Unstability problem of real analytic maps (Q6634556)

This paper gives examples of proper \(C^\omega\) maps that are \(C^\infty\) stable, but \(C^\omega\) unstable e.g., Whitney umbrella and swallowtail singularity. These exmaples contrast to the fact that the infinitesimal \(C^\infty\) stability is equivalent to the infinitesimal \(C^\omega\) stability and the fact that the infinitesimal \(C^\infty\) stability is equivalent to the \(C^\infty\) stability.