ABSTRACT
The first chapter illustrates stability methods used in the book: a priory estimation method, reduction to a system of delay differential equations of the first order and so-called W-method. All these methods are applied in the chapter to Minorsky equation. This equation was introduced by N. Minorsky in 1962 in the book “Nonlinear Oscillations” where he considered the problem of stabilizing the rolling of a ship by the activated tanks method in which ballast water is pumped from one position to another. A mathematical model, which solves this problem is based on a linear delay differential equation of the second order which is considered in the first chapter of the book. All three methods, which were applied in the chapter to Minorsky equation give different independent stability conditions. These conditions have an explicit form and are given in two variants: the first one does not contain delays, and the second one depends on delays. The second aim of the chapter is to give material for a one or two semester course for graduate students which can be called for example “Introduction to Stability of Linear Non-autonomous Functional Differential Equations”. In this course, you can consider delay differential equations of the first and the second order, and also integro-differential equations and equations with distributed delays. Sample syllabus of this course and a set of exercises also given the end of the chapter.
