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. 2013 Sep;23(3):033127. doi: 10.1063/1.4819165.
We modified the way in which the Universal Map is obtained in the regular dynamics to derive the Universal α-Family of Maps depending on a single parameter α>0, which is the order of the fractional derivative in the nonlinear fractional differential equation describing a system experiencing periodic kicks. We consider two particular α-families corresponding to the Standard and Logistic Maps. For fractional α<2 in the area of parameter values of the transition through the period doubling cascade of bifurcations from regular to chaotic motion in regular dynamics corresponding fractional systems demonstrate a new type of attractors--cascade of bifurcations type trajectories.
我们修改了常规动力学中获取通用图的方式，推导出了通用的α-映射族，该映射族依赖于单个参数α>0，该参数是描述经历周期性踢动的系统的非线性分数阶微分方程中的分数阶导数的阶数。我们考虑了对应于标准映射和 logistic 映射的两个特殊的α-族。对于分数阶α<2，在常规动力学中从规则运动到混沌运动的倍周期分岔通过的参数值区域中，相应的分数阶系统表现出一种新的吸引子类型——分岔类型轨迹的级联。
