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Existence of large-amplitude time-periodic breathers localized near a singlesite is proved for the discrete Klein--Gordon equation, in the case when thederivative of the on-site potential has a compact support. Breathers areobtained at small coupling between oscillators and under nonresonanceconditions. Our method is different from the classical anti-continuum limitdeveloped by MacKay and Aubry, and yields in general branches of breathersolutions that cannot be captured with this approach. When the couplingconstant goes to zero, the amplitude and period of oscillations at the excitedsite go to infinity. Our method is based on near-identity transformations,analysis of singular limits in nonlinear oscillator equations, and fixed-pointarguments.

Breather continuation from infinity in nonlinear oscillator chains