  Recently, concepts from the emerging field of tropical geometry have been
used to identify different scaling regimes in chemical reaction networks where
dimension reduction may take place. In this paper, we try to formalize these
ideas further in the context of planar polynomial ODEs. In particular, we
develop a theory of a tropical dynamical system, based upon a differential
inclusion, that has a set of discontinuities on a subset of the associated
tropical curve. The development is inspired by an approach of Peter Szmolyan
that uses the connection of tropical geometry with logarithmic paper. In this
paper, we define a phaseportrait, a notion of equivalence and characterize
structural stability. Furthermore, we demonstrate the results on several
examples, including a(n) (generalized) autocatalator model. Our main result is
that there are finitely many equivalence classes of structurally stable phase
portraits and we enumerate these ($15$ in total) in the context of the
generalized autocatalator model. We believe that the property of finitely many
structurally stable phase portraits underlines the potential of the tropical
approach, also in higher dimension, as a method to obtain and identify skeleton
models in chemical reaction networks in extreme parameter regimes.
