Generation of nonlocal fractional dynamical systems by fractional differential equations

TL;DR

This paper investigates the behavior of solution trajectories in one-dimensional fractional differential equations, revealing conditions under which they coincide or intersect, and explores how these systems generate nonlocal fractional dynamical systems.

Contribution

It clarifies the conditions for trajectory intersection in 1D FDEs and characterizes when fractional differential equations generate nonlocal dynamical systems.

Findings

In 1D FDEs, trajectories either coincide or do not intersect.

Higher-dimensional FDEs can have intersecting trajectories.

1D FDEs generate nonlocal fractional dynamical systems, unlike most higher-dimensional cases.

Abstract

We show that any two trajectories of solutions of a one-dimensional fractional differential equation (FDE) either coincide or do not intersect each other. In contrary, in the higher dimensional case, two different trajectories can meet. Furthermore, one-dimensional FDEs and triangular systems of FDEs generate nonlocal fractional dynamical systems, whereas a higher dimensional FDE does, in general, not generate a nonlocal dynamical system.