A Stability Testing Algorithm for Incommensurate Fractional Differential Equation Systems

TL;DR

This paper introduces a simplified stability testing algorithm for incommensurate fractional differential systems, applicable to linear and nonlinear cases, with a MATLAB implementation provided.

Contribution

It presents a novel, simpler algorithm for stability analysis of incommensurate fractional systems, extending to nonlinear cases with known techniques.

Findings

Algorithm effectively determines stability of fractional systems.

Applicable to both linear and nonlinear fractional differential equations.

MATLAB code implementation is provided.

Abstract

We consider the question of determining whether or not a given system of fractional-order differential equations is (asymptotically) stable. In particular, we admit systems where each constituent equation may have its own order, independent of the order of the other equations in the system, i.e. we discuss the so-called incommensurate case. Exploiting ideas based in numerical linear algebra, we present an algorithm that can be used to answer this question that is much simpler than known methods. We discuss in detail the case of linear problems where the ratios of orders are rational and indicate how known techniques can be used to apply our findings also to general nonlinear problems with arbitrary orders. A MATLAB implementation of the code is provided.