Decomposition and stability of linear singularly perturbed systems with two small parameters

In the domain W = {(t, epsilon(1), epsilon(2)) : t is an element of R, epsilon(1) > 0, epsilon(2) > 0}, we consider a linear singularly perturbed system with two small parameters. {(x) over dot(0) = A(00)x(0) + A(01)x(1) + A(02)x(2), epsilon(1)(x) over dot(1) = A(10)x(0) + A(11)x(1) + A(12)x(2), epsilon(1)epsilon(2)(x) over dot(2) = A(20)x(0) + A(21)x(1) + A(22)x(2), where x(0) is an element of Rn-0,Rn- x(1) is an element of R-n1, x(2) is an element of R-n2. In this paper, schemes of decomposition and splitting of the system into independent subsystems by using the integral manifolds method of fast and slow variables are investigated. We give the conditions under which the reduction principle is truthful to study the stability of zero solution of the original system.