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## Über dieses Buch

This book presents the recently introduced and already widely referred semi-discretization method for the stability analysis of delayed dynamical systems. Delay differential equations often come up in different fields of engineering, like feedback control systems, machine tool vibrations, 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. 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. The method can effectively be used to construct stability diagrams in the space of system parameters.

## Inhaltsverzeichnis

### Chapter 1. Introducing Delay in Linear Time-Periodic Systems

Abstract
Dynamical systems have been described with differential equations since the appearance of the differential calculus; Newton’s second law could be considered one of the first examples. A differential equation can serve as a model for how the rate of change of state depends on the present state of a system. However, the rate of change of state may depend on past states, too. It has been known for a long time that several problems can be described by models including past effects.
Tamás Insperger, Gábor Stépán

### Chapter 2. Stability Charts for Fundamental Delay-Differential Equations

Abstract
Simple scalar equations play an important role in understanding the main features of DDEs and the function of stability charts. Stability charts are diagrams constructed in the plane of two (or more) parameters of the system showing the stable and unstable domains or the numbers of unstable characteristic exponents/multipliers. In this chapter, some basic scalar equations are considered for which the stability charts can be constructed in closed form by a straightforward analysis of the characteristic equation.
Tamás Insperger, Gábor Stépán

### Chapter 3. Semi-discretization

Abstract
Stability analysis of DDEs with time-periodic coefficients requires the analysis of the eigenvalues of the infinite-dimensional monodromy operator. Generally, stability conditions cannot be given as closed-form functions of the system parameters (the delayed Mathieu equation in Section 2.4 is an exception), but numerical approximations can be used to derive stability properties. Semi-discretization is an efficient numerical method that provides a finite-dimensional matrix approximation of the infinite-dimensional monodromy matrix. This chapter presents the main concept of the semi-discretization method for general linear time-periodic DDEs following [123, 73, 126, 101, 133].
Tamás Insperger, Gábor Stépán

### Chapter 4. Newtonian Examples

Abstract
According to Newton’s second law, the acceleration of a particle is proportional to the net force acting on it. In cases, in which the net force depends on the actual position and on the actual velocity of the particle, the system is described by a second-order ODE (due to the velocity and the acceleration being the first and the second derivatives of the position, respectively). In cases, in which the net force depends on both the actual and some delayed values of the particle’s position and velocity, the system is described by a second-order DDE. Second-order systems are therefore often used in engineering to model dynamic behavior. In this chapter, some special second-order scalar DDEs are considered and analyzed by the semidiscretization method.
Tamás Insperger, Gábor Stépán

### Chapter 5. Engineering Applications

Abstract
Time-delay systems appear in several engineering problems, such as wheel shimmy [230, 257, 273, 274], car-following traffic models [213, 214, 215], feedback stabilization problems [194, 284, 252, 113], and machine tool chatter [280, 281, 255, 5]. In this chapter, engineering models are considered in which the time delay is coupled with parametric forcing. The first example to be discussed is the turning process with varying spindle speed, which is described by a DDE with time-periodic delay.
Tamás Insperger, Gábor Stépán

### Backmatter

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