Attracting Cycles and Isochrons

  • Arthur T. Winfree
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 12)


In Section A of this chapter we associate a phase with each state of an attracting limit-cycle oscillator during dynamics in the absence of any perturbing influence. In Section B a stimulus smoothly alters the trajectories so that phase changes in peculiar ways, even discontinuously. This analysis is intended to apply to smooth dynamics. Accordingly, in Section C references are compiled to models that violate this precondition and thus do not fall within the purview of this chapter.


State Space Latent Phase Relaxation Oscillator Slime Mold Phase Singularity 
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  1. 1.
    Unknown to this biologist before 1974, existence proofs for ordinary differential equations are to be found in Coddington and Levinson (1955, Theorem 2.2, p. 323); Hale (1963, Theorem 10.1, p. 94); Hale (1969, Theorem 2.1, p. 217); in Hirsch and Smale (1974, Theorem 13–3 p. 284–5), in Fenichel (1974, 1977) and, for functional differential equations, Hale (1977, Theorem 3.1, p. 242) where it is called “asymptotic phase. “Google Scholar
  2. 2.
    Conditions are more favorable in phase compromise experiments with chemical oscillators. When two volumes of reaction at distinct phases are mixed, each perturbs the other in a known and simple way, viz., taking a weighted average of concentrations. See Box F.Google Scholar
  3. 3.
    Which must exist, supposing the trajectories point inward from infinity, as they must in all real chemical systems (Wei, 1962).Google Scholar
  4. 4.
    Because Best chose parameters such that the steady-state is locally an attractor competing with the attracting cycle, the geometry was not quite so simple in detail. Resetting curves too close to the critical pulse magnitude are not either type, because each has a zone of old phase in which new phase is undefined. This happens because the system falls into the locally attracting steady-state after any such stimulus. Only along the boundary of this attractor basin does new phase behave in an honestly singular way. See contour maps in Winfree (1983b and 1987a.) A slight adjustment of the biasing current would presumably eliminate this peculiarity without affecting much else.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Arthur T. Winfree
    • 1
  1. 1.Department of Ecology and Evolutionary BiologyUniversity of ArizonaTucsonUSA

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