Stability of Linear Multiple‎ Different Order Caputo Fractional System ‎‏

Abstract

In this paper, we introduce a new equivalent system to the higher order Caputo fractional system ‎‎(CFS) . This equivalent system has multiple order Caputo fractional derivatives ‎‎(CFDs). These CFDs are lying between zero and one.‎ As well ‎as, we find the fundamental solution for linear CFS with multiple order CFDs. Also, we introduce new criteria of studying the stability (asymptotic stability) of the ‎linear CFS with multiple order CFDs. These criteria can be applied in three cases: ‎the first, all CFDs is lying between zero and one. The second, all CFDs are lying ‎between one and two. Finally, some of CFDs are lying between zero and one, ‎and the rest of these derivatives are lying between one and two. The criteria are ‎depending on the position of eigenvalues of the matrix system in the complex plane. ‎These criteria are considered as a generalized of the classical criteria which is ‎used to study the stability of linear first ODEs. Also, these criteria are considered ‎as generalized of the criteria which used to study the stability same order CFS in ‎case when all CFDs lying between zero and one, also in case when all CFDs lying ‎between one and two. Several examples are given to show the behavior of the ‎solution near the equilibrium point.‎

Keywords: Caputo fractional derivatives; Linear Caputo fractional system ‎; Fundamental solution Stability analysis.