Günter Radons (Hrsg.)
Delay Differential Equations
Retardierte Differentialgleichungen, Mehrschrittverfahren, Runge-Kutta-Methode mit Dense Output, Mackey-Glass-Gleichung, Lyapunov-Spektren, Fluktuierende Verzögerungen,Kolmogorov-Sinai-Entropie, Gaußisch korrelierte Zufallsfunktionen
122 Seiten, A5, Broschur
Verlag Wissenschaftliche Scripten
Within the research seminar during the winter semester 2006/2007 different, closely related topics have been investigated by the students and discussed in groups under guidance of Prof. Dr. G. Radons. The attained knowledge is presented in a compact form and provides a quick introduction into the topics. Appropriate topic related bibliography offers access to further deeping of knowledge. The focus was on delay differential equations. The category of these differential equations with a retarded argument of time requires an adjustment of the numerical methods, which are presented and illustrated. For this, linear multistep methods and a Runge-Kutta method with dense output are deduced. An overview of available programming libraries illustrates the implemented methods as well as their fi eld of application.For the Mackey-Glass Equation the chaotic behaviorof such delayed differential equationsin dependenceonthe parametersis outlined and the Lyapunov exponents are calculated. In another chapter a fl uctuation o f the delay is introduced and the behavior aswellasthe attractors of a simple map are analyzed. By using discretization methods, differential equations with time delay are investigated analytically and the spectrum of Lyapunov exponents is determined based on the transition matrices. Furthermore, the connection between the Lyapunov exponents and the Kolmogorov-Sinai entropy as well as the determination of the entropies of dynamical systems are explained. An additional chapter deals with different ways of generating random signals with Gaussian autocorrelation which may be used to simulate fl uctuating delay times.