Boolean networks are a valuable class of discrete dynamical systems models, but they remain fundamentally limited by their inability to capture multi-way interactions in their components. To remedy this limitation, we propose a model of Boolean hypernetworks, which generalize standard Boolean networks. Utilizing the bijection between hypernetworks and bipartite networks, we show how Boolean hypernetworks generalize standard Boolean networks. We derive ensembles of Boolean hypernetworks from standard random Boolean networks and simulate the dynamics of each. Our results indicate that several properties of Boolean network dynamics are affected by the addition of multi-way interactions, and that these additions can have stabilizing or destabilizing effects.
{"title":"Dynamical Properties of Random Boolean Hypernetworks","authors":"Kevin M. Stoltz, Cliff A. Joslyn","doi":"arxiv-2408.17388","DOIUrl":"https://doi.org/arxiv-2408.17388","url":null,"abstract":"Boolean networks are a valuable class of discrete dynamical systems models,\u0000but they remain fundamentally limited by their inability to capture multi-way\u0000interactions in their components. To remedy this limitation, we propose a model\u0000of Boolean hypernetworks, which generalize standard Boolean networks. Utilizing\u0000the bijection between hypernetworks and bipartite networks, we show how Boolean\u0000hypernetworks generalize standard Boolean networks. We derive ensembles of\u0000Boolean hypernetworks from standard random Boolean networks and simulate the\u0000dynamics of each. Our results indicate that several properties of Boolean\u0000network dynamics are affected by the addition of multi-way interactions, and\u0000that these additions can have stabilizing or destabilizing effects.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anna Gierzkiewicz, Rodrigo Gonçalves Schaefer, Piotr Zgliczyński
The infinite spin problem is a problem concerning the rotational behavior of total collision orbits in the $n$-body problem. The question makes also sense for partial collision. When a~cluster of bodies tends to a (partial) collision, then its normalized shape curve tends to the set of normalized central configurations, which in the planar case has $SO(2)$ symmetry. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation of some central configuration instead of a particular point on it. This is the emph{infinite spin problem}. We show that it is not possible if the limiting circle is isolated from other connected components of set of normalized central configuration. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on combination of the center manifold theorem with {L}ojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.
{"title":"No Infinite Spin for Partial Collisions converging to isolated CC on the plane","authors":"Anna Gierzkiewicz, Rodrigo Gonçalves Schaefer, Piotr Zgliczyński","doi":"arxiv-2408.16409","DOIUrl":"https://doi.org/arxiv-2408.16409","url":null,"abstract":"The infinite spin problem is a problem concerning the rotational behavior of\u0000total collision orbits in the $n$-body problem. The question makes also sense\u0000for partial collision. When a~cluster of bodies tends to a (partial) collision,\u0000then its normalized shape curve tends to the set of normalized central\u0000configurations, which in the planar case has $SO(2)$ symmetry. This leaves a\u0000possibility that the normalized shape curve tends to the circle obtained by\u0000rotation of some central configuration instead of a particular point on it.\u0000This is the emph{infinite spin problem}. We show that it is not possible if\u0000the limiting circle is isolated from other connected components of set of\u0000normalized central configuration. Our approach extends the method from recent\u0000work for total collision by Moeckel and Montgomery, which was based on\u0000combination of the center manifold theorem with {L}ojasiewicz inequality. To\u0000that we add a shadowing result for pseudo-orbits near normally hyperbolic\u0000manifold and careful estimates on the influence of other bodies on the cluster\u0000of colliding bodies.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paul Carter, Arjen Doelman, Peter van Heijster, Daniel Levy, Philip Maini, Erin Okey, Paige Yeung
We consider a Gatenby--Gawlinski-type model of invasive tumors in the presence of an Allee effect. We describe the construction of bistable one-dimensional traveling fronts using singular perturbation techniques in different parameter regimes corresponding to tumor interfaces with, or without, acellular gap. By extending the front as a planar interface, we perform a stability analysis to long wavelength perturbations transverse to the direction of front propagation and derive a simple stability criterion for the front in two spatial dimensions. In particular we find that in general the presence of the acellular gap indicates transversal instability of the associated planar front, which can lead to complex interfacial dynamics such as the development of finger-like protrusions and/or different invasion speeds.
