General Fractional Dynamics

TL;DR

This paper introduces the concept of general nonlocal maps derived from general fractional calculus, providing exact solutions for fractional systems with non-locality in time, advancing the understanding of nonlocal dynamics.

Contribution

It proposes the concept of general nonlocal maps as exact solutions of fractional equations with non-local kernels, expanding the framework of fractional dynamics.

Findings

Exact solutions for fractional differential equations with nonlocal kernels.

Derivation of general nonlocal maps without approximations.

Examples illustrating non-locality in time.

Abstract

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