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Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

In this paper, we show that, for scalar reaction-diffusion equations on the circle S 1 , the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non-linearity. In other words, we prove that in an appropriate functional space of non-linear terms in the equation, the set of functions, for which all equilibria and periodic orbits are hyperbolic, is a countable intersection of open dense sets. The main tools in the proof are the property of the lap number and the Sard-Smale Theorem.