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## Quasi-toric differential inclusions

 1 Department of Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI 53706, USA 2 Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI 53706, USA

* Corresponding author: craciun@math.wisc.edu

Received  October 2019 Revised  March 2020 Published  June 2020

Toric differential inclusions play a pivotal role in providing a rigorous interpretation of the connection between weak reversibility and the persistence of mass-action systems and polynomial dynamical systems. We introduce the notion of quasi-toric differential inclusions, which are strongly related to toric differential inclusions, but have a much simpler geometric structure. We show that every toric differential inclusion can be embedded into a quasi-toric differential inclusion and that every quasi-toric differential inclusion can be embedded into a toric differential inclusion. In particular, this implies that weakly reversible dynamical systems can be embedded into quasi-toric differential inclusions.

Citation: Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020181
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(a) Polyhedral fan in two dimensions. This fan has seven cones: three two-dimensional or maximal cones, three one-dimensional cones and one zero-dimensional cone. (b) Hyperplane-generated polyhedral fan in two dimensions. This fan has 13 cones: six two-dimensional cones, six one-dimensional cones and one cone of dimension zero. (c) Polyhedral fan in three dimensions. This fan has seven cones: three three-dimensional cones, three two-dimensional cones and one cone of dimension one. (d) Hyperplane-generated polyhedral fan in three dimensions. This fan has nine cones: four three-dimensional cones, four two-dimensional cones and one cone of dimension one. Cones in (a) and (b) are pointed, while cones in (c) and (d) are not pointed. Fans in (b) and (d) are hyperplane generated, while fans in (a) and (c) are not hyperplane generated
Right-hand side of a toric differential inclusion (denoted by $F_{\mathcal{F}, \delta}( \boldsymbol{X})$) for a hyperplane-generated fan $\mathcal{F}$. The red region represents the set of points for which $F_{\mathcal{F}, \delta}( \boldsymbol{X}) = \mathbb{R}^2$. For points outside the red region, the blue cones indicate $F_{\mathcal{F}, \delta}( \boldsymbol{X})$, which is not $\mathbb{R}^2$
Example of a quasi-toric differential inclusion that is not well-defined in the sense of Definition 5.2. Consider a point $\boldsymbol{X}$ labeled by a black dot in the figure. If we iterate through the steps of Definition 5.1, we get $dist( \boldsymbol{X}, C_1)\leq d_1$ and $dist( \boldsymbol{X}, \tilde{C}_1)\leq d_1$ in Step 1. It is not clear whether $F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) = C_1^o$ or $F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) = {\tilde{C}_1}^o$ and hence the notion of quasi-toric differential inclusion is not well-defined for this choice of $\boldsymbol{d} = (d_0, d_1)$
Right-hand side of a quasi-toric differential inclusion (denoted by $F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X})$) for a hyperplane-generated fan $\mathcal{F}$. The red circle represents the set of points for which $F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X}) = \mathbb{R}^2$. For points outside the red circle, the blue cones indicate $F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X})$. The numbers $d_0, d_1$ are chosen so that the quasi-toric differential inclusion generated by $\mathcal{F}$ and $\boldsymbol{d} = (d_0, d_1)$ is well-defined in the sense of Definition 5.2
Two dimensional illustration of Lemma 6.3
Two-dimensional illustration of Lemma 6.4
(a) RHS of a toric differential inclusion (denoted by $F_{\mathcal{F}, \delta}( \boldsymbol{X})$) for a hyperplane-generated fan $\mathcal{F}$. (b) RHS of a quasi-toric differential inclusion (denoted by $F_{\mathcal{F}, { \boldsymbol{d}}}( \boldsymbol{X})$) such that the toric differential inclusion given in part (a) can be embedded into this quasi-toric differential inclusion, i.e., $F_{\mathcal{F}, \delta}( \boldsymbol{X})\subseteq F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X})$ for every $\boldsymbol{X}\in\mathbb{R}^n$. As in the proof of Theorem 7.3, the vector $\boldsymbol{d}$ is constructed as follows: we set $d_1 = \delta$ and choose $d_0$ large enough ($d_0 = \lambda\alpha d_1$) so that the quasi-toric differential inclusion is well-defined
(a) RHS of a quasi-toric differential inclusion (denoted by $F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X})$) for a hyperplane-generated fan $\mathcal{F}$. (b) RHS of a toric differential inclusion (denoted by $F_{\mathcal{F}, \delta}( \boldsymbol{X})$) such that the quasi-toric differential inclusion given in part (a) can be embedded into this toric differential inclusion, i.e., $F_{\mathcal{F}, \boldsymbol{d}}( \boldsymbol{X})\subseteq F_{\mathcal{F}, \delta}( \boldsymbol{X})$ for every $\boldsymbol{X}\in\mathbb{R}^n$. As in the proof of Theorem 8.1, we choose $\delta = \max(d_0, d_1) = d_0$
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