Route to chaos in two-dimensional discrete parametric maps with bistable potentials

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.