Asymptotic Expansions of Eigenvalues of Periodic and Antiperiodic Boundary Problems for Singularly Perturbed Second Order Diﬀerential Equation with Turning Points - S. A. Kashchenko (Моделирование и анализ информационных систем, 2016)
For a second order equation with a small factor at the highest derivative the asymptotic behavior of all eigenvalues of periodic and antiperiodic problems is studied. The main assumption is that the coeﬃcient at the ﬁrst derivative in the equation is the sign of the variable so that turning points exist an algorithm for computing all coeﬃcients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coeﬃcients are deﬁned by coeﬃcient values of the original equation only in a neighborhood of turning points. Asymptotics for the length of Lyapunov zones of stability and instability was obtained. In particular, the problem of stability of solutions of second order equations with periodic coeﬃcients and small parameter at the highest derivative was solved