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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Local coordinates around limit cycles of partial differential equations
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by Arnold Stokes PDF
Proc. Amer. Math. Soc. 59 (1976), 225-231 Request permission


Given an isolated periodic solution (limit cycle) $p = p(x,t)$ of a parabolic differential equation in a variable $u = u(x,t)$, local coordinates $(s = s(t),\;w = w(x,t))$ are introduced so that $w = 0,\;s = t +$ constant corresponds to $u = p$, and the equations for $s,\;w$ are of the form $ds/dt = 1 +$ higher-order terms, and $w$ satisfies the variational equation for $p$ on a subspace of codimension one. It is indicated how the method applies to ordinary differential equations, as motivation, and to hyperbolic equations, as an obvious extension.
    P. Gould, Oscillations in nonlinear parabolic systems, Ph.D. thesis, Georgetown Univ., Washington, D.C., 1972. J. K. Hale, Orindary differential equations, Interscience, New York, 1969.
  • Jack K. Hale, Functional differential equations, Applied Mathematical Sciences, Vol. 3, Springer-Verlag New York, New York-Heidelberg, 1971. MR 0466837
  • Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
  • A. Stokes, Local coordinates around a limit cycle of a functional differential equation with applications, J. Differential Equations 24 (1977), no. 2, 153–172. MR 433002, DOI 10.1016/0022-0396(77)90141-3
  • Minoru Urabe, Geometric study of nonlinear autonomous oscillations, Funkcial. Ekvac. 1 (1958), 1–84. MR 99482
  • S. D. Èĭdel′man, Parabolicheskie sistemy, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0167726
  • Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 59 (1976), 225-231
  • MSC: Primary 35B10; Secondary 35K10
  • DOI:
  • MathSciNet review: 0412569