Pub Date : 2023-10-25DOI: 10.1080/14689367.2023.2270931
V. León, B. Scárdua
AbstractOne of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations [R. Moussu, Une démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 98–99 (1982), pp. 216–223]. In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with ‘many’ periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.Keywords: Foliationcenter singularityfirst integralintegrable form AcknowledgmentThe first author is grateful to Edital n°77/2022/PRPPG-PAAP-UNILA for partially supporting this research work.Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"On real center singularities of complex vector fields on surfaces","authors":"V. León, B. Scárdua","doi":"10.1080/14689367.2023.2270931","DOIUrl":"https://doi.org/10.1080/14689367.2023.2270931","url":null,"abstract":"AbstractOne of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations [R. Moussu, Une démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 98–99 (1982), pp. 216–223]. In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with ‘many’ periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.Keywords: Foliationcenter singularityfirst integralintegrable form AcknowledgmentThe first author is grateful to Edital n°77/2022/PRPPG-PAAP-UNILA for partially supporting this research work.Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"147 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134973489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1080/14689367.2023.2271407
Prohaska, Roland
We investigate three aspects of weak* convergence of the $n$-step distributions of random walks on finite volume homogeneous spaces $G/Gamma$ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesaro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesaro convergence towards Haar measure for uniquely ergodic random walks.
{"title":"Aspects of convergence of random walks on finite volume homogeneous spaces","authors":"Prohaska, Roland","doi":"10.1080/14689367.2023.2271407","DOIUrl":"https://doi.org/10.1080/14689367.2023.2271407","url":null,"abstract":"We investigate three aspects of weak* convergence of the $n$-step distributions of random walks on finite volume homogeneous spaces $G/Gamma$ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesaro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesaro convergence towards Haar measure for uniquely ergodic random walks.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135321234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-18DOI: 10.1080/14689367.2023.2257609
Artur O. Lopes, Jairo. K. Mengue
AbstractWe introduce a general IFS Bayesian method for getting posterior probabilities from prior probabilities, and also a generalized Bayes' rule, which will contemplate a dynamical, as well as a non-dynamical setting. Given a loss function l, we detail the prior and posterior items, their consequences and exhibit several examples. Taking Θ as the set of parameters and Y as the set of data (which usually provides random samples), a general IFS is a measurable map τ:Θ×Y→Y, which can be interpreted as a family of maps τθ:Y→Y,θ∈Θ. The main inspiration for the results we will get here comes from a paper by Zellner (with no dynamics), where Bayes' rule is related to a principle of minimization of information. We will show that our IFS Bayesian method which produces posterior probabilities (which are associated to holonomic probabilities) is related to the optimal solution of a variational principle, somehow corresponding to the pressure in Thermodynamic Formalism, and also to the principle of minimization of information in Information Theory. Among other results, we present the prior dynamical elements and we derive the corresponding posterior elements via the Ruelle operator of Thermodynamic Formalism; getting in this way a form of dynamical Bayes' rule.Keywords: Generalized Baye's ruleposterior probabilitygeneral IFS Bayesian methodminimization of informationholonomic probabilityThermodynamic Formalism Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 If there are more than one stationary ρ with respect to (l¯,ν,τ), then we get more than one possible posterior probability π2 observe the importance of considering dπ~=π~p(θ)dθdδy0, to get the below expression from (Equation32(32) supπ~holonomic∫[log(l(θ,y))+log(πa(θ))−log(φ(y))]dπ~+Hdθ(π~).(32) ), and that the supremum is over π~p(θ), which is a probability density function and no more a probability; furthermore, for the last term we take into account (Equation31(31) Hν(π)={−∫log(J)dπifdπ=J(θ,y)dν(θ)dρ(y)−∞ifπis not absolutely continuouswith respect toν×ρ(31) ).
