In this paper we firstly review how to textit{explicitly} solve a system of $3$ textit{first-order linear recursions }and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of $3$ textit{first-order nonlinear recursions} which also are textit{explicitly solvable}. These systems might be of interest for practitioners in textit{applied} sciences: they allow a complete display of their solutions, which may feature interesting behaviors, for instance be textit{completely periodic} ("isochronous systems", if the independent variable $n=0,1,2,3...$is considered a textit{ticking time}), or feature this property textit{only asymptotically} (astextit{ }$nrightarrow infty $).
{"title":"Interesting system of $3$ first-order recursions","authors":"Francesco Calogero","doi":"arxiv-2409.05074","DOIUrl":"https://doi.org/arxiv-2409.05074","url":null,"abstract":"In this paper we firstly review how to textit{explicitly} solve a system of\u0000$3$ textit{first-order linear recursions }and outline the main properties of\u0000these solutions. Next, via a change of variables, we identify a class of\u0000systems of $3$ textit{first-order nonlinear recursions} which also are\u0000textit{explicitly solvable}. These systems might be of interest for\u0000practitioners in textit{applied} sciences: they allow a complete display of\u0000their solutions, which may feature interesting behaviors, for instance be\u0000textit{completely periodic} (\"isochronous systems\", if the independent\u0000variable $n=0,1,2,3...$is considered a textit{ticking time}), or feature this\u0000property textit{only asymptotically} (astextit{ }$nrightarrow infty $).","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195898","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}
Ahmed Amr Morsy, Zhenwei Xu, Paolo Tiso, George Haller
Bolted joints can exhibit significantly nonlinear dynamics. Finite Element Models (FEMs) of this phenomenon require fine spatial discretizations, inclusion of nonlinear contact and friction laws, as well as geometric nonlinearity. Owing to the nonlinearity and high dimensionality of such models, full-order dynamic simulations are computationally expensive. In this work, we use the theory of Spectral Submanifolds (SSMs) to construct FEM-based data-driven Reduced Order Models (ROMs). The data used for constructing the model consists of a few transient trajectories of the full unforced system. Using this data, we obtain an SSM-reduced model that also predicts the forced nonlinear dynamics. We illustrate the method on a 187,920-dimensional FEM of the recent 2021 Tribomechadynamics benchmark structure. In this case, the SSM-based ROM is a 4-dimensional model that captures the internal resonance of the structure. The SSM-reduced model gives fast and accurate predictions of the experimental forced dynamics and allows to reproduce the local friction and contact stresses on the interfaces of the joint.
{"title":"Reducing Finite Element Models of Bolted Joints using Spectral Submanifolds","authors":"Ahmed Amr Morsy, Zhenwei Xu, Paolo Tiso, George Haller","doi":"arxiv-2409.05012","DOIUrl":"https://doi.org/arxiv-2409.05012","url":null,"abstract":"Bolted joints can exhibit significantly nonlinear dynamics. Finite Element\u0000Models (FEMs) of this phenomenon require fine spatial discretizations,\u0000inclusion of nonlinear contact and friction laws, as well as geometric\u0000nonlinearity. Owing to the nonlinearity and high dimensionality of such models,\u0000full-order dynamic simulations are computationally expensive. In this work, we\u0000use the theory of Spectral Submanifolds (SSMs) to construct FEM-based\u0000data-driven Reduced Order Models (ROMs). The data used for constructing the\u0000model consists of a few transient trajectories of the full unforced system.\u0000Using this data, we obtain an SSM-reduced model that also predicts the forced\u0000nonlinear dynamics. We illustrate the method on a 187,920-dimensional FEM of\u0000the recent 2021 Tribomechadynamics benchmark structure. In this case, the\u0000SSM-based ROM is a 4-dimensional model that captures the internal resonance of\u0000the structure. The SSM-reduced model gives fast and accurate predictions of the\u0000experimental forced dynamics and allows to reproduce the local friction and\u0000contact stresses on the interfaces of the joint.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195873","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}
Many motile microorganisms and bio-mimetic micro-particles have been successfully modeled as active rods - elongated bodies capable of self-propulsion. A hallmark of active rod dynamics under confinement is their tendency to accumulate at the walls. Unlike passive particles, which typically sediment and cease their motion at the wall, accumulated active rods continue to move along the wall, reorient, and may even escape from it. The dynamics of active rods at the wall and those away from it result in complex and non-trivial distributions. In this work, we examine the effects of wall curvature on active rod distribution by studying elliptical perturbations of tube-like microchannels, that is, the cylindrical confinement with a circular cross-section, common in both nature and various applications. By developing a computational model for individual active rods and conducting Monte Carlo simulations, we discovered that active rods tend to concentrate at locations with the highest wall curvature. We then investigated how the distribution of active rod accumulation depends on the background flow and orientation diffusion. Finally, we used a simplified mathematical model to explain why active rods preferentially accumulate at high-curvature locations.
