Semi-dynamical systems generated by autonomous Caputo fractional differential equations

TL;DR

This paper demonstrates that autonomous Caputo fractional differential equations of order (0,1) with Lipschitz continuous vector fields generate semi-dynamical systems in the space of continuous functions, contrasting with previous results on dynamical systems.

Contribution

It establishes that such fractional differential equations generate semi-dynamical systems in a function space, expanding understanding of their dynamical properties.

Findings

Generation of semi-dynamical systems in (0,1) fractional case

Contrast with non-generation of full dynamical systems in (0,1) case

Extension of dynamical systems theory to fractional differential equations

Abstract

An autonomous Caputo fractional differential equation of order $\alpha\in(0,1)$ in $\mathbb{R}^d$ whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space $\mathfrak{C}$ of continuous functions $f:\R^+\rightarrow \R^d$ with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong \& Tuan \cite{cong}, which showed that such equations do not, in general, generate a dynamical system on the space $\mathbb{R}^d$.