Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Stability Testing Algorithm for Incommensurate Fractional Differential Equation Systems

Kai Diethelm, Safoura Hashemishahraki

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.

A Stability Testing Algorithm for Incommensurate Fractional Differential Equation Systems

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.
Paper Structure (6 sections, 6 theorems, 26 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 6 theorems, 26 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Analytical Foundations
  3. The Basic Approach for Rational Quotients $\alpha_k / \alpha_\ell$
  4. Comparison with Existing Approaches
  5. Extension to Arbitrary $\alpha \in (0,1]^d$
  6. Examples

Key Result

theorem 1

Consider the differential equation system eq:lin-system. If all complex zeros of its characteristic function $\chi$ are in the open left half of the complex plane then all solutions $x$ of the system satisfy $\lim_{t \to \infty} x(t) = 0$.

Figures (2)

  • Figure 1: Location of the 78 finite eigenvalues $\mu$ for Example \ref{['ex:1']} with $\alpha = \alpha_1$. The blue rays separate the eigenvalues in category 3 from those in category 4.
  • Figure 2: The eight components of the numerical solution of the differential equation system from Example \ref{['ex:1']} with $\alpha = \alpha_1$ and initial condition $x(0) = (1, 0, -2, 0.5, -1, 1.5, -2, 0)^{\text{T}}$.

Theorems & Definitions (18)

  • remark 1
  • definition 1
  • theorem 1
  • theorem 2
  • remark 2
  • theorem 3
  • proof
  • remark 3
  • remark 4
  • theorem 4
  • ...and 8 more