Geodesic
Reducibility of linear systems with aftereffect
Trudy Instituta matematiki i mehaniki, Dynamical systems and control problems, Tome 11 (2005) no. 1, pp. 53-64

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It is shown that a linear system with aftereffect on each finite-dimensional subspace of solutions with finite Lyapunov indices is asymptotically similar under natural assumptions to a system of ordinary differential equations. A system with the right-hand side recurrent with respect to time is investigated in detail and a family of systems with aftereffect, whose space of solutions with finite Lyapunov indices is finite-dimensional, is constructed. The research is based on the conception of N. N. Krasovskii, according to which to every system with aftereffect there corresponds some dynamical system with infinite-dimensional phase space and a flow on it generated by solutions of the original system with aftereffect. 
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T. S. Bykova; E. L. Tonkov. Reducibility of linear systems with aftereffect. Trudy Instituta matematiki i mehaniki, Dynamical systems and control problems, Tome 11 (2005) no. 1, pp. 53-64. http://geodesic.mathdoc.fr/item/TIMM_2005_11_1_a5/