Volume 26, pp. 474-483, 2007.

Theory and numerics for multi-term periodic delay differential equations: small solutions and their detection

Neville J. Ford and Patricia M. Lumb

Abstract

In this paper we consider scalar linear periodic delay differential equations of the form $$x^{\prime }(t)=\sum _{j=0}^m{b}_j(t)x(t-jw),x(t)=\varphi (t)\text{ for }t\in [0,mw),t\ge mw(‡)$$ where $b_j$, $j=0,...,m$ are continuous periodic functions with period $w$. We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step-size is applied to approximate the solution to ($‡$) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

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Key words

delay differential equations, small solutions, super-exponential solutions, numerical methods

AMS subject classifications

34K28, 65P99, 37N30