# American Institute of Mathematical Sciences

May  2021, 26(5): 2343-2359. doi: 10.3934/dcdsb.2020181

## 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, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181
##### References:

show all references

##### References:
(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 $\delta = \max(d_0, d_1) = d_0$
 [1] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 [2] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031 [3] V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 [4] Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021013 [5] Moon Duchin, Tom Needham, Thomas Weighill. The (homological) persistence of gerrymandering. Foundations of Data Science, 2021  doi: 10.3934/fods.2021007 [6] Nadezhda Maltugueva, Nikolay Pogodaev. Modeling of crowds in regions with moving obstacles. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021066 [7] Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 [8] Kai Cai, Guangyue Han. An optimization approach to the Langberg-Médard multiple unicast conjecture. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021001 [9] Assia Boubidi, Sihem Kechida, Hicham Tebbikh. Analytical study of resonance regions for second kind commensurate fractional systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3579-3594. doi: 10.3934/dcdsb.2020247 [10] Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 [11] Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61 [12] Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043 [13] Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 [14] Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226 [15] Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021053 [16] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [17] Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409 [18] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006 [19] Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 [20] Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

2019 Impact Factor: 1.27