Time Lags in Biological Models pp 39-49 | Cite as

# Periodic Solutions

Chapter

## Abstract

In the following chapters I shall largely be concerned with the question of periodic behaviour brought about by lags. For distributed lags I shall rely on results obtained, either analytically or numerically, on periodic solutions of the linear chain equations. It therefore becomes necessary to look again at the relationship between the solutions of these equations and the solutions of the related integro-differential equation. So far as analytical results are concerned, and for the bulk of the numerical results also, it suffices to look at a single integro-differential equation. In the notation of Chapter 1, , where z is defined by , and the memory function G(u) is defined by .

$$ \frac{{dx}}{{dt}} = f\left( {x,z} \right) $$

(1)

$$ z = \int\limits_{ - \infty }^t {x\left( \tau \right)G\left( {t - \tau } \right)d\tau } $$

(2)

$$ G_a^P\left( u \right) = \frac{{{a^{p + 1}}{u^p}}}{{p!}}\exp \left( { - au} \right) $$

(3)

### Keywords

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© Springer-Verlag Berlin Heidelberg 1978