We consider a random walk on a homogeneous space $G/Lambda$ where $G$ is $mathrm{SO}(2,1)$ or $mathrm{SO}(3,1)$ and $Lambda$ is a lattice. The walk is driven by a probability measure $mu$ on $G$ whose support generates a Zariski-dense subgroup. We show that for every starting point $x in G/Lambda$ which is not trapped in a finite $mu$-invariant set, the $n$-step distribution $mu^{*n}*delta_{x}$ of the walk equidistributes toward the Haar measure. Moreover, under arithmetic assumptions on the pair $(Lambda, mu)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that $x$ may be high in a cusp or close to a finite orbit. Our approach is substantially different from that of Benoist-Quint, whose equidistribution statements only hold in Ces`aro average and are not quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes concerning the torus case, and that of Lindenstrauss-Mohammadi-Wang and Yang about the analogous problem for unipotent flows. A key new feature of our proof is the use of a new phenomenon which we call multislicing. The latter is a generalization of the discretized projection theorems `a la Bourgain and we believe it presents independent interest.
{"title":"Multislicing and effective equidistribution for random walks on some homogeneous spaces","authors":"Timothée Bénard, Weikun He","doi":"arxiv-2409.03300","DOIUrl":"https://doi.org/arxiv-2409.03300","url":null,"abstract":"We consider a random walk on a homogeneous space $G/Lambda$ where $G$ is\u0000$mathrm{SO}(2,1)$ or $mathrm{SO}(3,1)$ and $Lambda$ is a lattice. The walk\u0000is driven by a probability measure $mu$ on $G$ whose support generates a\u0000Zariski-dense subgroup. We show that for every starting point $x in G/Lambda$\u0000which is not trapped in a finite $mu$-invariant set, the $n$-step distribution\u0000$mu^{*n}*delta_{x}$ of the walk equidistributes toward the Haar measure.\u0000Moreover, under arithmetic assumptions on the pair $(Lambda, mu)$, we show\u0000the convergence occurs at an exponential rate, tempered by the obstructions\u0000that $x$ may be high in a cusp or close to a finite orbit. Our approach is substantially different from that of Benoist-Quint, whose\u0000equidistribution statements only hold in Ces`aro average and are not\u0000quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes concerning the torus\u0000case, and that of Lindenstrauss-Mohammadi-Wang and Yang about the analogous\u0000problem for unipotent flows. A key new feature of our proof is the use of a new\u0000phenomenon which we call multislicing. The latter is a generalization of the\u0000discretized projection theorems `a la Bourgain and we believe it presents\u0000independent interest.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195901","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 prove Ruelle's inequality for the geodesic flow in non-compact manifolds with Anosov geodesic flow and some assumptions on the curvature. In the same way, we obtain Pesin's formula for Anosov geodesic flow in non-compact manifolds with finite volume.
{"title":"Ruelle's inequality and Pesin's formula for Anosov geodesic flows in non-compact manifolds","authors":"Alexander Cantoral, Sergio Romaña","doi":"arxiv-2409.03207","DOIUrl":"https://doi.org/arxiv-2409.03207","url":null,"abstract":"In this paper, we prove Ruelle's inequality for the geodesic flow in\u0000non-compact manifolds with Anosov geodesic flow and some assumptions on the\u0000curvature. In the same way, we obtain Pesin's formula for Anosov geodesic flow\u0000in non-compact manifolds with finite volume.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195903","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 effects of noise on memory in a linear recurrent network are theoretically investigated. Memory is characterized by its ability to store previous inputs in its instantaneous state of network, which receives a correlated or uncorrelated noise. Two major properties are revealed: First, the memory reduced by noise is uniquely determined by the noise's power spectral density (PSD). Second, the memory will not decrease regardless of noise intensity if the PSD is in a certain class of distribution (including power law). The results are verified using the human brain signals, showing good agreement.
