title: Adjoint method provides phase response functions for delay-induced oscillations.
abstract: Limit-cycle oscillations induced by time delay are widely observed in various systems, but a systematic phase-reduction theory for them has yet to be developed. Here we present a practical theoretical framework to calculate the phase response function Z(θ), a fundamental quantity for the theory, of delay-induced limit cycles with infinite-dimensional phase space. We show that Z(θ) can be obtained as a zero eigenfunction of the adjoint equation associated with an appropriate bilinear form for the delay differential equations. We confirm the validity of the proposed framework for two biological oscillators and demonstrate that the derived phase equation predicts intriguing multimodal locking behavior.
authors.affiliation: Graduate School of Frontier Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwashi, Chiba 277-8563, Japan.
authors.forename: Kiyoshi
authors.initials: K
authors.lastname: Kotani
authors.forename: Ikuhiro
authors.initials: I
authors.lastname: Yamaguchi
authors.forename: Yutaro
authors.initials: Y
authors.lastname: Ogawa
authors.forename: Yasuhiko
authors.initials: Y
authors.lastname: Jimbo
authors.forename: Hiroya
authors.initials: H
authors.lastname: Nakao
authors.forename: G Bard
authors.initials: GB
authors.lastname: Ermentrout