Quasi-genericity of bifurcations to high dimensional invariant tori for maps
One considers a family of maps in a Banach space near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle, the other part of the spectrum being strictly inside the unit disc. One considers the set of parameter space where the truncated normal form of the maps has a bifurcation of a family of invariant \(T^ 1\)-circles into a family of invariant \(T^ 2\)-tori. One shows that this problem needs a 3 dimensional parameter unfolding. Making some assumptions one shows existence of ``clean'' bifurcations. The main tool is notion of the quasi-conjugacy to a constant for a matrix function on \(T^ 1\). This type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations from \(T^ n\) to \(T^{n+1}\) tori, and this needs an \((n+1)\)-dimensional parameter unfolding.
scientific article; zbMATH DE number 4105906
scientific article