Delayed Loss of Stability in Dynamic Bifurcations

In this paper, we investigate the delay in loss of stability in dynamic bifurcations, i.e., 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 asymptotics behavior of the Cauchy problem solution when the asymptotic stability condition is violated. And prove the asymptotic proximity of the solution of the singularly perturbed Cauchy problem when the asymptotic stability of the stationary point in the plane of ‘‘rapid motions’’ is violated to the solution of the limit system on a sufficiently large interval. 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 of a change in 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 is proved when the asymptotic stability of the a stationary point in the plane of ‘‘rapid motions’’ is violated to the solution of the limit system over a sufficiently large interval.