A constructive approach for investigating the stability of incommensurate fractional differential systems

Abstract

This paper is devoted to studying the asymptotic behaviour of solutions to generalized non-commensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional-order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of a noncommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a noncommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.

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References in corpus (4)

• Trapezoidal methods for fractional differential equations: theoretical and computational aspects
• Asymptotic behavior of solutions of linear multi-order fractional differential equation systems
• Asymptotic behaviour of solutions to non-commensurate fractional-order planar systems
• Qualitative analysis of solutions to mixed-order positive linear coupled systems with bounded or unbounded delays

Cited by in corpus (1)

• Stability Properties of Multi-Order Fractional Differential Systems in 3D