  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.
