The buffer property in a non-classical hyperbolic boundary-value problem from radiophysics

A mathematical model of a self-excited $RCL$-oscillator with a segment of a solenoid in the feedback loop is considered, which is the following boundary-value problem:
\begin{gather*} \frac{\partial^2}{\partial t^2} \biggl(u-\varkappa\frac{\partial^2u}{\partial x^2}\biggr) +\varepsilon\frac{\partial}{\partial t} \biggl(u-\varkappa\frac{\partial^2u}{\partial x^2}\biggr) =\frac{\partial^2u}{\partial x^2}\,, \\ \frac{\partial u}{\partial x}\bigg|_{x=1}=0, \qquad u\big|_{x=0}+(1+\varepsilon^2\gamma)u\big|_{x=1}-u^3\big|_{x=1}=0, \end{gather*}
where $0<\varepsilon\ll1$, and $\varkappa$ and $\gamma$ are positive parameters of order 1. For this boundary-value problem with suitably increased $\gamma$ and reduced $\varepsilon$ one proves the existence of an arbitrary prescribed finite number of stable cycles (solutions periodic in $t$).
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