  The texture of phase space and bifurcation diagrams of two-dimensional
discrete maps describing a lattice of interacting oscillators, confined in
on-site potentials with deformable double-well shapes, are examined. The two
double-well potentials considered belong to a family proposed by Dikand\'e and
Kofan\'e (A. M. Dikand\'e and T. C. Kofan\'e, Solid State Commun. vol. 89, p.
559, 1994), whose shapes can be tuned distinctively: one has a variable barrier
height and the other has variable minima positions. However the two
parametrized double-well potentials reduce to the $\phi^4$ substrate, familiar
in the studies of structural phase transitions in centro-symmetric crystals or
bistable processes in biophysics. It is shown that although the parametric maps
are area preserving their routes to chaos display different characteristic
features: the first map exhibits a cascade of period-doubling bifurcations with
respect to the potential amplitude, but period-halving bifurcations with
respect to the shape deformability parameter. On the other hand the first
bifurcation of the second map always coincides with the first pitchfork
bifurcation of the $\phi^4$ map. However, an increase of the deformability
parameter shrinks the region between successive period-doubling bifurcations.
The two opposite bifurcation cascades characterizing the first map, and the
shrinkage of regions between successive bifurcation cascades which is
characteristic of the second map, suggest a non-universal character of the
Feigenbaum-number sequences associate with the two discrete parametric
double-well maps.
