Interaction of the folded Whitney umbrella and swallowtail in slow-fast systems (Q744272)

The author considers a smooths slow-fast system defined by the equations \[ \varepsilon\dot s=S(s,x,y,\lambda),\quad \dot x= X(s,x,y,\lambda),\quad\dot y= Y(s,x,y,\lambda), \] where \(s\in\mathbb{R}\) is a fast variable, \(x,y\in\mathbb{R}\) are slow variables, \(\varepsilon,\lambda\in\mathbb{R}\) are constant parameters, \(\varepsilon\ll 1\). The graph of the first integral of this system is a singular surface in the three-dimensional space. ``The variation of the external parameter \(\lambda\) on which the system depends gives rise to transitions of this surface.'' The author has found a normal form and has presented ``figures of the transitions that describes the interaction between the swallowtail and folded Whitney umbrella on the graph of the first integral of a generic one-parameter family of such systems''.