## 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 $x'(t)=\sum_{j=0}^{m}b_j(t)x(t-jw), x(t)=\phi(t) \mbox{ for } t \in [0,mw), \mbox{ } t\ge mw \quad (\ddagger) \label{ddagger}$ 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 ($\ddagger$) 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

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