The synchronization of self-sustained rhythms plays important functional roles in
various real-world systems. Such rhythmic phenomena are often modeled as nonlinear
dynamical systems with asymptotically stable limit cycles. In analyzing the dynamics
of limit-cycle oscillators, phase reduction theory, which approximately describes
the oscillator using an asymptotic phase defined along the limit cycle, has played
a major role. Recently, Koopman operator theory has revealed that the asymptotic phase,
traditionally defined geometrically, actually corresponds to a Koopman eigenfunction
of the system, and that amplitudes of the oscillator can also be introduced. This
clarified that phase reduction is a typical case of global linearization and
dimensionality reduction based on Koopman eigenfunctions, and generalized the method
into phase-amplitude reduction. In this talk, after a brief introduction to the Koopman
operator viewpoint and phase-amplitude reduction, several applications to oscillator
networks, reaction-diffusion systems, and fluid systems, as well as data-driven
approaches will be presented.

16:15 - Tea