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We explain the emergence of zero field steps (ZFS) in a Frenkel-Kontorova (FK) model for a 1D annular chain being a model for an annular Josephson junction array. We demonstrate such steps for a case with a chain of 10 phase differences. We necessarily need the periodic boundary conditions. We propose a mechanism for the jump from  M  fluxons to    $$M+1$$  M + 1 in the chain. Graphic abstract

Description of zero field steps on the potential energy surface of a Frenkel-Kontorova model for annular Josephson junction arrays