{"title":"Boundary accumulations of active rods in microchannels with elliptical cross-section","authors":"Chase Brown, Mykhailo Potomkin, Shawn Ryan","doi":"arxiv-2409.04950","DOIUrl":"https://doi.org/arxiv-2409.04950","url":null,"abstract":"Many motile microorganisms and bio-mimetic micro-particles have been\u0000successfully modeled as active rods - elongated bodies capable of\u0000self-propulsion. A hallmark of active rod dynamics under confinement is their\u0000tendency to accumulate at the walls. Unlike passive particles, which typically\u0000sediment and cease their motion at the wall, accumulated active rods continue\u0000to move along the wall, reorient, and may even escape from it. The dynamics of\u0000active rods at the wall and those away from it result in complex and\u0000non-trivial distributions. In this work, we examine the effects of wall\u0000curvature on active rod distribution by studying elliptical perturbations of\u0000tube-like microchannels, that is, the cylindrical confinement with a circular\u0000cross-section, common in both nature and various applications. By developing a\u0000computational model for individual active rods and conducting Monte Carlo\u0000simulations, we discovered that active rods tend to concentrate at locations\u0000with the highest wall curvature. We then investigated how the distribution of\u0000active rod accumulation depends on the background flow and orientation\u0000diffusion. Finally, we used a simplified mathematical model to explain why\u0000active rods preferentially accumulate at high-curvature locations.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195897","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 addendum presents a relevant stronger consequence of the main theorem of the paper "Higher order stroboscopic averaged functions: a general relationship with Melnikov functions" [arXiv:2011.03663], EJQTDE No. 77 (2021).
{"title":"Addendum to Higher order stroboscopic averaged functions: a general relationship with Melnikov functions","authors":"Douglas D. Novaes","doi":"arxiv-2409.05912","DOIUrl":"https://doi.org/arxiv-2409.05912","url":null,"abstract":"This addendum presents a relevant stronger consequence of the main theorem of\u0000the paper \"Higher order stroboscopic averaged functions: a general relationship\u0000with Melnikov functions\" [arXiv:2011.03663], EJQTDE No. 77 (2021).","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195870","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 investigate the possibility that for any given reaction rate vector $k$ associated with a network $G$, there exists another network $G'$ with a corresponding reaction rate vector that reproduces the mass-action dynamics generated by $(G,k)$. Our focus is on a particular class of networks for $G$, where the corresponding network $G'$ is weakly reversible. In particular, we show that strongly endotactic two-dimensional networks with a two dimensional stoichiometric subspace, as well as certain endotactic networks under additional conditions, exhibit this property. Additionally, we establish a strong connection between this family of networks and the locus in the space of rate constants of which the corresponding dynamics admits globally stable steady states.
