  We present the results of linear stability of a damped coplanar double
pendulum and its non-linear motion, when the point of suspension is vibrated
sinusoidally in the vertical direction with amplitude $a$ and frequency $\omega
$. A double pendulum has two pairs of Floquet multipliers, which have been
calculated for various driving parameters. We have considered the stability of
a double pendulum when it is in any of its possible stationary states: (i) both
pendulums are either vertically downward or upward and (ii) one pendulum is
downward, and the other is upward. The damping is considered to be
velocity-dependent, and the driving frequency is taken in a wide range. A
double pendulum excited from its stable state shows both periodic and chaotic
motion. The periodic motion about its pivot may be either oscillatory or
rotational. The periodic swings of a driven double pendulum may be either
harmonic or subharmonic for lower values of $a$. The limit cycles corresponding
to the normal mode oscillations of a double pendulum of two equal masses are
squeezed into a line in its configuration space. For unequal masses, the
pendulum shows multi-period swings for smaller values of $a$ and damping, while
chaotic swings or rotational motion at relatively higher values of $a$. The
parametric driving may lead to stabilization of a partially or fully inverted
double pendulum.
