System of linear differential equations with degeneracy at a point (Q2518034)

Given the differential system \[ \varepsilon^h B(t)dx/dt=A(t,\varepsilon)x, \] where \(\varepsilon\in(0,\varepsilon_0)\) is a small parameter, \(h\geq1\) is an integer, \(A,B\) are \(n\times n\)-matrices. The authors discuss power-series solutions when the matrix \(A\) has certain power-series developments. The distinctive feature of the problem is that the coefficient \(B\) is singular at \(t=0\). One of the results reads as: Theorem 1. Assume: \(A(t,\varepsilon)=\sum_{s=0}^{\infty}\varepsilon^sA_s(t)\), \(A_s,B\in C^{\infty}\), \(A_s(0), B(0)\) have only real entries, \(\text{det }B(0)=0\), \(B(t)\) is non-singular for all \(t>0\), \((d/dt)(\det B)(0)\neq0\), \(\text{rank }B(0)=n-1\), the roots of the characteristic equation \(| A_0(t)-w(t)B(t)|=0\) are simple for \(t>0\) and \(n-1\) of them have finite limits as \(t\rightarrow0+\). Then the system has \(n-1\) solutions of the form \(x_i(t,\varepsilon)=u_i(t,\varepsilon)\exp(\varepsilon^{-h}\int_{0}^{t}\lambda_i(s,\varepsilon)ds)\) and one solution of the form \(x_{n}(t,\varepsilon)=t^{a^2(\varepsilon)/\varepsilon^h}u_{n}(t,\varepsilon)\exp(\varepsilon^{-h}\int_{0}^{t}\lambda_n(s,\varepsilon)ds)\), where the functions \(u_j,\lambda_j,a\) have also certain power-series expansions w.r.t. \(\varepsilon\). The work is based on techniques due to the research team of Professor A.M. Samoilenko from the Ukraine Academy of Science.