Gap modes in Arnold tongues and their topological origins

2025/05/21 by A. R. Brown, Hong Qin, Brown, Andrew +1
Psychology · #FOS: Physical sciences #Mathematical Physics (math-ph) #Other Condensed Matter (cond-mat.other) #Phonetics and Phonology Research #Plasma Physics (physics.plasm-ph)

paper · pdf · doi:10.48550/arxiv.2505.15007

openalex publication_date 2025/05/21 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

Gap modes in a modified Mathieu equation, perturbed by a Dirac delta potential, are investigated. It is proved that the modified Mathieu equation admits stable isolated gap modes with topological origins in the unstable regions of the Mathieu equation, which are known as Arnold tongues. The modes may be identified as localized electron wavefunctions in a 1D chain or as toroidal Alfvén eigenmodes. A generalization of this argument shows that gap modes can be induced in regimes of instability by localized potential perturbations for a large class of periodic Hamiltonians.

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