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In this work, we study certain topological propertms of a class of twodimentional ordinary differential systems x = A(t)x (-∞ < t < ∞) , (1) in which A = A(t) is a real-valued continuous matrix satisfying, for some positíve ω , the following conditions: (i) A(t + ω) = A(t), (∞ < t < ∞), (ii) ∫^ω₀ tr A(t)dt = 0). In chapter 0 ; we present certain results concerning the general theory of systems x = A(t)x where A = A(t) is an n x n periodic continuous matrix. In Chapter 1, we pose the above mentioned problem . We deal with examples in which the fundamental matrix U = U(t) is defined by U(0) = the identity matrix. In this case, the Jordan canonical form of U(ω) is given by one of the following cases: (1) (1 0) (0 1) (2) (1 1 ) (0 1) (3) (-1 0) (0 -1) (4) (-1 1) (0 -1) (5) (a +ib 0) (0 a-ib) (b ≠ 0 a² + b² =1) (6) (c 0) (0 c^-1) (0 < |c| < 1). In Chapter II, we characterize certain sets of matrices in which the behavior of the solutions of (1) is strongly related to its characteristic multipliers, that is, to the eigenvalues of U(ω) . By using results proved in this chapter, we obtain certain topological properties of the above mentioned sets of matrices.