TY - JOUR

T1 - Non-linear Volterra IDE, infinite systems and normal forms of ODE

AU - Goltser, Ya

AU - Litsyn, E.

PY - 2008/3/15

Y1 - 2008/3/15

N2 - We study the Volterra integro-differential equation in Rn({star operator} )frac(d x, d t) = X (t, x, ∫0t 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, ∞) Cj Fj (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 ({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.

AB - We study the Volterra integro-differential equation in Rn({star operator} )frac(d x, d t) = X (t, x, ∫0t 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, ∞) Cj Fj (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 ({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.

UR - http://www.scopus.com/inward/record.url?scp=38349099007&partnerID=8YFLogxK

U2 - 10.1016/j.na.2006.12.036

DO - 10.1016/j.na.2006.12.036

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AN - SCOPUS:38349099007

SN - 0362-546X

VL - 68

SP - 1553

EP - 1569

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 6

ER -