Numerical Dissipativity Of Runge-Kutta Methods For Delay Differential Equations

Delay differential dynamical systems have appeared in areas of applications as diverse as neural networks, optics, biology, and automatic control theory. There are many significant achievements on basic theory of solution, stability, periodic solution and bifurcation of delay differential dynamical systems.Numerical simulation is one of the most important methods to understand dynamical properties of delay differential dynamical systems. A numerical method which is convergent in a finite interval does not yield the same asymptotic behavior as the underlying differential system. Therefore, it is very important to understand the dynamics of numerical methods for solving delay differential equations.The purpose of this thesis is to study dissipativity property of Runge-Kutta methods solving delay differential equations. It containsâ‘  Numerical dissipativity of Runge-Kutta methods for linear delay differential equations with variable coefficients. A sufficient condition is given to ensure that a class of linear delay differential equations is dissipative. We then apply Runge-Kutta methods combined with Lagrange interpolation procedure to these dissipative equations and investigate the numerical dissipativity of numerical solution. Also, we give several examples to illustrate our theoretical result.â‘¡ Numerical dissipativity of Runge-Kutta methods for nonlinear nonautonomous delay differential equations with variable coefficients. We study the numerical dissipativity of (k, l)-algebraically stable Runge-Kutta methods and give some numerical examples to demonstrate our theory.