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Numerical stability of nonlinear delay differential equations of neutral type. (English) Zbl 0980.65077
The paper is devoted to the stability analysis of both the true solutions and the numerical approximations for systems of nonlinear neutral delay differential equations (NDDEs) of the form $$y'(t)=F(t,y(t),G(t,y(t-\tau (t)),y'(t-\tau (t))))$$, $$t\geq t_0$$, $$y(t)=g(t)$$, $$t\leq t_0$$, where $$F$$ and $$G$$ are complex continuous vector functions, $$g(t)$$ is a $$C^1$$-continuous complex-valued function and $$\tau (t)$$ is a continuous delay function such that $$\tau (t)\geq \tau_0 (t)>0$$ and $$\alpha (t)=t-\tau (t)$$ is increasing $$\forall t\geq t_0$$. The main result is an extension of the results recently obtained by the authors [BIT 39, No. 1, 1-24 (1999; Zbl 0917.65071)] for the linear case. This is accomplished by considering a suitable reformulation of the given system, which transforms it into a nonlinear differential system coupled with an algebraic functional recursion. Numerical processes preserving the qualitative properties of the solutions are also investigated.
Reviewer: A.Dishliev (Sofia)

##### MSC:
 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L05 Numerical methods for initial value problems 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010) 34K40 Neutral functional-differential equations
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