We study the propagation properties of abstract linear Schr"odinger equations of the form $ipartial_tpsi = H_0psi+V(t)psi$, where $H_0$ is a self-adjoint operator and $V(t)$ a time-dependent potential. We present explicit sufficient conditions ensuring that if the initial state $psi_0$ has spectral support in $(-infty,0]$ with respect to a reference self-adjoint operator $phi$, then, for some $c>0$ independent of $psi_0$ and all $tne0$, the solution $psi_t$ remains spectrally supported in $(-infty,c|t|]$ with respect to $phi$, up to an $O(|t|^{-n})$ remainder in norm. The main condition is that the multiple commutators of $H_0$ and $phi$ are uniformly bounded in operator norm up to the $(n+1)$-th order. We then apply the abstract theory to a class of nonlocal Schr"odinger equations on $mathbb{R}^d$, proving that any solution with compactly supported initial state remains approximately supported, up to a polynomially suppressed tail in $L^2$-norm, inside a linearly spreading region around the initial support for all $tne0$.
{"title":"Spectral localization estimates for abstract linear Schrödinger equations","authors":"Jingxuan Zhang","doi":"arxiv-2409.10873","DOIUrl":"https://doi.org/arxiv-2409.10873","url":null,"abstract":"We study the propagation properties of abstract linear Schr\"odinger\u0000equations of the form $ipartial_tpsi = H_0psi+V(t)psi$, where $H_0$ is a\u0000self-adjoint operator and $V(t)$ a time-dependent potential. We present\u0000explicit sufficient conditions ensuring that if the initial state $psi_0$ has\u0000spectral support in $(-infty,0]$ with respect to a reference self-adjoint\u0000operator $phi$, then, for some $c>0$ independent of $psi_0$ and all $tne0$,\u0000the solution $psi_t$ remains spectrally supported in $(-infty,c|t|]$ with\u0000respect to $phi$, up to an $O(|t|^{-n})$ remainder in norm. The main condition\u0000is that the multiple commutators of $H_0$ and $phi$ are uniformly bounded in\u0000operator norm up to the $(n+1)$-th order. We then apply the abstract theory to\u0000a class of nonlocal Schr\"odinger equations on $mathbb{R}^d$, proving that any\u0000solution with compactly supported initial state remains approximately\u0000supported, up to a polynomially suppressed tail in $L^2$-norm, inside a\u0000linearly spreading region around the initial support for all $tne0$.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250407","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 show that, given a real or complex hyperbolic metric $g_0$ on a closed manifold $M$ of dimension $ngeq 3$, there exists a neighborhood $mathcal U$ of $g_0$ in the space of negatively curved metrics such that for any $gin mathcal U$, the topological entropy and Liouville entropy of $g$ coincide if and only if $g$ and $g_0$ are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result of the hyperbolic rank near complex hyperbolic metrics.
{"title":"Katok's entropy conjecture near real and complex hyperbolic metrics","authors":"Tristan Humbert","doi":"arxiv-2409.11197","DOIUrl":"https://doi.org/arxiv-2409.11197","url":null,"abstract":"We show that, given a real or complex hyperbolic metric $g_0$ on a closed\u0000manifold $M$ of dimension $ngeq 3$, there exists a neighborhood $mathcal U$\u0000of $g_0$ in the space of negatively curved metrics such that for any $gin\u0000mathcal U$, the topological entropy and Liouville entropy of $g$ coincide if\u0000and only if $g$ and $g_0$ are homothetic. This provides a partial answer to\u0000Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we\u0000obtain a local rigidity result of the hyperbolic rank near complex hyperbolic\u0000metrics.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250363","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}
Sergey Bezuglyi, Palle E. T. Jorgensen, Olena Karpel, Jan Kwiatkowski
Bratteli diagrams with countably infinite levels exhibit a new phenomenon: they can be horizontally stationary. The incidence matrices of these horizontally stationary Bratteli diagrams are infinite banded Toeplitz matrices. In this paper, we study the fundamental properties of horizontally stationary Bratteli diagrams. In these diagrams, we provide an explicit description of ergodic tail invariant probability measures. For a certain class of horizontally stationary Bratteli diagrams, we prove that all ergodic tail invariant probability measures are extensions of measures from odometers. Additionally, we establish conditions for the existence of a continuous Vershik map on the path space of a horizontally stationary Bratteli diagram.
