Abstract
Abstract We show that, when the delay is an integer multiple of the forcing period, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word series techniques to obtain high-order averaged equations for differential equations without delay.
Highlights
We show that, when the delay is an integer multiple of the forcing period, it is possible to obtain high-order averaged systems of periodically forced systems of delay differential equations with constant delay
Assume that (8)–(9) is to be studied in a bounded interval of the form 0 ≤ t ≤ Lτ for a suitable integer L > 0. (This hypothesis is made at this stage for mathematical convenience to avoid systems of infinitely many differential equations; the averaged systems that we will find will be valid for all t ≥ 0.) We introduce the functions
We have showed that, when the delay is a multiple of the forcing period, it is possible to extend to periodically forced, constant delay problems, the word series approach to the systematic derivation of high-order averaged systems
Summary
We show that, when the delay is an integer multiple of the forcing period, it is possible to obtain high-order averaged systems of periodically forced systems of delay differential equations with constant delay. The well-known theory of averaging [14, 20] studies the reduction, by means of time-dependent changes of variables, of (nonautonomous) forced systems of differential equations to autonomous time-independent systems (averaged systems). In this paper we show that, for periodically forced differential systems with constant delay, it is possible to obtain high-order averaged systems by an application of the word series results in [6,7,8,9,10, 15, 18, 19].
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