Osculatory Dynamics: Framework for the Analysis of Oscillatory Systems

Intractable phase dynamics often challenge our understanding of complex oscillatory systems, hindering the exploration of synchronisation, chaos, and emergent phenomena across diverse fields. We introduce a novel conceptual framework for phase analysis, using the osculating circle to construct a co-moving coordinate system, which allows us to define a unique phase of the system. This coordinate independent, geometrical technique allows dissecting intricate local phase dynamics, even in regimes where traditional methods fail. Our methodology enables the analysis of a wider range of complex systems which were previously deemed intractable.