{"title":"How noise affects memory in linear recurrent networks","authors":"JingChuan Guan, Tomoyuki Kubota, Yasuo Kuniyoshi, Kohei Nakajima","doi":"arxiv-2409.03187","DOIUrl":"https://doi.org/arxiv-2409.03187","url":null,"abstract":"The effects of noise on memory in a linear recurrent network are\u0000theoretically investigated. Memory is characterized by its ability to store\u0000previous inputs in its instantaneous state of network, which receives a\u0000correlated or uncorrelated noise. Two major properties are revealed: First, the\u0000memory reduced by noise is uniquely determined by the noise's power spectral\u0000density (PSD). Second, the memory will not decrease regardless of noise\u0000intensity if the PSD is in a certain class of distribution (including power\u0000law). The results are verified using the human brain signals, showing good\u0000agreement.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195908","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 work, we introduce a novel concept of magic billiard games and analyse their properties in the case of elliptical boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic, and polynomial forms. A topological description of those billiards is given using Fomenko graphs.
{"title":"Magic Billiards: the Case of Elliptical Boundaries","authors":"Vladimir Dragović, Milena Radnović","doi":"arxiv-2409.03158","DOIUrl":"https://doi.org/arxiv-2409.03158","url":null,"abstract":"In this work, we introduce a novel concept of magic billiard games and\u0000analyse their properties in the case of elliptical boundaries. We provide\u0000explicit conditions for periodicity in algebro-geometric, analytic, and\u0000polynomial forms. A topological description of those billiards is given using\u0000Fomenko graphs.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195902","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 introduce Lyapunov-type results to investigate the stability of the trivial solution of a Stieltjes dynamical system. We utilize prolongation results to establish the global existence of the maximal solution. Using Lyapunov's second method, we establish results of (uniform) stability and (uniform) asymptotic stability by employing a Lyapunov function. Additionally, we present examples and real-life applications to study asymptotic stability of equilibria in two population dynamics models.
{"title":"Prolongation of solutions and Lyapunov stability for Stieltjes dynamical systems","authors":"Lamiae Maia, Noha El Khattabi, Marlène Frigon","doi":"arxiv-2409.03408","DOIUrl":"https://doi.org/arxiv-2409.03408","url":null,"abstract":"In this article, we introduce Lyapunov-type results to investigate the\u0000stability of the trivial solution of a Stieltjes dynamical system. We utilize\u0000prolongation results to establish the global existence of the maximal solution.\u0000Using Lyapunov's second method, we establish results of (uniform) stability and\u0000(uniform) asymptotic stability by employing a Lyapunov function. Additionally,\u0000we present examples and real-life applications to study asymptotic stability of\u0000equilibria in two population dynamics models.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195907","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}
Steady states of the Swift--Hohenberg equation are studied. For the associated four--dimensional ODE we prove that on the energy level $E=0$ two smooth branches of even periodic solutions are created through the saddle-node bifurcation. We also show that these orbits satisfy certain geometric properties, which implies that the system has positive topological entropy for an explicit and wide range of parameter values of the system. The proof is computer-assisted and it uses rigorous computation of bounds on certain Poincar'e map and its higher order derivatives.
{"title":"Continuation and bifurcations of periodic orbits and symbolic dynamics in the Swift-Hohenberg equation","authors":"Jakub Czwórnóg, Daniel Wilczak","doi":"arxiv-2409.03036","DOIUrl":"https://doi.org/arxiv-2409.03036","url":null,"abstract":"Steady states of the Swift--Hohenberg equation are studied. For the\u0000associated four--dimensional ODE we prove that on the energy level $E=0$ two\u0000smooth branches of even periodic solutions are created through the saddle-node\u0000bifurcation. We also show that these orbits satisfy certain geometric\u0000properties, which implies that the system has positive topological entropy for\u0000an explicit and wide range of parameter values of the system. The proof is computer-assisted and it uses rigorous computation of bounds on\u0000certain Poincar'e map and its higher order derivatives.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195906","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 study a system of reaction-diffusion equations in which the diffusivities are widely separated. We report on the discovery of families of spatially periodic canard solutions that emerge from {em singular Turing bifurcations}. The emergence of these spatially periodic canards asymptotically close to the Turing bifurcations, which are reversible 1:1 resonant Hopf bifurcations in the spatial ODE system, is an analog in spatial dynamics of the emergence of limit cycle canards in the canard explosions that occur asymptotically close to Hopf bifurcations in time-dependent ODEs. In the full PDE system, we show that for most parameter values under study the Turing bifurcation is sub-critical, and we present the results of some direct numerical simulations showing that several of the different types of spatial canard patterns are attractors in the prototypical PDE. To support the numerical discoveries, we use geometric desingularization and geometric singular perturbation theory to demonstrate the existence of these families of spatially periodic canards. Crucially, in the singular limit, we study a novel class of {em reversible folded singularities}. In particular, there are two reversible folded saddle-node bifurcations of type II (RFSN-II), each occurring asymptotically close to a Turing bifurcation. We derive analytical formulas for these singularities and show that their canards play key roles in the observed families of spatially periodic canard solutions. Then, for an interval of values of the bifurcation parameter further below the Turing bifurcation and RFSN-II point, the spatial ODE also has spatially periodic canard patterns, however these are created by a reversible folded saddle (instead of the RFSN-II). It also turns out that there is an interesting scale invariance, so that some components of some spatial canards exhibit nearly self-similar dynamics.
