通过$q$ -产品概括逻辑图
Generalising the logistic map through the $q$-product
摘要：
我们研究逻辑斯蒂映射的推广为$ x_{n+1}=1-ax_{n}\otimes_{q_{map}} x_{n}$（$-1 \le x_{n} \le 1$，$0<a\le2$），其中$\otimes_q$代表普通乘积的推广，称为$q$ -乘积[Borges，E. P. Physica A { \bf 340}，95（2004）]。在极限$q\to 1$中恢复了通常的乘积，从而恢复了通常的Logistic映射。帐篷映射也是$q_{map}\to\infty$的特殊情况。这个（和其他）代数算子的推广已经在非广延的统计力学范围内得到了广泛的应用（见C。Tsallis，{ \em Introduction to Nonextensive Statistical Mechanics }，Springer，NY，2009）。重点分析了混沌边缘的$q_{map}>1$，特别是第一临界点$a_c$，它依赖于$q_{map}$的值.分叉图，初始条件的敏感性，分形维数和熵增长率进行评估，在$a_c(q_{map})$，并与非广泛的统计力学的连接进行了探讨。
英文摘要：
We investigate a generalisation of the logistic map as $ x_{n+1}=1-ax_{n}\otimes_{q_{map}} x_{n}$ ($-1 \le x_{n} \le 1$, $0<a\le2$) where $\otimes_q$ stands for a generalisation of the ordinary product, known as $q$-product [Borges, E.P. Physica A {\bf 340}, 95 (2004)]. The usual product, and consequently the usual logistic map, is recovered in the limit $q\to 1$, The tent map is also a particular case for $q_{map}\to\infty$. The generalisation of this (and others) algebraic operator has been widely used within nonextensive statistical mechanics context (see C. Tsallis, {\em Introduction to Nonextensive Statistical Mechanics}, Springer, NY, 2009). We focus the analysis for $q_{map}>1$ at the edge of chaos, particularly at the first critical point $a_c$, that depends on the value of $q_{map}$. Bifurcation diagrams, sensitivity to initial conditions, fractal dimension and rate of entropy growth are evaluated at $a_c(q_{map})$, and connections with nonextensive statistical mechanics are explored.
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