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The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval
Dynamics of oscillations in a multidimensional delay differential system
1.  School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv 
$\dot x_i(t)=F_i(x_1(t),\ldots,x_n(t),t)$ sign $x_i(th_i),\quad i=1,\ldots,n,$
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. Nonzero delays do not allow such a stabilization, but cause oscillations around zero level in any variable. We introduce integralvalued 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.
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