{"title":"Les Canards de Turing","authors":"Theodore Vo, Arjen Doelman, Tasso J. Kaper","doi":"arxiv-2409.02400","DOIUrl":"https://doi.org/arxiv-2409.02400","url":null,"abstract":"In this article, we study a system of reaction-diffusion equations in which\u0000the diffusivities are widely separated. We report on the discovery of families\u0000of spatially periodic canard solutions that emerge from {em singular Turing\u0000bifurcations}. The emergence of these spatially periodic canards asymptotically\u0000close to the Turing bifurcations, which are reversible 1:1 resonant Hopf\u0000bifurcations in the spatial ODE system, is an analog in spatial dynamics of the\u0000emergence of limit cycle canards in the canard explosions that occur\u0000asymptotically close to Hopf bifurcations in time-dependent ODEs. In the full\u0000PDE system, we show that for most parameter values under study the Turing\u0000bifurcation is sub-critical, and we present the results of some direct\u0000numerical simulations showing that several of the different types of spatial\u0000canard patterns are attractors in the prototypical PDE. To support the numerical discoveries, we use geometric desingularization and\u0000geometric singular perturbation theory to demonstrate the existence of these\u0000families of spatially periodic canards. Crucially, in the singular limit, we\u0000study a novel class of {em reversible folded singularities}. In particular,\u0000there are two reversible folded saddle-node bifurcations of type II (RFSN-II),\u0000each occurring asymptotically close to a Turing bifurcation. We derive\u0000analytical formulas for these singularities and show that their canards play\u0000key roles in the observed families of spatially periodic canard solutions.\u0000Then, for an interval of values of the bifurcation parameter further below the\u0000Turing bifurcation and RFSN-II point, the spatial ODE also has spatially\u0000periodic canard patterns, however these are created by a reversible folded\u0000saddle (instead of the RFSN-II). It also turns out that there is an interesting\u0000scale invariance, so that some components of some spatial canards exhibit\u0000nearly self-similar dynamics.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195909","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}
Noise-induced phenomena in high-dimensional dynamical systems were investigated from a random dynamical systems point of view. In a class of generalized H'enon maps, which are randomly perturbed delayed logistic maps, with monotonically increasing noise levels, we observed (i) an increase in the number of positive Lyapunov exponents from 4 to 5, and the emergence of characteristic periods at the same time, and (ii) a decrease in the number of positive Lyapunov exponents from 4 to 3, and an increase in Kolmogorov--Sinai entropy at the same time. Our results imply that simple concepts of noise-induced phenomena, such as noise-induced chaos and/or noise-induced order, may not describe those analogue in high dimensional dynamical systems, owing to coexistence of noise-induced chaos and noise-induced order.
