Singularities in Dynamics: A Catastrophic Viewpoint

The article is about singularities of dynamical systems, a notion which is far from being as clear as for smooth maps. To give an idea of it, the beginning of “catastrophe theory” for dynamical systems (including very recent results) is developed, first in parallel with the well-known analogous theory for potentials, showing that dynamical versions of the Morse lemma with parameters already lead to difficult open problems. The second part of the paper makes it clear that the short-sighted view that singularities in dynamics are rest points and periodic orbits cannot resist serious investigation since many other specifically dynamical “singularities” are born from statics. A homage to René Thom, the whole article is written in the language of stratifications of function spaces – even though it deals with semi-local phenomena.