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

We study the Volterra integro-differential equation in RnRnequation(⋆ )dxdt=X(t,x,∫0tK(t,s)g(x(s)ds)). We establish a connection between system (⋆) with a kernel of the form equation(⋆ ⋆)K(t,s)=∑j=1∞CjFj(t)Gj(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 (⋆) and (⋆⋆), 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.