The Nonlinear Stability And Dissipativity Of Linear Multistep Methods Of Delay Differential Equations

Delay differential equations (DDEs) are widely used in the fields of Ecology , Environmental Science , Economics, Electrical power Engineering ,Automation and so on, so it turns very essential to make research on the numerical solution of DDEs. In recent years, this research has attracted many scholars'attention. In this paper, we focus on the nonlinear stability and dissipativity properties of linear multistepmethods for initial value problems of DDEs with a variable lag and for nonlinear neutral DDEs.In the first chapter, we recall some examples of DDEs and survey the development of the stability for theoretical and numerical solutions of DDEs varied from linear scalar model equation, DDEs with constant delay, nonlinear DDEs with a variable lag to neutral DDEs. Later, we review the research results on the dissipativity of neutral DDEs . Then, some new problems are introduced.When considering the nonlinear stability of DDEs with a variable lag ,we select a special class of linear multistep methods-linearθ? methods,and we obtain a series of results on nonlinear stability and asymptotic stability under the assumption that the delay satisfies Lipschitz conditions with the least Lipschitz constant less than one; About the dissipativity ,we discuss DDEs with a bounded lag under some certain conditions in scalar product. The numerical dissipativity of linear multistep methods is studied.In the research of dissipativity of nonlinear neutral DDEs ,we first discuss the dissipativity of the analytical silution. Then, a class of linear multistep methods are applied and some results on numerical dissipativity are obtained.Finally, we make some numerical experiments about the stability of linearθ? methods of nonlinear DDEs with a variable delay and the dissipativity for nonlinear neutral DDEs , which confirm our theoretical results.