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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, whereas 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 that 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 a nonlinear oscillator, is a new theoretical toolkit to understand the morphology of various confined systems, such as coated …