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Journal of Mathematical Biology

, Volume 1, Issue 3, pp 259–273 | Cite as

Isochrons and phaseless sets

  • J. Guckenheimer
Article

Summary

Winfree has developed mathematical models for his phase resetting experiments on biological clocks. These models lead him to ask a number of mathematical questions concerning dynamical systems. This paper deals with these mathematical questions. In Winfree's terminology we show the existence of isochrons and establish some of their properties.

Keywords

Vector Field Periodic Orbit Invariant Manifold Unstable Manifold Stable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Dieudonné, J.: Foundations of Modern Analysis, Vol. 1. Academic Press.Google Scholar
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    Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds, mimeographed. U. C. Berkeley 1974.Google Scholar
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    Irwin, M.: A Classification of Elementary Cycles. Topology9, 35–48 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
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    Peixoto, M.: Structural Stability on Two Dimensional Manifolds. Topology1, 101–120 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
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    Smale, S.: Differentiable Dynamical Systems. Bulletin Am. Math. Soc.73, 747–817 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
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    Spanier, E.: Algebraic Topology. McGraw-Hill.Google Scholar
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    Winfree, A.: Patterns of Phase Compromise in Biological Cycles. Journ. of Math. Biol.1, 73–95 (1974).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • J. Guckenheimer
    • 1
  1. 1.Division of Natural SciencesUniversity of CaliforniaSanta CruzUSA

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