Non-linear Volterra IDE, infinite systems and normal forms of ODE

We study the Volterra integro-differential equation in Rⁿ({star operator} )frac(d x, d t) = X (t, x, ∫₀^t K (t, s) g (x (s) d s)) . We establish a connection between system ({star operator}) with a kernel of the form ({star operator} {star operator})K (t, s) = underover(∑, j = 1, ∞) C_j F_j (t) G_j (s) and a countable system of ordinary differential equations. Such a reduction allows use of results obtained earlier for the countable systems of differential equations in the study of integro-differential equations. In this paper we discuss problems related to the stability of systems ({star operator}) and ({star operator}{star operator}), as well as applications of the method of normal forms to solving some problems in the qualitative theory of integro-differential equations. In particular, it can be employed for the study of critical cases of stability and bifurcation problems in integro-differential equations.