Analytic And Numerical Dissipativity Of Delay Functional Differential Equations With Bounded Lag

Delay functional differential dynamical systems have appeared in many disciplines of natural sciences, engineering technology and social sciences. They have extensive applications in various fields such as nuclear physics, circuit signal system, ecology, environmental sciences, power system, neural networks and automatic control theory. Many real-life phenomena can be modeled by delay functional differential dynamical systems whose analytic solution can seldom obtained in general. So numerical simulation has become one of the most important methods to understand dynamical properties of the delay functional differential dynamical system. Effective and reliable computational methods must mimic the dynamical properties of the delay functional differential dynamical system.The purpose of this thesis is to study dissipativity of delay functional differential dynamical systems and their corresponding numerical dissipativity of one-leg θ-method.We first establish a sufficient condition under which a class of delay functional differential dynamical systems is dissipative. Then we construct a one-leg θ-method to solve such dissipative dynamical systems and prove that the one-leg θ-method is dissipative if θ ∈ [1/2, 1].We consider another class of delay functional differential dynamical systems and establish a sufficient condition under which such delay functional differential dynamical systems are dissipative. Then we use the same one-leg 0-method to solve such dissipative dynamical systems and prove that it is dissipative if θ = 1.Some numerical examples are given to illustrate the theoretical analysis.