{"title":"Horizontally stationary generalized Bratteli diagrams","authors":"Sergey Bezuglyi, Palle E. T. Jorgensen, Olena Karpel, Jan Kwiatkowski","doi":"arxiv-2409.10084","DOIUrl":"https://doi.org/arxiv-2409.10084","url":null,"abstract":"Bratteli diagrams with countably infinite levels exhibit a new phenomenon:\u0000they can be horizontally stationary. The incidence matrices of these\u0000horizontally stationary Bratteli diagrams are infinite banded Toeplitz\u0000matrices. In this paper, we study the fundamental properties of horizontally\u0000stationary Bratteli diagrams. In these diagrams, we provide an explicit\u0000description of ergodic tail invariant probability measures. For a certain class\u0000of horizontally stationary Bratteli diagrams, we prove that all ergodic tail\u0000invariant probability measures are extensions of measures from odometers.\u0000Additionally, we establish conditions for the existence of a continuous Vershik\u0000map on the path space of a horizontally stationary Bratteli diagram.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250410","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 propose a delayed cytokine-enhanced viral infection model incorporating saturation incidence and immune response. We compute the basic reproduction numbers and introduce a convex cone to discuss the impact of non-negative initial data on solutions. By defining appropriate Lyapunov functionals and employing LaSalle's invariance principle, we investigate the stability of three equilibria: the disease-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium. We establish conditions under which these equilibria are globally asymptotically stable. Numerical analyses not only corroborate the theoretical results but also reveal that intervention in virus infection can be achieved by extending the delay period.
{"title":"A cytokine-enhanced viral infection model with CTL immune response, distributed delay and saturation incidence","authors":"Xiaodong Cao, Songbo Hou, Xiaoqing Kong","doi":"arxiv-2409.10223","DOIUrl":"https://doi.org/arxiv-2409.10223","url":null,"abstract":"In this paper, we propose a delayed cytokine-enhanced viral infection model\u0000incorporating saturation incidence and immune response. We compute the basic\u0000reproduction numbers and introduce a convex cone to discuss the impact of\u0000non-negative initial data on solutions. By defining appropriate Lyapunov\u0000functionals and employing LaSalle's invariance principle, we investigate the\u0000stability of three equilibria: the disease-free equilibrium, the\u0000immunity-inactivated equilibrium, and the immunity-activated equilibrium. We\u0000establish conditions under which these equilibria are globally asymptotically\u0000stable. Numerical analyses not only corroborate the theoretical results but\u0000also reveal that intervention in virus infection can be achieved by extending\u0000the delay period.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250409","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}
This article presents a new scheme for studying the dynamics of a quintic wave equation with nonlocal weak damping in a 3D smooth bounded domain. As an application, the existence and structure of weak, strong, and exponential attractors for the solution semigroup of this equation are obtained. The investigation sheds light on the well-posedness and long-time behavior of nonlinear dissipative evolution equations with nonlinear damping and critical nonlinearity.
{"title":"Dynamics of the quintic wave equation with nonlocal weak damping","authors":"Feng Zhou, Hongfang Li, Kaixuan Zhu, Xinyu Mei","doi":"arxiv-2409.10035","DOIUrl":"https://doi.org/arxiv-2409.10035","url":null,"abstract":"This article presents a new scheme for studying the dynamics of a quintic\u0000wave equation with nonlocal weak damping in a 3D smooth bounded domain. As an\u0000application, the existence and structure of weak, strong, and exponential\u0000attractors for the solution semigroup of this equation are obtained. The\u0000investigation sheds light on the well-posedness and long-time behavior of\u0000nonlinear dissipative evolution equations with nonlinear damping and critical\u0000nonlinearity.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"143 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250412","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}
Let $G$ be a discrete, countably infinite group and $H$ a subgroup of $G$. If $H$ acts continuously on a compact metric space $X$, then we can induce a continuous action of $G$ on $prod_{Hbackslash G}X$ where $Hbackslash G$ is the collection of right-cosets of $H$ in $G$. This process is known as the co-induction. In this article, we will calculate the maximal pattern entropy of the co-induction. If $[G:H] < +infty$ we will show that the $H$ action is null if and only if the co-induced action of $G$ is null. Also, we will discuss an example where $H$ is a proper subgroup of $G$ with finite index where the maximal pattern entropy of the $H$ action is equal to the co-induced action of $G$. If $[G:H] = +infty$ we will show that the maximal pattern entropy of the co-induction is always $+infty$ given the $H$-system is not trivial.