{"title":"Deformations of acid-mediated invasive tumors in a model with Allee effect","authors":"Paul Carter, Arjen Doelman, Peter van Heijster, Daniel Levy, Philip Maini, Erin Okey, Paige Yeung","doi":"arxiv-2408.16172","DOIUrl":"https://doi.org/arxiv-2408.16172","url":null,"abstract":"We consider a Gatenby--Gawlinski-type model of invasive tumors in the\u0000presence of an Allee effect. We describe the construction of bistable\u0000one-dimensional traveling fronts using singular perturbation techniques in\u0000different parameter regimes corresponding to tumor interfaces with, or without,\u0000acellular gap. By extending the front as a planar interface, we perform a\u0000stability analysis to long wavelength perturbations transverse to the direction\u0000of front propagation and derive a simple stability criterion for the front in\u0000two spatial dimensions. In particular we find that in general the presence of\u0000the acellular gap indicates transversal instability of the associated planar\u0000front, which can lead to complex interfacial dynamics such as the development\u0000of finger-like protrusions and/or different invasion speeds.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well known in general relativity that trajectories of Hamiltonian systems lift to geodesics of pp-wave spacetimes, an example of a more general phenomenon known as the "Eisenhart lift." We review and expand upon the benefits of this correspondence for dynamical systems theory. One benefit is the use of curvature and conjugate points to study the stability of Hamiltonian systems. Another benefit is that this lift unfolds a Hamiltonian system into a family of ODEs akin to a moduli space. One such family arises from the conformal invariance of lightlike geodesics, by which any Hamiltonian system unfolds into a "conformal class" of non-diffeomorphic ODEs with solutions in common. By utilizing higher-index versions of pp-waves, a similar lift and conformal class are shown to exist for certain second-order complex ODEs. Another such family occurs by lifting to a Riemannian metric that is dual to a pp-wave, a process that in certain cases yields a "square root" for the Hamiltonian. We prove a two-point boundary result for the family of ODEs arising from this lift, as well as the existence of a constant of the motion generalizing conservation of energy.
{"title":"The Eisenhart Lift and Hamiltonian Systems","authors":"Amir Babak Aazami","doi":"arxiv-2408.16139","DOIUrl":"https://doi.org/arxiv-2408.16139","url":null,"abstract":"It is well known in general relativity that trajectories of Hamiltonian\u0000systems lift to geodesics of pp-wave spacetimes, an example of a more general\u0000phenomenon known as the \"Eisenhart lift.\" We review and expand upon the\u0000benefits of this correspondence for dynamical systems theory. One benefit is\u0000the use of curvature and conjugate points to study the stability of Hamiltonian\u0000systems. Another benefit is that this lift unfolds a Hamiltonian system into a\u0000family of ODEs akin to a moduli space. One such family arises from the\u0000conformal invariance of lightlike geodesics, by which any Hamiltonian system\u0000unfolds into a \"conformal class\" of non-diffeomorphic ODEs with solutions in\u0000common. By utilizing higher-index versions of pp-waves, a similar lift and\u0000conformal class are shown to exist for certain second-order complex ODEs.\u0000Another such family occurs by lifting to a Riemannian metric that is dual to a\u0000pp-wave, a process that in certain cases yields a \"square root\" for the\u0000Hamiltonian. We prove a two-point boundary result for the family of ODEs\u0000arising from this lift, as well as the existence of a constant of the motion\u0000generalizing conservation of energy.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the dynamics of a piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in $mathbb{R}^2$. The geometric map of study may be analogized to the billiard map with a different reflection rule so that each iteration is a contraction over the space, thereby providing asymptotic behavior of interest. Our study emphasizes the behavior of periodic orbits generated by the map, with description of their geometry and bifurcation behavior. We establish that for any initial point in the space, the orbit will converge to a fixed point or periodic orbit, and we demonstrate that there exists an infinite variety of periodic orbits the orbits may converge to, dependent on the parameters of the underlying space.
