Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the lower order terms which really govern behaviour around the point of interest.
However, there are cases where we cannot throw out the higher order terms, because all the lower order terms are zero, and it is the higher order terms that are non-zero, thus governing behaviour.
I think one example of such a phenomenon is the 2D "elastic buckling beam" -- the beam (under the action of a force) can buckle either left or right, or simply be crushed straight down, and thus the "bifurcation diagram" for its behaviour has a so-called three-pronged "pitchfork bifurcation", corresponding to: buckling to the left, buckling to the right, and crushing. In the case of the elastic buckling beam, the ODE governing its behaviour has a Taylor series approximation where the order 0, 1 and 2 terms are zero, but the order 3 terms and above are non-zero. Thus, the ODE can roughly be written as a cubic curve, with at least 3 "solutions" corresponding to buckling to the left, right, and crushing.
What are some other physical examples corresponding to situations where only the higher order Taylor series terms survive?