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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 2701 - 2727, Volume 32, Issue 8, August 2012      doi:10.3934/dcds.2012.32.2701

 
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A. R. Humphries - Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada (email)
O. A. DeMasi - Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada (email)
F. M. G. Magpantay - Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada (email)
F. Upham - Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada (email)

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.

Keywords:  Delay differential equations, state-dependent delays, Hopf bifurcations, periodic solutions, bistability, tori, period-doubling.
Mathematics Subject Classification:  Primary: 34K18, 34K13, 34K28.

Received: June 2011;      Revised: October 2011;      Available Online: March 2012.

 References