{"title":"A piecewise contractive map on triangles","authors":"Samuel Everett","doi":"arxiv-2408.16019","DOIUrl":"https://doi.org/arxiv-2408.16019","url":null,"abstract":"We study the dynamics of a piecewise map defined on the set of three pairwise\u0000nonparallel, nonconcurrent lines in $mathbb{R}^2$. The geometric map of study\u0000may be analogized to the billiard map with a different reflection rule so that\u0000each iteration is a contraction over the space, thereby providing asymptotic\u0000behavior of interest. Our study emphasizes the behavior of periodic orbits\u0000generated by the map, with description of their geometry and bifurcation\u0000behavior. We establish that for any initial point in the space, the orbit will\u0000converge to a fixed point or periodic orbit, and we demonstrate that there\u0000exists an infinite variety of periodic orbits the orbits may converge to,\u0000dependent on the parameters of the underlying space.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study a well known two-step anaerobic digestion model in a configuration of two chemostats in series. This model is an eight-dimensional system of ordinary differential equations. Since the reaction system has a cascade structure, we show that the eight-order model can be reduced to a four-dimensional one. Using general growth rates, we provide an in-depth mathematical analysis of the asymptotic behavior of the system. First, we determine all the steady states of the model where there can be more than fifteen equilibria with a non-monotonic growth rate. Then, the necessary and sufficient conditions of existence and local stability of all steady states are established according to the operating parameters: the dilution rate, the input concentrations of the two nutrients, and the distribution of the total process volume considered. The operating diagrams are then analyzed theoretically to describe the asymptotic behavior of the process according to the four control parameters. There can be seventy regions with rich behavior where the system may exhibit bistability or tristability with the coexistence of both microbial species in the two bioreactors.
{"title":"Analysis of anaerobic digestion model with two serial interconnected chemostats","authors":"Thamer Hmidhi, Radhouane Fekih-Salem, Jérôme Harmand","doi":"arxiv-2408.04984","DOIUrl":"https://doi.org/arxiv-2408.04984","url":null,"abstract":"In this paper, we study a well known two-step anaerobic digestion model in a\u0000configuration of two chemostats in series. This model is an eight-dimensional\u0000system of ordinary differential equations. Since the reaction system has a\u0000cascade structure, we show that the eight-order model can be reduced to a\u0000four-dimensional one. Using general growth rates, we provide an in-depth\u0000mathematical analysis of the asymptotic behavior of the system. First, we\u0000determine all the steady states of the model where there can be more than\u0000fifteen equilibria with a non-monotonic growth rate. Then, the necessary and\u0000sufficient conditions of existence and local stability of all steady states are\u0000established according to the operating parameters: the dilution rate, the input\u0000concentrations of the two nutrients, and the distribution of the total process\u0000volume considered. The operating diagrams are then analyzed theoretically to\u0000describe the asymptotic behavior of the process according to the four control\u0000parameters. There can be seventy regions with rich behavior where the system\u0000may exhibit bistability or tristability with the coexistence of both microbial\u0000species in the two bioreactors.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we investigate some relations between dynamical and algebraic properties of semigroups of entire maps with applications to semigroups of formal series. We show that two entire maps fixing the origin share the set of preperiodic points, whenever these maps generate a semigroup which contains neither free nor free abelian non-cyclic subsemigroups and one of the maps has the origin as a superattracting fixed point. We show that a subgroup of formal series generated by rational elements is amenable, whenever contains no free non-cyclic subsemigroup generated by rational elements. We prove that a left-amenable semigroup S of entire maps admits a invariant probability measure for a continuous extension of S on the Stone-Cech compactification of the complex plane. Finally, given an entire map f, we associate a semigroup S such that f admits no ergodic fixed point of the Ruelle operator, whenever every finitely generated subsemigroup of S admits a left-amenable Ruelle representation.
