Phase-Amplitude Reduction of Limit Cycling Systems

  • Sho ShirasakaEmail author
  • Wataru Kurebayashi
  • Hiroya Nakao
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 484)


Optimization and control of collective dynamics in rhythmic systems have attracted increasing interest, and various methods have been developed on the basis of the phase reduction framework for limit cycle oscillators. The phase reduction relies on the notion of the isochron, which represents the set of system states that share the same asymptotic phase. However, the phase reduction does not take into account the amplitude degrees of freedom representing deviations of the system state from the limit cycle attractor, to which some rich and nontrivial transient behaviors of the oscillators are attributed. In this chapter, after a brief introduction of the phase reduction framework, a phase-amplitude reduction framework that is applicable to transient dynamics far from the limit cycle attractor is formulated. A rigorous theoretical background for defining the amplitudes is provided by the notion of the isostable, which naturally complements the isochron in the sense that both of them can be understood from a unified viewpoint of the spectral properties of the Koopman operator. The utility of the proposed phase-amplitude reduction framework is illustrated by evaluating the optimal injection timing of a weak control input that efficiently suppresses deviations of the system state from the limit cycle attractor.



S. S. acknowledges financial support from Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 18H06478. W. K. acknowledges financial support from JSPS KAKENHI Grants No. 16K16125 No. 17H03279. H. N. acknowledges financial support from JSPS KAKENHI Grants No. 16K13847, No. 17H03279, No. 18K03471, No. 18H03287, and JST CREST Grant No. JPMJCR1913.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Sho Shirasaka
    • 1
    Email author
  • Wataru Kurebayashi
    • 2
  • Hiroya Nakao
    • 3
  1. 1.Department of Information and Physical SciencesOsaka UniversitySuitaJapan
  2. 2.Center for Data Science Education and ResearchShiga UniversityHikoneJapan
  3. 3.Department of Systems and Control EngineeringTokyo Institute of TechnologyMeguroJapan

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