergodiclike, limit) to zero.