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
