MARKUS-YAMABE CONJECTURE FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS DAVID CHEBAN Abstract. The aim of this paper is the study of the problem of global asymp- totic stability of trivial solutions of non-autonomous dynamical systems (both with continuous and discrete time). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). In particularly, we present some new results for non-autonomous version of Markus-Yamabe con- jecture. 1. Introduction 1.1. Markus–Yamabe conjecture (MYC) [33]. Consider the differential equa- tion (1) u ′ = f (u) and suppose that the Jacobian f ′ (u) of f has only eigenvalues with negative real part for all u. The Markus Yamabe conjecture is that if f (0) = 0, then 0 is a globally asymptotically stable solution for (1). It is easy to prove MYC for n = 1. In the two-dimensional case the affirmative answer to MYC was obtained in the works [15, 17, 16] (see also the references therein). In the work [9] (see also [10, 11] and the references therein) is given a polynomial counterexample to the Markus–Yamabe conjecture. If n> 2 there are also some additional conditions forcing the Markus–Yamabe conjecture. For example if f ′ (u) is negative definite for all u ∈ R n the conjecture was proved in [19, 20] (see also [26, 27, 33]). For triangular systems MYC was proved in [33]. 1.2. The discrete Markus–Yamabe conjecture (DMYC) [12, 41]. Let f be a C 1 mapping from R n into itself such that f (0) = 0 and for all u ∈ R n ,f ′ (u) has all its eigenvalues with modulus less than one. Then 0 is a globally asymptotically stable solution of the difference equation (2) u(n + 1) = f (u(n)). In his book [29] J. P. LaSalle proves the DMYC for n = 1. The discrete Markus– Yamabe conjecture is true only for planar maps (see [12] and also the references therein) and the answer to the question is yes only in the case of planar polynomial Date : September 4, 2013. 1991 Mathematics Subject Classification. 37B25, 37B55, 39A11, 39C10, 39C55. Key words and phrases. Global asymptotic stability; attractor; non-autonomous dynamical systems; Markus-Yamabe problem. 1