{"title":"Realizations through Weakly Reversible Networks and the Globally Attracting Locus","authors":"Samay Kothari, Jiaxin Jin, Abhishek Deshpande","doi":"arxiv-2409.04802","DOIUrl":"https://doi.org/arxiv-2409.04802","url":null,"abstract":"We investigate the possibility that for any given reaction rate vector $k$\u0000associated with a network $G$, there exists another network $G'$ with a\u0000corresponding reaction rate vector that reproduces the mass-action dynamics\u0000generated by $(G,k)$. Our focus is on a particular class of networks for $G$,\u0000where the corresponding network $G'$ is weakly reversible. In particular, we\u0000show that strongly endotactic two-dimensional networks with a two dimensional\u0000stoichiometric subspace, as well as certain endotactic networks under\u0000additional conditions, exhibit this property. Additionally, we establish a\u0000strong connection between this family of networks and the locus in the space of\u0000rate constants of which the corresponding dynamics admits globally stable\u0000steady states.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195874","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 ( K ) be a number field. We provide quantitative estimates for the size of the Zsigmondy set of an integral ideal sequence generated by iterating a polynomial function (varphi(z) in K[z]) at a wandering point (alpha in K.)
让 ( K ) 是一个数域。我们提供了在一个游走点 (alpha inK.)上迭代apolynomial函数 (varphi(z) in K[z])所产生的积分理想序列的Zsigmondy集合大小的定量估计值。
{"title":"Quantitative Estimates for the Size of the Zsigmondy Set in Arithmetic Dynamics","authors":"Yang Gao, Qingzhong Ji","doi":"arxiv-2409.04710","DOIUrl":"https://doi.org/arxiv-2409.04710","url":null,"abstract":"Let ( K ) be a number field. We provide quantitative estimates for the size\u0000of the Zsigmondy set of an integral ideal sequence generated by iterating a\u0000polynomial function (varphi(z) in K[z]) at a wandering point (alpha in\u0000K.)","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195875","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}
A. Lins Neto presented in [Lins-Neto,2002] a $1$-dimensional family of degree four foliations on the complex projective plane $mathcal{F}_{t in overline{mathbb{C}}}$ with non-degenerate singularities of fixed analytic type, whose set of parameters $t$ for which $mathcal{F}_t$ is an elliptic pencil is dense and countable. In [McQuillan,2001] and [Guillot,2002], M. McQuillan and A. Guillot showed that the family lifts to linear foliations on the abelian surface $E times E$, where $E = mathbb{C}/Gamma$, $Gamma = < 1 , tau>$ and $tau$ is a primitive 3rd root of unity, the parameters for which $mathcal{F}_t$ are elliptic pencils being $tin mathbb{Q}(tau) cup {infty}$. In [Puchuri,2013], the second author gave a closed formula for the degree of the elliptic curves of $mathcal{F}_t$ a function of $t in mathbb{Q}(tau)$. In this work we determine degree, positions and multiplicities of singularities of the elliptic curves of $mathcal{F}_t$, for any given $t in mathbb{Z}(tau)$ in algorithmical way implemented in Python. And also we obtain the explicit expressions for the generators of the elliptic pencils, using the Singular software. Our constructions depend on the effect of quadratic Cremona maps on the family of foliations $mathcal{F}_t$.
{"title":"Effective Integrability of Lins Neto's Family of Foliations","authors":"Liliana Puchuri, Luís Gustavo Mendes","doi":"arxiv-2409.04336","DOIUrl":"https://doi.org/arxiv-2409.04336","url":null,"abstract":"A. Lins Neto presented in [Lins-Neto,2002] a $1$-dimensional family of degree\u0000four foliations on the complex projective plane $mathcal{F}_{t in\u0000overline{mathbb{C}}}$ with non-degenerate singularities of fixed analytic\u0000type, whose set of parameters $t$ for which $mathcal{F}_t$ is an elliptic\u0000pencil is dense and countable. In [McQuillan,2001] and [Guillot,2002], M.\u0000McQuillan and A. Guillot showed that the family lifts to linear foliations on\u0000the abelian surface $E times E$, where $E = mathbb{C}/Gamma$, $Gamma = < 1\u0000, tau>$ and $tau$ is a primitive 3rd root of unity, the parameters for which\u0000$mathcal{F}_t$ are elliptic pencils being $tin mathbb{Q}(tau) cup\u0000{infty}$. In [Puchuri,2013], the second author gave a closed formula for the\u0000degree of the elliptic curves of $mathcal{F}_t$ a function of $t in\u0000mathbb{Q}(tau)$. In this work we determine degree, positions and\u0000multiplicities of singularities of the elliptic curves of $mathcal{F}_t$, for\u0000any given $t in mathbb{Z}(tau)$ in algorithmical way implemented in Python.\u0000And also we obtain the explicit expressions for the generators of the elliptic\u0000pencils, using the Singular software. Our constructions depend on the effect of\u0000quadratic Cremona maps on the family of foliations $mathcal{F}_t$.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195899","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 survey the distributional properties of progressively dilating sets under projection by covering maps, focusing on manifolds of constant sectional curvature. In the Euclidean case, we review previously known results and formulate some generalizations, derived as a direct byproduct of recent developments on the problem of Fourier decay of finite measures. In the hyperbolic setting, we consider a natural upgrade of the problem to unit tangent bundles; confining ourselves to compact hyperbolic surfaces, we discuss an extension of our recent result with Ravotti on expanding circle arcs, establishing a precise asymptotic expansion for averages along expanding translates of homogeneous curves.
