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 S1, 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 nonlinear 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. 
Romain Joly (IF), Genevi\`eve Raugel (LM-Orsay)