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Local coordinates around limit cycles of partial differential equations


Author: Arnold Stokes
Journal: Proc. Amer. Math. Soc. 59 (1976), 225-231
MSC: Primary 35B10; Secondary 35K10
DOI: https://doi.org/10.1090/S0002-9939-1976-0412569-8
MathSciNet review: 0412569
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Abstract: 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.


References [Enhancements On Off] (What's this?)

  • [1] P. Gould, Oscillations in nonlinear parabolic systems, Ph.D. thesis, Georgetown Univ., Washington, D.C., 1972.
  • [2] J. K. Hale, Orindary differential equations, Interscience, New York, 1969.
  • [3] -, Functional differential equations, Springer-Verlag, New York, 1971. MR 0466837 (57:6711)
  • [4] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 30 #1270. MR 0171038 (30:1270)
  • [5] A. Stokes, Local coordinates around a limit cycle of a functional differential equation with applications, J. Differential Equations (to appear). MR 0433002 (55:5981)
  • [6] M. Urabe, Geometric study of nonlinear autonomous oscillations, Funkcial. Ekvac. 1 (1958), 1-84. MR 20 #5921. MR 0099482 (20:5921)
  • [7] S. D. Èĭdel'man, Parabolic systems, ``Nauka", Moscow, 1964; English transl., Scripta Technica, London; North-Holland, Amsterdam; Noordhoff, Groningen, 1969. MR 29 #4998; 40 #6023. MR 0167726 (29:4998)
  • [8] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 31 #6062. MR 0181836 (31:6062)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0412569-8
Article copyright: © Copyright 1976 American Mathematical Society

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