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

A. Yu. Kolesov ; N. Kh. Rozov

Matematičeskie zametki, Tome 69 (2001) no. 6, pp. 866-875

Résumé

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.