Asymptotic solutions of a system of differential equations with an analytic nonlinearity (Q881233)

In this paper a method is proposed which provides an asymptotic solution for the singularly perturbed Cauchy problem with analytic nonlinearity for the case when the characteristic equation has one multiple root with one multiple elementary divisor. The system under consideration is of the form: \[ \varepsilon^h{\frac{dx}{dt}}=A(t,\varepsilon)x+f(t,\varepsilon,x) \] with initial condition \[ x(0,\varepsilon)=x_0, \] where \(\varepsilon>0\) is a small parameter, \(h\in{\mathbb{N}}\); \(f(t,\varepsilon,x),x(t,\varepsilon),x_0\) are \(n\)-measurable vectors; \(A(t,\varepsilon)\) is an \(n\times n\)-matrix. Under suitable assumptions concerning the analyticity of the nonlinearities \(f(t,\varepsilon,x)\) and of the matrix \(A(t,\varepsilon)\), expressions for the formal solution of the Cauchy problem are derived.