On a tropicalization of planar polynomial ODEs with finitely many structurally stable phase portraits

TL;DR

This work formulates a tropical dynamical framework for planar polynomial ODEs by transforming to $(u,v)$ coordinates and describing the limit as $oldsymbol{\\epsilon} o 0$ via a differential inclusion with a set-valued vector-field supported on tropical curves. It introduces a new equivalence notion based on polygonal orbit homotopy, proves existence of piecewise-affine solutions, and shows that for fixed degree $N$ there are finitely many structurally stable phase portraits; the generalized autocatalator example yields exactly $15$ equivalence classes. These results establish a global, combinatorial lens (via tropical geometry and a crossing graph) to classify skeleton dynamics in extreme parameter regimes, with potential extensions to higher dimensions and connections to slow-fast dynamics for small $oldsymbol{\epsilon}$. The framework offers a pathway to skeletonizing complex reaction networks and may inform how continuous models persist or bifurcate under extreme scaling, enabling systematic enumerations of phase portraits in polynomial systems.

Abstract

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.

Figures (32)

• Figure 1: In (a): The upper envelope of the set of points $(\operatorname{deg} F_{i,j},\alpha_{i,j})$. The subdivision $\mathcal{S}$ of the Newton polygon is the projection of the envelope. In (b): The subdivision $\mathcal{S}$ and the dual tropical curve $\mathcal{T}$. The subdivision fixes $\mathcal{T}$ up to homotopy and the tropical edges of $\mathcal{T}$ are perpendicular to the edges of $\mathcal{S}$, see Proposition \ref{['proposition:dualS']}.
• Figure 2: Illustrations of sliding and the set-valued vector-field $\mathbf{trop}(u,v)$, $(u,v)\in \mathcal{E}_{i,j}$. The blue curves show orbits of the differential inclusion $(\dot u,\dot v)\in \mathbf{trop}(u,v)$. In particular, for Filippov sliding ((a) and (b)), there is a unique vector ${\mathbf{d}}_{\mathrm{Fp}}$, the Filippov sliding-vector, tangent to the switching manifold. For transversal (not tangential) nullcline sliding (c), in a point which is not atropical singularity, there is also a unique vector ${\mathbf{d}}_{\mathrm{nc}}$ that is tangent to the switching manifold. The direction is also indicated by a little (red) triangle, a convention we will also use in the subsequent figures. In (d) (tangential nullcline sliding), only ${\mathbf{d}}_i$ and ${\mathbf{d}}_j$ in $\mathbf{trop}(u,v)$ are tangent to $\mathcal{E}_{i,j}$ at $(u,v)\in \mathcal{E}_{i,j}$.
• Figure 3: Illustration of crossing.
• Figure 4: Phase portrait of (\ref{['eq:auto']}) for $\mu=2$ and ${\color{black}{\theta}}=0.001$, using two different scalings: $(x,y)$ in (a) and $(x,{\color{black}{\theta}} y)$ in (b). The limit cycle $\gamma_{\color{black}{\theta}}$(in red) is unbounded in the $(x,y)$-coordinates as ${\color{black}{\theta}}\to 0$ but bounded in $(x,{\color{black}{\theta}} y)$.
• Figure 5: In (a): The graph of the tropical polynomial $F_{\max}$ associated with the tropical monomials in (\ref{['eq:tropauto']}). In (b): The (labelled) subdivision associated with monomial in (\ref{['eq:tropauto']}). In all figures, $\alpha=\frac{1}{4}$. Finally, in (c): The tropical phase portrait with a source $\mathcal{Q}_{1346}$ and a sliding limit cycle (in red).

Theorems & Definitions (93)

• Remark 2.1
• Definition 2.2
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Remark 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Remark 3.4