{"title":"Topological Sequence Entropy of co-Induced Systems","authors":"Dakota M. Leonard","doi":"arxiv-2409.10745","DOIUrl":"https://doi.org/arxiv-2409.10745","url":null,"abstract":"Let $G$ be a discrete, countably infinite group and $H$ a subgroup of $G$. If\u0000$H$ acts continuously on a compact metric space $X$, then we can induce a\u0000continuous action of $G$ on $prod_{Hbackslash G}X$ where $Hbackslash G$ is\u0000the collection of right-cosets of $H$ in $G$. This process is known as the\u0000co-induction. In this article, we will calculate the maximal pattern entropy of\u0000the co-induction. If $[G:H] < +infty$ we will show that the $H$ action is null\u0000if and only if the co-induced action of $G$ is null. Also, we will discuss an\u0000example where $H$ is a proper subgroup of $G$ with finite index where the\u0000maximal pattern entropy of the $H$ action is equal to the co-induced action of\u0000$G$. If $[G:H] = +infty$ we will show that the maximal pattern entropy of the\u0000co-induction is always $+infty$ given the $H$-system is not trivial.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250406","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 Kron reduction is used in power grid modeling when the analysis can -- supposedly -- be restricted to a subset of nodes. Typically, when one is interested in the phases' dynamics, it is common to reduce the load buses and focus on the generators' behavior. The rationale behind this reduction is that voltage phases at load buses adapt quickly to their neighbors' phases and, at the timescale of generators, they have virtually no dynamics. We show that the dynamics of the Kron-reduced part of a network can have a significant impact on the dynamics of the non-reduced buses. Therefore, Kron reduction should be used with care and, depending on the context, reduced nodes cannot be simply ignored. We demonstrate that the noise in the reduced part can unexpectedly affect the non-reduced part, even under the assumption that nodal disturbances are independent. Therefore, the common assumption that the noise in the non-reduced part is uncorrelated may lead to inaccurate assessments of the grid's behavior. To cope with such shortcomings of the Kron reduction, we show how to properly incorporate the contribution of the reduced buses into the reduced model using the Mori-Zwanzig formalism.
{"title":"Nontrivial Kron Reduction for Power Grid Dynamics Modeling","authors":"Laurent Pagnier, Robin Delabays, Melvyn Tyloo","doi":"arxiv-2409.09519","DOIUrl":"https://doi.org/arxiv-2409.09519","url":null,"abstract":"The Kron reduction is used in power grid modeling when the analysis can --\u0000supposedly -- be restricted to a subset of nodes. Typically, when one is\u0000interested in the phases' dynamics, it is common to reduce the load buses and\u0000focus on the generators' behavior. The rationale behind this reduction is that\u0000voltage phases at load buses adapt quickly to their neighbors' phases and, at\u0000the timescale of generators, they have virtually no dynamics. We show that the\u0000dynamics of the Kron-reduced part of a network can have a significant impact on\u0000the dynamics of the non-reduced buses. Therefore, Kron reduction should be used\u0000with care and, depending on the context, reduced nodes cannot be simply\u0000ignored. We demonstrate that the noise in the reduced part can unexpectedly\u0000affect the non-reduced part, even under the assumption that nodal disturbances\u0000are independent. Therefore, the common assumption that the noise in the\u0000non-reduced part is uncorrelated may lead to inaccurate assessments of the\u0000grid's behavior. To cope with such shortcomings of the Kron reduction, we show\u0000how to properly incorporate the contribution of the reduced buses into the\u0000reduced model using the Mori-Zwanzig formalism.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250411","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}
Hibiki Kato, Miki U Kobayashi, Yoshitaka Saiki, James A. Yorke
Intermittent switchings between weakly chaotic (laminar) and strongly chaotic (bursty) states are often observed in systems with high-dimensional chaotic attractors, such as fluid turbulence. They differ from the intermittency of a low-dimensional system accompanied by the stability change of a fixed point or a periodic orbit in that the intermittency of a high-dimensional system tends to appear in a wide range of parameters. This paper considers a case where the skeleton of a laminar state $L$ exists as a proper chaotic subset $S$ of a chaotic attractor $X$, that is, $S subsetneq X$. We characterize such a laminar state $L$ by a chaotic saddle $S$, which is densely filled with periodic orbits of different numbers of unstable directions. This study demonstrates the presence of chaotic saddles underlying intermittency in fluid turbulence and phase synchronization. Furthermore, we confirm that chaotic saddles persist for a wide range of parameters. Also, a kind of phase synchronization turns out to occur in the turbulent model.
