Edelman M
Department of Physics, Stern College at Yeshiva University, 245 Lexington Ave., New York, New York 10016, USA and Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, New York 10012, USA.
Chaos. 2014 Jun;24(2):023137. doi: 10.1063/1.4885536.
In this paper, the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo standard α-Families of maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories, and dependence of properties on the memory parameter α) and differences in properties of falling factorial- and power-law memory maps are investigated.
在本文中，作者以分数阶和分数阶差分Caputo标准α-映射族为例，比较了可用分数阶微分方程和分数阶差分方程描述的系统的行为。作者表明，分数阶差分映射（具有降阶乘律记忆的系统）的性质与分数阶映射（具有幂律记忆的系统）的性质相似。研究了降阶乘律和幂律记忆映射在性质上的相似之处（吸引子类型、轨迹的幂律收敛、分岔级联和间歇性分岔级联类型轨迹的存在，以及性质对记忆参数α的依赖性）和差异。
