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

We study the Volterra integro-differential equation in RnRn equation(⋆ ) View the MathML sourcedxdt=X(t,x,∫0tK(t,s)g(x(s)ds)). Turn MathJax on We establish a connection between system (⋆) with a kernel of the form equation(⋆ ⋆) View the MathML sourceK(t,s)=∑j=1∞CjFj(t)Gj(s) Turn MathJax on 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.