Local Stable Manifold theorem for fractional systems revisited

TL;DR

This paper revisits and corrects the local stable manifold theorem for fractional systems, providing an analytical foundation that was previously limited despite extensive numerical research in fractional dynamical systems.

Contribution

It corrects the asymptotic expansion of Mittag-Leffler functions with matrix arguments and proves the local stable manifold theorem for fractional systems.

Findings

Corrected the asymptotic expansion of Mittag-Leffler functions with matrix argument.

Proved the local stable manifold theorem for fractional systems.

Enhanced analytical understanding of fractional differential equations.

Abstract

The subject of fractional calculus has witnessed rapid development over past few decades. In particular the area of fractional differential equations has received considerable attention. Several theoretical results have been obtained and powerful numerical methods have been developed. In spite of the extensive numerical simulations that have been carried out in the area of fractional order dynamical systems, analytical results obtained are very few. In pursuance to this, present authors have extended local stable manifold theorem in case of fractional systems \cite{deshpande2016local}. Cong et al. \cite{cong2016stable} have pointed out discrepancies in the asymptotic expansion of two-parameter Mittag-Leffler functions with matrix argument (\textit{cf.} Lemma 4 part 2 of article \cite{deshpande2016local}). In the present communication we give the corrected expansion of the same and prove the local stable manifold theorem by following the same approach given in \cite{deshpande2016local}.