  Phase reduction framework for limit-cycling systems based on isochrons has
been used as a powerful tool for analyzing rhythmic phenomena. Recently, the
notion of isostables, which complements the isochrons by characterizing
amplitudes of the system state, i.e., deviations from the limit-cycle
attractor, has been introduced to describe transient dynamics around the limit
cycle [Wilson and Moehlis, Phys. Rev. E 94, 052213 (2016)]. In this study, we
introduce a framework for a reduced phase-amplitude description of transient
dynamics of stable limit-cycling systems. In contrast to the preceding study,
the isostables are treated in a fully consistent way with the Koopman operator
analysis, which enables us to avoid discontinuities of the isostables and to
apply the framework to system states far from the limit cycle. We also propose
a new, convenient bi-orthogonalization method to obtain the response functions
of the amplitudes, which can be interpreted as an extension of the adjoint
covariant Lyapunov vector to transient dynamics in limit-cycling systems. We
illustrate the utility of the proposed reduction framework by estimating
optimal injection timing of external input that efficiently suppresses
deviations of the system state from the limit cycle in a model of a biochemical
oscillator.
