Asymptotic integration of a system of differential equations of third order (Q2711765)

The author deals with a 3-dimensional system of the form \(\varepsilon \dot x=A(t)x,\) where \(\varepsilon \) is a small parameter and the elements of the matrix \(A(t)=\{a_{ij}(t)\}_{i,j=1}^{3}\) are infinitely differentiable functions on \([0,L]\). The case is studied where the matrix \(A(t)\) has eigenvalue \(\lambda_0(t)\) with elementary divisor of multiplicity three and thus the Jordan normal form of \(A(t)\) consists of single 3-dimensional Jordan cell \(J_3(t)\).NEWLINENEWLINENEWLINELet \(V(t)\) be a matrix such that \(V^{-1}(t)A(t)V(t)=J_3(t)\). Then the change of variables \(x=V(t)y\) transforms the system under consideration to the following one: \(\varepsilon \dot y=D(t,\varepsilon)y\equiv(J_3(t)+\varepsilon V^{-1}(t)V(t))y\). The author constructs a formal solution to this system in the form NEWLINE\[NEWLINEy(t,\varepsilon)=U(t,\varepsilon,\varepsilon)\exp\left({1\over \varepsilon }\int_{0}^{t}\lambda(\tau,\varepsilon,\varepsilon) d\tau \right),NEWLINE\]NEWLINE with NEWLINE\[NEWLINEU(t,\varepsilon,\varepsilon)=\sum_{s =0}^{\infty }\varepsilon ^{2s/3}U_s(t,\varepsilon),\quad \lambda(t,\varepsilon, \varepsilon)=\rho_1(t,\varepsilon)+\sum_{s=1}^{\infty }\varepsilon ^{(2s+1)/ 3}\lambda_s(t,\varepsilon),NEWLINE\]NEWLINE \(\rho_1(t,\varepsilon)=\lambda_0(t)+O(\varepsilon ^{1/3})\) is an eigenvalue of \(D(t,\varepsilon)\), and \(U_s(t,\varepsilon), \lambda_s(t,\varepsilon)\) are analytical in \(\varepsilon \). The asymptotic property of such a solution is established under the assumption that \(\text{Re}(\rho_1(t,\varepsilon))\leq 0\).NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].