Dynamics of a delay differential equation with multiple state-dependent delays

A. R. Humphries; O. A. DeMasi; F. M. G. Magpantay; F. Upham

doi:10.3934/dcds.2012.32.2701

Article Contents

2012, Volume 32, Issue 8: 2701-2727. Doi: 10.3934/dcds.2012.32.2701

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Dynamics of a delay differential equation with multiple state-dependent delays

1.
Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9

Received: May 31, 2011

Revised: September 30, 2011

Published: July 2012

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Abstract

We study the dynamics of a linear scalar delay differential equation $$\epsilon \dot{u}(t)=-\gamma u(t)-\sum_{i=1}^N\kappa_i u(t-a_i-c_iu(t)),$$ which has trivial dynamics with fixed delays ($c_i=0$). We show that if the delays are allowed to be linearly state-dependent ($c_i\ne0$) then very complex dynamics can arise, when there are two or more delays. We present a numerical study of the bifurcation structures that arise in the dynamics, in the non-singularly perturbed case, $\epsilon=1$. We concentrate on the case $N=2$ and $c_1=c_2=c$ and show the existence of bistability of periodic orbits, stable invariant tori, isola of periodic orbits arising as locked orbits on the torus, and period doubling bifurcations.

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Mathematics Subject Classification: Primary: 34K18, 34K13, 34K28.

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