Numerical Oscillations Analysis for Nonlinear Delay Differential Equations in Physiological Control Systems

Abstract

This paper deals with the oscillations of numerical solutions for the nonlinear delay differential equations in physiological control systems. The exponential $\theta$-method is applied to $p^{\prime }(t)={\beta }_0{\omega }^{\mu }p(t-\tau )/{\mathrm{(\omega }}^{\mu }+p^{\mu }(t-\tau ))-\gamma p(t)$ and it is shown that the exponential $\theta$-method has the same order of convergence as that of the classical $\theta$-method. Several conditions under which the numerical solutions oscillate are derived. Moreover, it is proven that every nonoscillatory numerical solution tends to positive equilibrium of the continuous system. Finally, the main results are illustrated with numerical examples.