在本文中，我们研究了全映射半群的动力学性质和代数性质之间的一些关系，并将其应用于形式数列半群。我们证明，只要两个固定原点的全映射生成的半群既不包含自由非循环子半群，也不包含自由非循环子半群，且其中一个映射以原点为超吸引定点，那么这两个全映射就共享前周期点集。我们证明，只要不包含由有理元素生成的自由非循环子半群，由有理元素生成的形式数列子群就是可解的。我们证明了全映射的左可门半群 S 在复平面的 Stone-Cechcompactification 上对 S 的连续扩展具有不变量概率度量。最后，给定一个全映射 f，我们关联了一个半群 S，只要 S 的每一个有限生成的子半群都承认左可门 Ruelle 表示，那么 f 就不承认 Ruelleoperator 的遍历定点。
{"title":"On the amenability of semigroups of entire maps and formal power series","authors":"C. Cabrera, P. Dominguez, P. Makienko","doi":"arxiv-2408.05180","DOIUrl":"https://doi.org/arxiv-2408.05180","url":null,"abstract":"In this article, we investigate some relations between dynamical and\u0000algebraic properties of semigroups of entire maps with applications to\u0000semigroups of formal series. We show that two entire maps fixing the origin\u0000share the set of preperiodic points, whenever these maps generate a semigroup\u0000which contains neither free nor free abelian non-cyclic subsemigroups and one\u0000of the maps has the origin as a superattracting fixed point. We show that a\u0000subgroup of formal series generated by rational elements is amenable, whenever\u0000contains no free non-cyclic subsemigroup generated by rational elements. We\u0000prove that a left-amenable semigroup S of entire maps admits a invariant\u0000probability measure for a continuous extension of S on the Stone-Cech\u0000compactification of the complex plane. Finally, given an entire map f, we\u0000associate a semigroup S such that f admits no ergodic fixed point of the Ruelle\u0000operator, whenever every finitely generated subsemigroup of S admits a\u0000left-amenable Ruelle representation.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"127 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We classify singular holomorphic vector fields in two-dimensional complex space admitting a (Levi-nonflat) real-analytic invariant 3-fold through the singularity. In this way, we complete the classification of infinitesimal symmetries of real-analytic Levi-nonflat hypersurfaces in complex two-space. The classification of holomorphic vector fields obtained in the paper has very interesting overlaps with the recent Lombardi-Stolovitch classification theory for holomorphic vector fields at a singularity. In particular, we show that most of the resonances arising in Lombardi-Stolovitch theory do not occur under the presence of (Levi-nonflat) integral manifolds.
{"title":"Holomorphic vector fields with real integral manifolds","authors":"Martin Kolář, Ilya Kossovskiy, Bernhard Lamel","doi":"arxiv-2408.05186","DOIUrl":"https://doi.org/arxiv-2408.05186","url":null,"abstract":"We classify singular holomorphic vector fields in two-dimensional complex\u0000space admitting a (Levi-nonflat) real-analytic invariant 3-fold through the\u0000singularity. In this way, we complete the classification of infinitesimal\u0000symmetries of real-analytic Levi-nonflat hypersurfaces in complex two-space.\u0000The classification of holomorphic vector fields obtained in the paper has very\u0000interesting overlaps with the recent Lombardi-Stolovitch classification theory\u0000for holomorphic vector fields at a singularity. In particular, we show that\u0000most of the resonances arising in Lombardi-Stolovitch theory do not occur under\u0000the presence of (Levi-nonflat) integral manifolds.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an abstract viewpoint, but in applied problems one piece of the map often has degenerate range, corresponding to a zero determinant. This provides simplification, yet even in two dimensions the dynamics can be incredibly rich. The purpose of this paper is to determine broadly how the dynamics of the two-dimensional border-collision normal form with a zero determinant differs for different values of its parameters. We identify parameter regions of period-adding, period-incrementing, mode-locking, and component doubling of chaotic attractors, and characterise the dominant bifurcation boundaries. The intention is for the results to enable border-collision bifurcations in mathematical models to be analysed more easily and effectively, and we illustrate this with a flu epidemic model and two stick-slip friction oscillator models. We also describe three novel bifurcation structures that remain to be explored.
