Asymptotics of the Cauchy problem solution in the case of instability of a stationary point in the plane of "rapid motions"

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. 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. Moreover, the singularly perturbed Cauchy problem has an additional singularity, namely, the corresponding unperturbed equation has a non-smooth solution in the investigated extended domain. More exactly, the solution of the corresponding unperturbed equation has poles in the complex plane. Therefore, the Cauchy problem under consideration can be called bisingular in the terminology introduced by Academician A.M. Il'in.
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.
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 bisingularly perturbed Cauchy problem in the case of a change in the 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 negative fractional power with respect to a small parameter, which is characteristic of bisingularly perturbed equations or equations with turning points.
The obtained results can find applications in chemical kinetics, in the study of Ziegler's pendulum, etc.