  Spatially confined rigid membranes reorganize their morphology in response to
the imposed constraints. A crumpled elastic sheet presents a complex pattern of
random folds focusing the deformation energy while compressing a membrane
resting on a soft foundation creates a regular pattern of sinusoidal wrinkles
with a broad distribution of energy. Here, we study the energy distribution for
highly confined membranes and show the emergence of a new morphological
instability triggered by a period-doubling bifurcation. A periodic
self-organized focalization of the deformation energy is observed provided an
up-down symmetry breaking, induced by the intrinsic nonlinearity of the
elasticity equations, occurs. The physical model, exhibiting an analogy with
parametric resonance in nonlinear oscillator, is a new theoretical toolkit to
understand the morphology of various confined systems, such as coated materials
or living tissues, e.g., wrinkled skin, internal structure of lungs, internal
elastica of an artery, brain convolutions or formation of fingerprints.
Moreover, it opens the way to new kind of microfabrication design of
multiperiodic or chaotic (aperiodic) surface topography via self-organization.
