Fractional-delay control of a weakly nonlinear locally resonant metastructure: Passive gaps,attenuation persistence,and synchronous optical-edge response

Abstract

We study a weakly nonlinear locally resonant metastructure in which each host cell is coupled to a local resonator and the attachment channel is augmented by fractional-order memory and delayed feedback. For the passive infinite lattice, we derive the Bloch dispersion relation in closed form and prove the exact open locally resonant band gap. For the passive finite ring, we diagonalize the linear dynamics by the discrete Fourier transform, identify the exact discrete acoustic and optical spectra, and prove the corresponding discrete-gap theorem. For the controlled linear lattice, interpreted strictly in the steady-state harmonic sense, we prove persistence of every compact real-frequency interval strictly inside the passive gap for the harmonic compatibility determinant under sufficiently small fractional-delay control, and we derive a local first-order complex edge-shift formula for simple passive harmonic edge zeros. For the nonlinear forced problem, we identify the exact synchronous optical manifold associated with the q = 0 optical edge and reduce the dynamics exactly to a scalar fractional-delay Duffing-type equation for the host–resonator relative motion. On that manifold, we derive first-harmonic amplitude, phase, backbone, fold, and cusp relations together with a complete algebraic classification of multistability for the reduced amplitude equation, and periodic Fourier collocation recovers the predicted coexistence windows. A supplementary weak-symmetry-breaking computation shows that the reduced optical predictions remain accurate under mild forcing heterogeneity. The controlled linear result is therefore exact at the level of the real-frequency harmonic determinant, whereas the nonlinear conclusions are restricted to the synchronous optical branch and to the first-harmonic closure used there.