{"title":"Noise-induced order in high dimensions","authors":"Huayan Chen, Yuzuru Sato","doi":"arxiv-2409.02498","DOIUrl":"https://doi.org/arxiv-2409.02498","url":null,"abstract":"Noise-induced phenomena in high-dimensional dynamical systems were\u0000investigated from a random dynamical systems point of view. In a class of\u0000generalized H'enon maps, which are randomly perturbed delayed logistic maps,\u0000with monotonically increasing noise levels, we observed (i) an increase in the\u0000number of positive Lyapunov exponents from 4 to 5, and the emergence of\u0000characteristic periods at the same time, and (ii) a decrease in the number of\u0000positive Lyapunov exponents from 4 to 3, and an increase in Kolmogorov--Sinai\u0000entropy at the same time. Our results imply that simple concepts of\u0000noise-induced phenomena, such as noise-induced chaos and/or noise-induced\u0000order, may not describe those analogue in high dimensional dynamical systems,\u0000owing to coexistence of noise-induced chaos and noise-induced order.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195913","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}
Kai Chenga, Iason Papaioannoua, MengZe Lyub, Daniel Straub
We present a new surrogate model for emulating the behavior of complex nonlinear dynamical systems with external stochastic excitation. The model represents the system dynamics in state space form through a sparse Kriging model. The resulting surrogate model is termed state space Kriging (S2K) model. Sparsity in the Kriging model is achieved by selecting an informative training subset from the observed time histories of the state vector and its derivative with respect to time. We propose a tailored technique for designing the training time histories of state vector and its derivative, aimed at enhancing the robustness of the S2K prediction. We validate the performance of the S2K model with various benchmarks. The results show that S2K yields accurate prediction of complex nonlinear dynamical systems under stochastic excitation with only a few training time histories of state vector.
{"title":"State Space Kriging model for emulating complex nonlinear dynamical systems under stochastic excitation","authors":"Kai Chenga, Iason Papaioannoua, MengZe Lyub, Daniel Straub","doi":"arxiv-2409.02462","DOIUrl":"https://doi.org/arxiv-2409.02462","url":null,"abstract":"We present a new surrogate model for emulating the behavior of complex\u0000nonlinear dynamical systems with external stochastic excitation. The model\u0000represents the system dynamics in state space form through a sparse Kriging\u0000model. The resulting surrogate model is termed state space Kriging (S2K) model.\u0000Sparsity in the Kriging model is achieved by selecting an informative training\u0000subset from the observed time histories of the state vector and its derivative\u0000with respect to time. We propose a tailored technique for designing the\u0000training time histories of state vector and its derivative, aimed at enhancing\u0000the robustness of the S2K prediction. We validate the performance of the S2K\u0000model with various benchmarks. The results show that S2K yields accurate\u0000prediction of complex nonlinear dynamical systems under stochastic excitation\u0000with only a few training time histories of state vector.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195910","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 describe a mechanism for transport of energy in a mechanical system consisting of a pendulum and a rotator subject to a random perturbation. The perturbation that we consider is the product of a Hamiltonian vector field and a scalar, continuous, stationary Gaussian process with H"older continuous realizations, scaled by a smallness parameter. We show that for almost every realization of the stochastic process, there is a distinguished set of times for which there exists a random normally hyperbolic invariant manifold with associated stable and unstable manifolds that intersect transversally, for all sufficiently small values of the smallness parameter. We derive the existence of orbits along which the energy changes over time by an amount proportional to the smallness parameter. This result is related to the Arnold diffusion problem for Hamiltonian systems, which we treat here in the random setting.
{"title":"Energy Transport in Random Perturbations of Mechanical Systems","authors":"Anna Maria Cherubini, Marian Gidea","doi":"arxiv-2409.03132","DOIUrl":"https://doi.org/arxiv-2409.03132","url":null,"abstract":"We describe a mechanism for transport of energy in a mechanical system\u0000consisting of a pendulum and a rotator subject to a random perturbation. The\u0000perturbation that we consider is the product of a Hamiltonian vector field and\u0000a scalar, continuous, stationary Gaussian process with H\"older continuous\u0000realizations, scaled by a smallness parameter. We show that for almost every\u0000realization of the stochastic process, there is a distinguished set of times\u0000for which there exists a random normally hyperbolic invariant manifold with\u0000associated stable and unstable manifolds that intersect transversally, for all\u0000sufficiently small values of the smallness parameter. We derive the existence\u0000of orbits along which the energy changes over time by an amount proportional to\u0000the smallness parameter. This result is related to the Arnold diffusion problem\u0000for Hamiltonian systems, which we treat here in the random setting.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195904","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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