Asymptotics of the solution of the singularly perturbed Cauchy problem in the case of a change in the stability, when the eigenvalues have poles

In this paper, the Cauchy problem for a normal system of two linear inhomogeneous ordinary differential equations with a small parameter at the derivative is considered. The coefficient matrix of the linear part of the system has complex conjugate eigenvalues. These eigenvalues have poles in the complex plane. The real parts of the complex conjugate eigenvalues in the considered interval change signs from negative to positive ones. A singularly perturbed Cauchy problem is investigated in the case of instability, i.e., when the asymptotic stability condition is violated.
The aim of the research is to construct the principal term of the asymptotic behavior of the Cauchy problem solution when the asymptotic stability condition is violated and to prove that the solution of the singularly perturbed Cauchy problem is asymptotically close to the solution of the limit system on a sufficiently large interval when the asymptotic stability of the stationary point in the plane of “rapid motions” is violated.
In the study, methods of the stationary phase, saddle point, successive approximations, and L.S. Pontryagin's idea — the transition to a complex plane — are applied.
An asymptotic estimate is obtained for the solution of a singularly perturbed Cauchy problem in the case where the asymptotic stability of a stationary point in the plane of “rapid motions” is violated. The principal term of the asymptotic expansion of the solution is constructed. It has a positive power with respect to a small parameter. The asymptotic proximity of the solution of the singularly perturbed Cauchy problem to the solution of the limit system on a sufficiently large interval is proved when the asymptotic stability of the stationary point in the plane of “rapid motions” is violated.
The obtained results can find applications in chemical kinetics, in the study of Ziegler's pendulum, etc.