Study On Stability And Dynamic Response Of Time Delay Systems With Complex Coefficients

Delay differential equations with complex coefficients (Complex-DDEs for short) are widely used in dynamics analysis of biology, physics and nonlinear systems. Particularly, they serve as basic and important mathematical models in rotor systems and optical systems. Compared with real time delay systems, modeling with Complex-DDEs, not only puts the result system in a simpler form, but also enables simplification of computation in dynamic analysis when proper method is adopted. However, most of the current methods for dynamics analysis are designed for Real delay differential equation (Real-DDEs), and only few of those methods can be applied for Complex-DDEs directly. Therefore, in real applications, Complex-DDEs are usually transformed into a set of coupled Real-DDEs, which increases the complexity and computation of analysis as a result of the increasing of system dimensions. For that reason, this paper improves and generalizes the current methods so that the modified method can be used for Complex-DDEs directly. On the other hand, new methods for Complex-DDEs are also presented based on analysis of some particular problems and special systems. The main contents of this paper are as follows:The first charter presents an introduction regarding applications, research overview of Complex-DDEs and some methods which can be applied for dynamics analysis of Complex-DDEs directly. The shortcomings of those methods, as well as the way against the shortcomings are also concluded in this chapter.In the second charter, we begin with an intensive study on the Lambert W function that has been successfully applied for the stability analysis of time-delay systems, where the roots locations of the Lambert W function are obtained. Then by applying the results to the stability analysis of time-delay systems, new criteria that are independent of the Lambert W function are achieved for asymptotical stability and robust stability of some time-delay systems. The new stability criteria have two advantages over the available stability conditions: one is that they are dependent of the elementary functions so that the stability and robust stability can be tested simply even by hand, and the other is that they generalize or improve some current criteria such as Hayes Theorem from real domain to complex domain and improve the robust stability criterion of Shinozaki-Mori. By using the time-delay feedbacks which also have decoupling rules, we study the robust control of an unstable single degree of freedom system with uncertain parameters. The method performs so simply that the control parameters can be obtained easily.The stability switch presented by Lee and Hsu can be applied for Complex-DDEs directly, but one should suffer the complexity of analysis and computation when using it. On the other hand, there are some simple conclusions of stability switch method for Real-DDEs. This motivates us to improve those methods so that they can be both simple and widely used. In this paper, we present new effective stability criteria for linear Complex-DDEs with a single delay, commensurate delays, and delay-dependent parameters respectively. New stability criteria generalize some known results in literature.Most of the current methods in stability analysis can only verify whether a DDE is stable, but they can not tell how fast a DDE drives to its stable solution. In the fourth charter, we improve the Hassard stability criterion such that it can analyze the stability of Complex-DDEs and DDEs of neutral type directly. Then by using the improved Hassard stability criterion, a simple algorithm is presented for calculating the real part of the right most root of the characteristic equation. Moreover, the algorithm is not restricted by the condition that the characteristic equation has no pure root.When a perturbation method is adopted for resonance analysis of a nonlinear oscillator with time delay feedback, an amplitude-frequency equation, which is a Complex-DDE with delay-dependent coefficients, may be obtained. A non-trivial solution of the amplitude-frequency equation corresponds to a non-trivial periodic solution of the original dynamical system with external excitation. Thus, the stability analysis of the periodic solution resulted from resonance can be carried out by the stability analysis of the equilibrium of the delayed amplitude-frequency equation. In the fifth charter, a simple algorithm is presented to analyze the stability of periodic solutions of the nonlinear oscillator, focusing on the calculation of the rightmost characteristic roots. This algorithm, with no approximation in the characteristic equation, gives quite accurate stability results of periodic solutions of time-delay systems, even with very large delay.The last chapter is the summary and outlook of this paper.