It has been observed in laboratory experiments that when nonlinear
dispersive waves are forced periodically from one end of undisturbed
stretch of the medium of propagation, the signal eventually becomes
temporally periodic at each spatial point. The observation has been
confirmed mathematically in the context of the damped Kortewg-de
Vries (KdV) equation and the damped Benjamin-Bona-Mahony (BBM)
equation. In this paper we intend to show the same results hold for
the pure KdV equation (without the damping terms) posed on a bounded
domain. Consideration is given to the initial-boundary-value problem
$ u_t +u_x +$uux$ +$uxxx$=0, \quad u(x,0)
=\phi (x), \qquad $ 0 < x < 1, t > 0, (*)
$ u(0,t) =h (t), \qquad
u(1,t)=0, \qquad u_x (1,t) =0, \quad t>0. $
It is shown that if the boundary forcing $h$ is periodic with
small amplitude, then the small amplitude solution $u$ of (*)
becomes eventually time-periodic. Viewing (*) (without the initial
condition ) as an infinite-dimensional dynamical system in the
Hilbert space $L^2(0,1)$, we also demonstrate that for a
given periodic boundary forcing with small amplitude, the system
(*) admits a (locally) unique limit cycle, or forced
oscillation, which is locally exponentially stable. A list of open
problems are included for the interested readers to conduct further
investigations.
{{journal.titleEn}} 
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