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March  2004, 11(2&3): 557-576. doi: 10.3934/dcds.2004.11.557

Dynamics of oscillations in a multi-dimensional delay differential system


School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv

Received  December 2002 Revised  March 2004 Published  June 2004

We consider a system of delay differential equations

$\dot x_i(t)=F_i(x_1(t),\ldots,x_n(t),t)-$ sign $x_i(t-h_i),\quad i=1,\ldots,n,$

with positive constant delays $h_1,...,h_n$ and perturbations $F_1,...,F_n$ absolutely bounded by a constant less than 1. This is a model of a negative feedback controller of relay type intended to bring the system to the origin. Non-zero delays do not allow such a stabilization, but cause oscillations around zero level in any variable. We introduce integral-valued relative frequencies of zeroes of the solution components, and show that they always decrease to some limit values. Moreover, for any prescribed limit relative frequencies, there exists at least an $n$-parametric family of solutions realizing these oscillation frequencies. We also find sufficient conditions for the stability of slow oscillations, and show that in this case there exist absolute frequencies of oscillations.

Citation: Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557

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