The Buffer Phenomenon in One-Dimensional Piecewise Linear Mapping in Radiophysics

We study the problem of attractors of the two-dimensional mapping
$$ (u,v)\to(v,-(1-\mu)u-F(v)),\qquad F(v)=\begin{cases} \hphantom{-}q_1&\text{for }v>0, \\ \hphantom{-}0&\text{for }v=0, \\ -q_2&\text{for }v<0, \end{cases} $$
where $0<\mu\ll1$ and $q_1,q_2>0$. This mapping is the mathematical model of a self-excited oscillator with relay amplifier and a part of the long transmission line without distortions in the feedback circuit. We prove that, in the system under study, there coexist stable cycles with arbitrarily large periods as the parameter $\mu$ decreases properly. We also show that the total number of these cycles increases without bound as $\mu\to0$, i.e., the buffer phenomenon is realized.