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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0412569-8
PII: S 0002-9939(1976)0412569-8
Article copyright: © Copyright 1976 American Mathematical Society