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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?)

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Article copyright: © Copyright 1976 American Mathematical Society