{"title":"A laminar chaotic saddle within a turbulent attractor","authors":"Hibiki Kato, Miki U Kobayashi, Yoshitaka Saiki, James A. Yorke","doi":"arxiv-2409.08870","DOIUrl":"https://doi.org/arxiv-2409.08870","url":null,"abstract":"Intermittent switchings between weakly chaotic (laminar) and strongly chaotic\u0000(bursty) states are often observed in systems with high-dimensional chaotic\u0000attractors, such as fluid turbulence. They differ from the intermittency of a\u0000low-dimensional system accompanied by the stability change of a fixed point or\u0000a periodic orbit in that the intermittency of a high-dimensional system tends\u0000to appear in a wide range of parameters. This paper considers a case where the\u0000skeleton of a laminar state $L$ exists as a proper chaotic subset $S$ of a\u0000chaotic attractor $X$, that is, $S subsetneq X$. We characterize such a\u0000laminar state $L$ by a chaotic saddle $S$, which is densely filled with\u0000periodic orbits of different numbers of unstable directions. This study\u0000demonstrates the presence of chaotic saddles underlying intermittency in fluid\u0000turbulence and phase synchronization. Furthermore, we confirm that chaotic\u0000saddles persist for a wide range of parameters. Also, a kind of phase\u0000synchronization turns out to occur in the turbulent model.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250417","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}
Anton-David Almasan, Sergey Shvydun, Ingo Scholtes, Piet Van Mieghem
We study human mobility networks through timeseries of contacts between individuals. Our proposed Random Walkers Induced temporal Graph (RWIG) model generates temporal graph sequences based on independent random walkers that traverse an underlying graph in discrete time steps. Co-location of walkers at a given node and time defines an individual-level contact. RWIG is shown to be a realistic model for temporal human contact graphs, which may place RWIG on a same footing as the Erdos-Renyi (ER) and Barabasi-Albert (BA) models for fixed graphs. Moreover, RWIG is analytically feasible: we derive closed form solutions for the probability distribution of contact graphs.
{"title":"Generating Temporal Contact Graphs Using Random Walkers","authors":"Anton-David Almasan, Sergey Shvydun, Ingo Scholtes, Piet Van Mieghem","doi":"arxiv-2409.08690","DOIUrl":"https://doi.org/arxiv-2409.08690","url":null,"abstract":"We study human mobility networks through timeseries of contacts between\u0000individuals. Our proposed Random Walkers Induced temporal Graph (RWIG) model\u0000generates temporal graph sequences based on independent random walkers that\u0000traverse an underlying graph in discrete time steps. Co-location of walkers at\u0000a given node and time defines an individual-level contact. RWIG is shown to be\u0000a realistic model for temporal human contact graphs, which may place RWIG on a\u0000same footing as the Erdos-Renyi (ER) and Barabasi-Albert (BA) models for fixed\u0000graphs. Moreover, RWIG is analytically feasible: we derive closed form\u0000solutions for the probability distribution of contact graphs.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250419","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}
Guilherme S. Costa, Marcel Novaes, Marcus A. M. de Aguiar
Synchronization is an important phenomenon in a wide variety of systems comprising interacting oscillatory units, whether natural (like neurons, biochemical reactions, cardiac cells) or artificial (like metronomes, power grids, Josephson junctions). The Kuramoto model provides a simple description of these systems and has been useful in their mathematical exploration. Here we investigate this model in the presence of two characteristics that may be important in applications: an external periodic influence and higher-order interactions among the units. The combination of these ingredients leads to a very rich bifurcation scenario in the dynamics of the order parameter that describes phase transitions. Our theoretical calculations are validated by numerical simulations.
{"title":"Bifurcations in the Kuramoto model with external forcing and higher-order interactions","authors":"Guilherme S. Costa, Marcel Novaes, Marcus A. M. de Aguiar","doi":"arxiv-2409.08736","DOIUrl":"https://doi.org/arxiv-2409.08736","url":null,"abstract":"Synchronization is an important phenomenon in a wide variety of systems\u0000comprising interacting oscillatory units, whether natural (like neurons,\u0000biochemical reactions, cardiac cells) or artificial (like metronomes, power\u0000grids, Josephson junctions). The Kuramoto model provides a simple description\u0000of these systems and has been useful in their mathematical exploration. Here we\u0000investigate this model in the presence of two characteristics that may be\u0000important in applications: an external periodic influence and higher-order\u0000interactions among the units. The combination of these ingredients leads to a\u0000very rich bifurcation scenario in the dynamics of the order parameter that\u0000describes phase transitions. Our theoretical calculations are validated by\u0000numerical simulations.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250418","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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