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 incommensurate 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 incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate 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. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.