Planar arrays of magnetoinductively coupled rf SQUIDs (Superconducting QUantum Interference Devices) belong to the emergent class of superconducting metamaterials that encompass the Josephson effect. These SQUID-based metamaterials acquire their electromagnetic properties from the resonant characteristics of their constitutive elements, i.e., the individual rf SQUIDs. In its simplest version, an rf SQUID consists of a superconducting ring interrupted by a Josephson junction. We investigate numerically the response of a two-dimensional rf SQUID metamaterial with respect to the driving frequency of an externally applied alternating magnetic field in the presence of disorder arising from critical current fluctuations of the Josephson elements; in effect, the resonance frequencies of individual SQUIDs are distributed randomly around a mean value. Bistability is observed in the current amplitude–frequency curves both in ordered and disordered SQUID metamaterials; moreover, bistability is favored by disorder through the improvement of synchronization between SQUID oscillators. Relatively weak disorder widens significantly the bistability region by helping the system to self-organize and leads to nearly homogeneous states that change smoothly with varying driving frequency. Also, the total current of the metamaterial is enhanced, compared with that of uncoupled SQUIDs, through the synergetic action of coupling and synchronization. The existence of simultaneously stable states that provide either high or low total current, allows the metamaterial to exhibit different magnetic responses that correspond to different values of the effective magnetic permeability. At low power of the incident field, high current amplitude states exhibit extreme diamagnetic properties corresponding to negative magnetic permeability in a narrow frequency interval.

Intense work on discrete breathers or intrinsic localized modes in recent years has revealed a wealth of new properties of classical energy localization. Relaxation and mobility in particular may be two of the critical links with biomolecular processes. We review some of the basic discrete breather properties that we think are pertinent to biomolecules and make conjectures as to their possible biological utility.

The dynamics of a metastable beam of particles ‘‘injected’’ into artificially excited resonance islands in the Fermilab Tevatron is studied. Evolution of the captured beam distribution is measured by an externally detectable coherent signal. Dynamical partitions observed in the control space of amplitude and tune of a focusing modulation are consistent with an equivalent mode of an ensemble of driven gravity pendula. The model’s single parameter, the island tune ${\mathit{Q}}_{\mathit{I}}$ of small oscillations, is inferred as the central frequency of resonant detrapping under this weak drive.

The conditions leading to flat dispersionless frequency bands in truly one-dimensional parity-time ( $${\mathscr{P}}{\mathscr{T}}$$P T ) symmetric metamaterials comprised of split‐ring resonators (SRRs) arranged in a binary pattern are obtained analytically. In this paradigmatic system, in which the SRRs are coupled through both electric and magnetic dipole-dipole forces, flat-bands may arise from tailoring its natural parameters (such as, e.g., the coupling coefficients between SRRs) and not from geometrical effects. For sets of parameters which values are tailored to flatten the upper band of the spectrum, the solution of the corresponding quadratic eigenvalue problem reveals the existence of compact, two-site localized eigenmodes. Numerical simulations confirm the existence and the dynamic stability of such modes, which can be formed through the evolution of single-site initial excitations without disorder or nonlinearity.