General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

中文翻译：

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一般分数动力学

广义分数动力学 (GFDynamics) 可以被视为一门跨学科的科学，其中通过使用广义分数阶微积分、具有广义分数积分 (GFI) 和导数 (GFD) 的方程或广义非定域来研究线性和非线性动力学系统的非局部性质。离散时间映射。GFDynamics 意味着研究和获得关于非局域性的一般形式的结果，可以用一般形式的算子核来描述，而不是通过其特定的实现和表示。本文提出了“一般非局部映射”的概念；这些是离散点 GFI 和 GFD 方程的精确解。在这些映射中，非局部性由属于 Sonin 和 Luchko 内核对集的算子内核确定。这些类型的核用于初始方程的一般分数积分和导数。使用一般分数阶微积分，我们考虑了具有一般时间非定域性的分数系统，这些分数系统由具有一般分数运算符和周期踢的方程描述。具有任意阶数的 GFI 和 GFD 的方程也用于导出一般的非局部映射。获得了这些带踢腿的一般分数阶微分和积分方程的精确解。这些具有离散时间点的精确解用于推导出没有近似值的一般非局部映射。描述了一些时间上的非定域性示例。它们由具有一般分数运算符和周期踢的方程描述。具有任意阶数的 GFI 和 GFD 的方程也用于导出一般的非局部映射。获得了这些带踢腿的一般分数阶微分和积分方程的精确解。这些具有离散时间点的精确解用于推导出没有近似值的一般非局部映射。描述了一些时间上的非定域性示例。它们由具有一般分数运算符和周期踢的方程描述。具有任意阶数的 GFI 和 GFD 的方程也用于导出一般的非局部映射。获得了这些带踢腿的一般分数阶微分和积分方程的精确解。这些具有离散时间点的精确解用于推导出没有近似值的一般非局部映射。描述了一些时间上的非定域性示例。这些具有离散时间点的精确解用于推导出没有近似值的一般非局部映射。描述了一些时间上的非定域性示例。这些具有离散时间点的精确解用于推导出没有近似值的一般非局部映射。描述了一些时间上的非定域性示例。