  We are concerned with the existence and multiplicity of nontrivial
time-periodic viscosity solutions to \[ \partial_t w(x,t) + H( x,\partial_x
w(x,t),w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty). \] We find that
there are infinitely many nontrivial time-periodic viscosity solutions with
different periods when $\frac{\partial H}{\partial u}(x,p,u)\leqslant-\delta<0$
by analyzing the asymptotic behavior of the dynamical system $(C(\mathbb{S}
,\mathbb{R}),\{T_t\}_{t\geqslant 0})$, where $\{T_t\}_{t\geqslant 0}$ was
introduced in \cite{WWY1}. Moreover, in view of the convergence of
$T_{t_n}\varphi$, we get the existence of nontrivial periodic points of $T_t$,
where $\varphi$ are initial data satisfying certain properties. This is a
long-time behavior result for the solution to the above equation with initial
data $\varphi$. At last, as an application, we describe to readers a
bifurcation phenomenon for \[ \partial_t w(x,t) + H( x,\partial_x
w(x,t),\lambda w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty), \] when
the sign of the parameter $\lambda$ varies. The structure of the unit circle
$\mathbb{S}$ plays an essential role here. The most important novelty is the
discovery of the nontrivial recurrence of $(C(\mathbb{S}
,\mathbb{R}),\{T_t\}_{t\geqslant 0})$.
