Bifurcations of continuous dynamical systems under extreme and new transversality conditions

For a family of continuous-time dynamical systems, in [F. Balibrea, A. Martinez, J.C. Valverde, Local bifurcations of one-dimensional continuous dynamical systems under higher order conditions, in press] we gave sufficient conditions of higher order in terms of the first non-zero partial derivative with respect to the state variable to have (local) generic bifurcations of equilibria. Inspired by these results, here we study what happens when those conditions are extreme in the sense that all those partial derivatives vanish. Besides we deal with some others transversality conditions for the family to undergo the same bifurcations and give some counterexamples proving that, in absence of analyticity of the function which defines the system, the phenomenon could be unexpectedly different.