Annales Univ. Sci. Budapest., Sect. Comp. 47 (2018) 211–226 EXPONENTIAL DICHOTOMY AND THE STABILITY OF LINEAR SYSTEMS Szilvia Cs´ asz´ ar and S´ andor Kov´ acs (Budapest, Hungary) Communicated by Ferenc Schipp (Received March 30, 2018; accepted June 16, 2018) Abstract. This paper examines the relation of the exponential dichotomy and the stability concepts for systems of linear differential equations. We are going to show some relationship between the studied concepts, more precisely we are presenting how the stability of a linear non-autonomous system is investigated with the help of the exponential dichotomy. Fur- thermore we are going to show how the stable and unstable subspace of an exponentially dichotomic system can be specified using the definition of the exponential dichotomy. 1. Introduction The asymptotic behaviour of the solutions and the stability of the equilib- rium points is an important element in the investigation of systems of differen- tial equations. As it is well known the equilibrium point ξ of the autonomous system (1.1) ˙ x = f ◦ x is asymptotically stable, provided the Jacobian f ′ (ξ ) is Hurwitz-stable, where f ∈ C 1 (Ω, R n ) with a domain Ω ⊂ R n and ξ ∈ Ω. Key words and phrases : Exponential dichotomy, stability. 2010 Mathematics Subject Classification : 34D09, 34D20.