The book presents the recently introduced and already widely cited semi-discretization method for the stability analysis of delayed dynamical systems with parametric excitation. Delay-differential equations often come up in different fields of engineering, such as feedback control systems, machine tool vibrations, and balancing/stabilization with reflex delay. The behavior of such systems is often counter-intuitive and closed form analytical formulas can rarely be given even for the linear stability conditions. The same holds for parametrically excited systems. If parametric excitation is coupled with the delay effect, then the governing equation is a delay-differential equation with time-periodic coefficients, and the stability properties are even more intriguing. The semi-discretization method is a simple but efficient method that is based on the discretization with respect to the delayed term and the periodic coefficients only. This discretization results in a system of ordinary differential equations that can be solved using standard techniques, which are part of basic engineering curriculums. The method can effectively be used to construct stability charts in the space of system parameters. These charts provide a useful tool for engineers, since they present an overview on the effects of system parameters on the local dynamics of the system. The book presents the application of the method to different engineering problems, such as dynamics of turning and milling processes with constant and with varying spindle speeds, stick balancing with reflex delay, force control processes in the presence of feedback delay, and stabilization using time-periodic control gains.The book is designed for graduate and PhD students as well as researchers working in the field of delayed dynamical systems with application to mechanical, electrical and chemical engineering, control theory, biomechanics, population dynamics, neuro-physiology, and climate research.

From the reviews:“The goal of this book is to present the semi-discretization method for the stability analysis of linear periodic delay systems. … The book is very well written and organized. … is very interesting from both theoretical and applied points of view. … a good list of references to the relevant literature is included. It is a good source to introduce Ph.D. students to the subject, and also a useful reference book for researchers in many areas where time-delay systems play an important role … .” (Eduardo Liz, Mathematical Reviews, Issue 2012 f)“This monograph presents a numerical technique, called the semi-discretization method, for the stability analysis of linear time-periodic and time-delay systems. Partial differential equations describing the systems are discretized only along the spatial coordinates while the time coordinate is unchanged. The book is based mostly on the author’s research work over the last 10 years. … The book is addressed to graduate and PhD students as well to scientists working in the field of mechanical, electrical and chemical engineering, control theory biomechanics etc.” (Tadeusz Kaczorek, Zentralblatt MATH, Vol. 1245, 2012)

Stability diagrams for some basic delay systems appearing in engineering problems are summarized in the introduction The semi-discretization method is presented in a simple and clear way with examples Different approaches for the application of the semi-discretization method is presented considering the rate convergence and numerical efficiency Basic engineering models described by time-periodic and time-delayed systems are presented Includes supplementary material: sn.pub/extras