Osculatory Dynamics: Framework for the Analysis of Oscillatory Systems

TL;DR

This paper introduces a geometric framework using osculating circles to analyze phase dynamics in complex oscillatory systems, enabling insights into systems previously considered intractable.

Contribution

The authors develop a novel, coordinate-independent geometric method for phase analysis based on osculating circles, expanding analysis capabilities for complex oscillatory systems.

Findings

Allows analysis of complex systems with intractable phase dynamics

Provides a coordinate-independent technique for local phase analysis

Enables exploration of synchronization and chaos in new regimes

Abstract

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.