Dissipativity of θ-Methods for Nonlinear Delay Differential Equations

This paper concerns dissipativity of one-leg theta-methods and linear theta-methods for nonlinear delay differential equations (DDEs). Firstly, we obtain the absorbing set generated by the numerical methods and then prove that the methods can inherit the dissipativity of underlying system. It is shown that the radius of the absorbing set generated by the discrete system goes to that generated by the continuous system as the step size goes to zero. The estimate of the radius is sharp in this sense. Secondly, because the model considered is a very broad class of differential equations with or without delays, the main results obtained provide a unified treatment for ordinary differential equations (ODEs) and DDEs with constant delays and variable delays (including bounded and unbounded variable delays). In particular, the results are also new even in the case of ODEs. Finally, numerical experiments are given to support our theoretical results.