Asymptotic reduction of a singularly perturbed system of differential equations with regular singularity to the diagonal form (Q2740290)

The authors study the system of the form \(\varepsilon t\dot x=A(t)x\) where \(\varepsilon >0\) is a small parameter, \(A(t)\in C^\infty([0,L] \mapsto \mathbb R^{n\times n})\). The roots \(\omega_j(t)\), \(j=1,\ldots,n\), of the characteristic equation \(\det(A(t)-\omega \text{Id})=0\), are assumed to be simple for all \(t\in [0,L]\). From this it follows that \(A(t)\) is similar to a diagonal matrix \(W(t)\). Let NEWLINE\[NEWLINE\Lambda_0(t):=(W(t)-W(0))/t:=\text{diag}(\lambda _{01}(t),\ldots,\lambda _{0n}(t)).NEWLINE\]NEWLINE The next assumption is that \(\omega_j(0)+t\text{Re} \lambda _{0j}(t)\leq 0\) and \(\text{Re} \lambda _{0j}<0\) for all \(t\in (0,L]\), \(j=1,\ldots,n\). Under these restrictions it is shown that the solution \(x(t,\varepsilon)\) satisfying the initial condition \(x(\varepsilon,\varepsilon)=x_0\) can be represented in the form NEWLINE\[NEWLINEx(t,\varepsilon)= U_m(t,\varepsilon)t^{W(0)/\varepsilon }\exp\left(\int_{0}^{t}\Lambda_m(s,\varepsilon) ds\right)U_m^{-1}(\varepsilon,\varepsilon)x_0+O(\varepsilon^m)NEWLINE\]NEWLINE with \(U_m(t,\varepsilon)=\sum_{k=0}^{m}\varepsilon^kU_k(t)\), \(U_k(t)\in C^\infty([0,L] \mapsto \mathbb R^{n\times n})\).NEWLINENEWLINENEWLINESee also the papers of \textit{M. I. Shkil'} and \textit{G. V. Zavizion} [Ukr. Math. Zh. J. 51, No. 12, 1917-1928 (1999); translation from Ukr. Mat. Zh. 51, No. 12, 1694-1703 (1999; Zbl 0948.34032)], \textit{O. I. Kocherga} and \textit{V. P. Yakovets'} [Dopov. Akad. Nauk Ukr. 1999, No. 5, 34-39 (1998; Zbl 0937.34042]), and \textit{V. N. Bobochko} [Ukr. Mat. Zh. 50, No. 7, 867-876 (1998; Zbl 0914.34058)].