Symmetries in bifurcation theory : the appropriate context

Many phenomena in nature can be modeled by differential equations depending on parameters that are being varied continuously. We say that a given solution undergoes a bifurcation with respect to a given parameter if the qualitative behaviour of the system changes arbitrarily close to this solution when the parameter is varied across a critical value. Bifurcation problems can achieve a very high level of complexity because nature is complex. Several assumptions can be made in order to introduce considerable simplifications without going too far from reality. In this thesis we are mainly concerned in setting the problem in a symmetric context and showing that this is a realistic assumption that makes analysis much simpler. We want to emphasize that a lot of behaviour can be much easier to understand and predict when the appropriate symmetry context has been set. The message in part I of this thesis is that the full set of symmetries is not always obvious. We give examples of phenomena that are modeled by partial differential equations on rectangular domains and show that these problems have more than rectangular symmetry. Such hidden symmetries are found by embedding our problem into a larger one satisfying periodic boundary conditions and then consider all the symmetries that satisfy the original boundary conditions. In part II we study the behaviour of an electric circuit which can be modeled by a 3-dimensional system of ordinary differential equations. We begin by analysing this system under a symmetry assumption. Then in order to be more realistic we break the symmetry with a small perturbation. Most of the results for the asymmetric system are obtained by numerical and experimental search since a rigorous analysis became much harder. We observe a smooth change in qualitative behaviour by increasing the symmetry breaking perturbation. There is no dramatic change and we conclude that the original symmetry assumption was convenient and not misleading