分数阶导数微分方程与记忆泛映射
Differential Equations with Fractional Derivative and Universal Map with Memory
摘要：
从具有Riemann-Liouville和Caputo分数阶导数的非线性微分方程得到具有长期记忆的离散映射。这些映射是著名的普适映射的推广。记忆意味着它们现在的状态是由所有过去的状态以特殊形式的权重决定的。利用Cauchy型问题与第二类非线性Volterra型积分方程的等价性，得到了分数阶微分方程的离散映射。对于Riemann-Liouville和Caputo分数阶导数的情形，给出了考虑一般初始条件的具有记忆的泛映射的一般形式。
英文摘要：
Discrete maps with long-term memory are obtained from nonlinear differential equations with Riemann-Liouville and Caputo fractional derivatives. These maps are generalizations of the well-known universal map. The memory means that their present state is determined by all past states with special forms of weights. To obtain discrete map from fractional differential equations, we use the equivalence of the Cauchy-type problems and to the nonlinear Volterra integral equations of second kind. General forms of the universal maps with memory, which take into account general initial conditions, for the cases of the Riemann-Liouville and Caputo fractional derivatives, are suggested.
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