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General Fractional Dynamics

Vasily E. Tarasov

TL;DR

This work extends fractional calculus to a general kernel framework (Sonin and Luchko) to define $I^{t}_{(M)}$ and $D^{t}_{(K)}$ that describe nonlocal memory in time. It derives exact discrete-time nonlocal maps from continuous equations with general fractional operators and periodic kicks, enabling nonlocal dynamics to be represented without approximation. Key contributions include formalizing the Sonin and Luchko kernel sets, establishing fundamental theorems for arbitrary-order GFC, introducing generalized momenta, and presenting universal nonlocal maps parameterized by an arbitrary nonlinear function $\mathcal{G}(t,X)$, with potential to uncover novel attractors and chaotic regimes. The framework provides a rigorous bridge between continuous nonlocal models and exact discrete-time representations, with wide applicability across physics, biology, and economics, and builds a foundation for analyzing nonlocal temporal dynamics via discrete mappings inspired by Edelman’s approach.

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.

General Fractional Dynamics

TL;DR

This work extends fractional calculus to a general kernel framework (Sonin and Luchko) to define and that describe nonlocal memory in time. It derives exact discrete-time nonlocal maps from continuous equations with general fractional operators and periodic kicks, enabling nonlocal dynamics to be represented without approximation. Key contributions include formalizing the Sonin and Luchko kernel sets, establishing fundamental theorems for arbitrary-order GFC, introducing generalized momenta, and presenting universal nonlocal maps parameterized by an arbitrary nonlinear function , with potential to uncover novel attractors and chaotic regimes. The framework provides a rigorous bridge between continuous nonlocal models and exact discrete-time representations, with wide applicability across physics, biology, and economics, and builds a foundation for analyzing nonlocal temporal dynamics via discrete mappings inspired by Edelman’s approach.

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.
Paper Structure (8 sections, 12 theorems, 225 equations)

This paper contains 8 sections, 12 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Equations of General Fractional Dynamical Systems
  3. To General Fractional Dynamics
  4. Examples Non-Locality in form of Sonin Kernels
  5. General Fractional Dynamics of Arbitrary Orders
  6. General Fractional Calculus of Arbitrary Orders
  7. General Momenta of Arbitrary Orders
  8. Conclusion

Key Result

Theorem 1

Let $K(t) \in C_{-1,0}(0,\infty)$ and $X(t) \in C^1_{-1}(0,\infty)$. Then the integro-differential equation has the solution if $Tn<t<T(n+1)$, where $M(t) \in C_{-1,0}(0,\infty)$ is a function that is associated kernel to the kernel $K(t)$, i.e. the functions $K(t)$, $M(t)$ form a mutually associated pair of Sonin's kernels.

Theorems & Definitions (39)

  • Example 1
  • Example 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 29 more