The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem
$$ u_{tt}+\varepsilon u_t+(1+\varepsilon\alpha\cos 2\tau)\sin u =\varepsilon\sigma^2u_{xx}, \qquad u_x|_{x=0}=u_x|_{x=\pi}=0, $$
, where $0<\varepsilon\ll1$, $\tau=(1+\varepsilon\delta)t$, $\alpha,\sigma>0$, and the sign of $\delta$ is arbitrary. It is proved that for an appropriate choice of the external parameters $\alpha$ and $\delta$ and for sufficiently small $\sigma$ the number of exponentially stable solutions $2\pi$-periodic in $\tau$ can be made equal to an arbitrary predefined number.