  We consider a finite number of coupled oscillators as an adaptation of the
Kuramoto model of populations of oscillators. The synchronized solutions are
characterized by an integer $m$, the winding number, and a second integer $l$.
Synchronized solutions of type ($m$, $l=0$) are all stable, and an explicit
perturbative expression for these for large values of the coupling constant $K$
is presented. For low $K$, these solutions appear at certain specific values,
each merging with a solution of type ($m$, $l=1$), both these solutions being
stable for $K$ close to the relevant value. The ($m$, 0) solution continues to
be stable for larger $K$, while the ($m$, 1) solution, on continuation in $K$
becomes unstable and then merges with a new type (with a different $m$ and l)
of unstable solution. The ($m$,0) type solutions for large $K$ are in the
nature of phase waves traveling round the ring. All the stable synchronized
solutions are entrained by an external periodic driving, provided that the
driving frequency is sufficiently close to the frequency of the synchronized
population. A perturbative approach is outlined for the construction of the
entrained solutions. For a given amplitude of driving, there is a certain
maximum detuning between the two up to which the entrained solution persists.
The question of stability of the entrained solution is addressed. The simplest
situation involving three oscillators is investigated in details where the
onset of synchronization is seen to occur through a tangent bifurcation at some
critical value of $K$. Immediately before the appearance of the first
synchronized solution, the system exhibits intermittent chaos with a typical
scaling for the duration of the laminar phase. Under a periodic driving with an
appropriately limited detuning, there occurs entrainment of the chaotic
solution.