{"title":"The distribution of dilating sets: a journey from Euclidean to hyperbolic geometry","authors":"Emilio Corso","doi":"arxiv-2409.04611","DOIUrl":"https://doi.org/arxiv-2409.04611","url":null,"abstract":"We survey the distributional properties of progressively dilating sets under\u0000projection by covering maps, focusing on manifolds of constant sectional\u0000curvature. In the Euclidean case, we review previously known results and\u0000formulate some generalizations, derived as a direct byproduct of recent\u0000developments on the problem of Fourier decay of finite measures. In the\u0000hyperbolic setting, we consider a natural upgrade of the problem to unit\u0000tangent bundles; confining ourselves to compact hyperbolic surfaces, we discuss\u0000an extension of our recent result with Ravotti on expanding circle arcs,\u0000establishing a precise asymptotic expansion for averages along expanding\u0000translates of homogeneous curves.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195876","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 investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions anihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the inartistic connection between these operators and local analytic first integrals. To illustrate the duality the case of quadratic vector fields is considered in detail.
{"title":"Integrability of polynomial vector fields and a dual problem","authors":"Tatjana Petek, Valery Romanovski","doi":"arxiv-2409.04322","DOIUrl":"https://doi.org/arxiv-2409.04322","url":null,"abstract":"We investigate the integrability of polynomial vector fields through the lens\u0000of duality in parameter spaces. We examine formal power series solutions\u0000anihilated by differential operators and explore the properties of the\u0000integrability variety in relation to the invariants of the associated Lie\u0000group. Our study extends to differential operators on affine algebraic\u0000varieties, highlighting the inartistic connection between these operators and\u0000local analytic first integrals. To illustrate the duality the case of quadratic\u0000vector fields is considered in detail.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195900","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 pay attention to a weaker version of Walters's question on the existence of non-uniform cocycles for uniquely ergodic minimal dynamical systems on non-degenerate connected spaces. We will classify such dynamical systems into three classes: not totally uniquely ergodic; totally uniquely ergodic but not topological weakly mixing; totally uniquely ergodic and topological weakly mixing. We will give an affirmative answer to such question for the first two classes. Also, we will show the existence of such dynamical systems in the first class with arbitrary topological entropy.
{"title":"Non-uniform Cocycles for Some Uniquely Ergodic Minimal Dynamical Systems on Connected Spaces","authors":"Wanshan Lin, Xueting Tian","doi":"arxiv-2409.03310","DOIUrl":"https://doi.org/arxiv-2409.03310","url":null,"abstract":"In this paper, we pay attention to a weaker version of Walters's question on\u0000the existence of non-uniform cocycles for uniquely ergodic minimal dynamical\u0000systems on non-degenerate connected spaces. We will classify such dynamical\u0000systems into three classes: not totally uniquely ergodic; totally uniquely\u0000ergodic but not topological weakly mixing; totally uniquely ergodic and\u0000topological weakly mixing. We will give an affirmative answer to such question\u0000for the first two classes. Also, we will show the existence of such dynamical\u0000systems in the first class with arbitrary topological entropy.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195914","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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