  The bifurcation structure of coupled periodically driven double-well Duffing
oscillators is investigated as a function of the strength of the driving force
$f$ and its frequency $\Omega$. We first examine the stability of the steady
state in linear response, and classify the different types of bifurcation
likely to occur in this model. We then explore the complex behaviour associated
with these bifurcations numerically. Our results show many striking departures
from the behaviour of coupled driven Duffing Oscillators with single
well-potentials, as characterised by Kozlowski et al \cite{k1}. In addition to
the well known routes to chaos already encountered in a one-dimensional Duffing
oscillator, our model exhibits imbricated period-doubling of both types,
symmetry-breaking, sudden chaos and a great abundance of Hopf bifurcations,
many of which occur more than once for a given driving frequency. We explore
the chaotic behaviour of our model using two indicators, namely Lyapunov
exponents and the power spectrum. Poincar\'e cross-sections and phase portraits
are also plotted to show the manifestation of coexisting periodic and chaotic
attractors including the destruction of $T^2$ tori doubling.
