Phase shifts inside Arnold tongues of weakly coupled oscillators

In this paper, we investigate phase synchronization phenomena in weakly coupled oscillators, with a particular focus on the phase shifts that occur within Arnold tongues. Using a proposed theoretical approach, we provide proof of the existence of the corresponding cycle manifold near zero coupling, along with a detailed derivation of its shape. This allows us to explore the conditions under which phase-shift synchronization arises. We employ the implicit function theorem in an appropriate Banach space to establish the existence of the cycle manifold and provide a methodology to study in-phase and anti-phase synchrony in systems governed by both ordinary and delay differential equations. We introduce a numerical technique for cycle continuation that reveals the shift structure of Arnold tongues in coupling and parameter space near the cusps, offering new insights into the dynamics of phase-shifted coupled oscillators. This framework is applied to a classic model of two coupled circle oscillators, coupled van der Pol oscillators, a model of two interneurons, and a two-level interneuronal network to better understand and demonstrate the numerical continuation methodology, which allows for the study of phase synchronization in neuroscience and other fields. The proposed methods not only advance the theoretical understanding of synchronization but also offer practical computational tools for studying complex oscillatory systems.