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

Abstract

We study the Volterra integro-differential equation in R n (⋆ ) d x d t = X ( t , x , ∫ 0 t K ( t , s ) g ( x ( s ) d s ) ) . We establish a connection between system (⋆) with a kernel of the form (⋆ ⋆) K ( t , s ) = ∑ 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 (⋆) 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.