{"title":"The two-dimensional border-collision normal form with a zero determinant","authors":"David J. W. Simpson","doi":"arxiv-2408.04790","DOIUrl":"https://doi.org/arxiv-2408.04790","url":null,"abstract":"The border-collision normal form is a piecewise-linear family of continuous\u0000maps that describe the dynamics near border-collision bifurcations. Most prior\u0000studies assume each piece of the normal form is invertible, as is generic from\u0000an abstract viewpoint, but in applied problems one piece of the map often has\u0000degenerate range, corresponding to a zero determinant. This provides\u0000simplification, yet even in two dimensions the dynamics can be incredibly rich.\u0000The purpose of this paper is to determine broadly how the dynamics of the\u0000two-dimensional border-collision normal form with a zero determinant differs\u0000for different values of its parameters. We identify parameter regions of\u0000period-adding, period-incrementing, mode-locking, and component doubling of\u0000chaotic attractors, and characterise the dominant bifurcation boundaries. The\u0000intention is for the results to enable border-collision bifurcations in\u0000mathematical models to be analysed more easily and effectively, and we\u0000illustrate this with a flu epidemic model and two stick-slip friction\u0000oscillator models. We also describe three novel bifurcation structures that\u0000remain to be explored.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marie Dorchain, Wilfried Segnou, Riccardo Muolo, Timoteo Carletti
We hereby develop the theory of Turing instability for reaction-diffusion systems defined on m-directed hypergraphs, the latter being generalization of hypergraphs where nodes forming hyperedges can be shared into two disjoint sets, the head nodes and the tail nodes. This framework encodes thus for a privileged direction for the reaction to occur: the joint action of tail nodes is a driver for the reaction involving head nodes. It thus results a natural generalization of directed networks. Based on a linear stability analysis we have shown the existence of two Laplace matrices, allowing to analytically prove that Turing patterns, stationary or wave-like, emerges for a much broader set of parameters in the m-directed setting. In particular directionality promotes Turing instability, otherwise absent in the symmetric case. Analytical results are compared to simulations performed by using the Brusselator model defined on a m-directed d-hyperring as well as on a m-directed random hypergraph.
后者是超图（hypergraphs）的广义化，在超图中，形成超桥的节点可以共享为两个不相交的集合，即头部节点和尾部节点。因此，这一框架为反应的发生提供了一个有利的方向：尾节点的联合行动是涉及头节点的反应的驱动力。因此，它是有向网络的自然概括。在线性稳定性分析的基础上，我们证明了两个拉普拉斯矩阵的存在，从而可以分析证明图灵模式（静态或波浪式）在 m 定向环境中出现的参数范围更广。尤其是方向性促进了图灵不稳定性，而对称情况下则不存在这种现象。分析结果与使用布鲁塞尔器模型（Brusselator model）在 m 向 d 型超环和 m 向随机超图上定义的模拟结果进行了比较。
{"title":"Impact of directionality on the emergence of Turing patterns on m-directed higher-order structures","authors":"Marie Dorchain, Wilfried Segnou, Riccardo Muolo, Timoteo Carletti","doi":"arxiv-2408.04721","DOIUrl":"https://doi.org/arxiv-2408.04721","url":null,"abstract":"We hereby develop the theory of Turing instability for reaction-diffusion\u0000systems defined on m-directed hypergraphs, the latter being generalization of\u0000hypergraphs where nodes forming hyperedges can be shared into two disjoint\u0000sets, the head nodes and the tail nodes. This framework encodes thus for a\u0000privileged direction for the reaction to occur: the joint action of tail nodes\u0000is a driver for the reaction involving head nodes. It thus results a natural\u0000generalization of directed networks. Based on a linear stability analysis we\u0000have shown the existence of two Laplace matrices, allowing to analytically\u0000prove that Turing patterns, stationary or wave-like, emerges for a much broader\u0000set of parameters in the m-directed setting. In particular directionality\u0000promotes Turing instability, otherwise absent in the symmetric case. Analytical\u0000results are compared to simulations performed by using the Brusselator model\u0000defined on a m-directed d-hyperring as well as on a m-directed random\u0000hypergraph.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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