The texture of phase-space and bifurcation diagrams of two-dimensional discrete maps describing a lattice of interacting oscillators, confined in bistable potentials with deformable double-well shapes, are examined. There are considered two bistable potentials that belong to a family of hyperbolic nonlinear on-site potentials whose double-well shapes can be tuned differently: one has a variable barrier height and the other has variable minima positions. However, the two hyperbolic double-well potentials reduce to the well-known canonical ϕ 4 field, familiar in the studies of structural phase transitions in centro-symmetric crystals and 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 ϕ 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 associated with the two discrete parametric double-well maps.
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