{"title":"The generalized IFS Bayesian method and an associated variational principle covering the classical and dynamical cases","authors":"Artur O. Lopes, Jairo. K. Mengue","doi":"10.1080/14689367.2023.2257609","DOIUrl":"https://doi.org/10.1080/14689367.2023.2257609","url":null,"abstract":"AbstractWe introduce a general IFS Bayesian method for getting posterior probabilities from prior probabilities, and also a generalized Bayes' rule, which will contemplate a dynamical, as well as a non-dynamical setting. Given a loss function l, we detail the prior and posterior items, their consequences and exhibit several examples. Taking Θ as the set of parameters and Y as the set of data (which usually provides random samples), a general IFS is a measurable map τ:Θ×Y→Y, which can be interpreted as a family of maps τθ:Y→Y,θ∈Θ. The main inspiration for the results we will get here comes from a paper by Zellner (with no dynamics), where Bayes' rule is related to a principle of minimization of information. We will show that our IFS Bayesian method which produces posterior probabilities (which are associated to holonomic probabilities) is related to the optimal solution of a variational principle, somehow corresponding to the pressure in Thermodynamic Formalism, and also to the principle of minimization of information in Information Theory. Among other results, we present the prior dynamical elements and we derive the corresponding posterior elements via the Ruelle operator of Thermodynamic Formalism; getting in this way a form of dynamical Bayes' rule.Keywords: Generalized Baye's ruleposterior probabilitygeneral IFS Bayesian methodminimization of informationholonomic probabilityThermodynamic Formalism Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 If there are more than one stationary ρ with respect to (l¯,ν,τ), then we get more than one possible posterior probability π2 observe the importance of considering dπ~=π~p(θ)dθdδy0, to get the below expression from (Equation32(32) supπ~holonomic∫[log(l(θ,y))+log(πa(θ))−log(φ(y))]dπ~+Hdθ(π~).(32) ), and that the supremum is over π~p(θ), which is a probability density function and no more a probability; furthermore, for the last term we take into account (Equation31(31) Hν(π)={−∫log(J)dπifdπ=J(θ,y)dν(θ)dρ(y)−∞ifπis not absolutely continuouswith respect toν×ρ(31) ).","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135149030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-09DOI: 10.1080/14689367.2023.2254260
Xiaona Fang, Ran Lu
{"title":"Conditional Brin-Katok's entropy formula for monotonic partitions on Feldman-Katok metric","authors":"Xiaona Fang, Ran Lu","doi":"10.1080/14689367.2023.2254260","DOIUrl":"https://doi.org/10.1080/14689367.2023.2254260","url":null,"abstract":"","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"192 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136107574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-21DOI: 10.1080/14689367.2023.2236946
Fang Xu, Leiye Xu
{"title":"Discrete spectrum for group actions","authors":"Fang Xu, Leiye Xu","doi":"10.1080/14689367.2023.2236946","DOIUrl":"https://doi.org/10.1080/14689367.2023.2236946","url":null,"abstract":"","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47593743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-18DOI: 10.1080/14689367.2023.2236952
Naveenkumar Yadav
{"title":"Weaker forms of specification for maps on uniform spaces","authors":"Naveenkumar Yadav","doi":"10.1080/14689367.2023.2236952","DOIUrl":"https://doi.org/10.1080/14689367.2023.2236952","url":null,"abstract":"","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48909294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-14DOI: 10.1080/14689367.2023.2234845
L. Hoang
{"title":"Long-time behaviour of solutions of superlinear systems of differential equations","authors":"L. Hoang","doi":"10.1080/14689367.2023.2234845","DOIUrl":"https://doi.org/10.1080/14689367.2023.2234845","url":null,"abstract":"","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48784957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-10DOI: 10.1080/14689367.2023.2228737
J. Llibre, C. Valls
A centre of a differential system in the plane is a singular point having a neighbourhood U such that is filled of periodic orbits. A global centre is a centre such that is filled of periodic orbits. To determine if a given differential system has a centre is in general a difficult problem, but it is even harder to know if it has a global centre. In the present paper we deal with the class of polynomial differential systems of the form (1) where P and Q are homogeneous polynomials of degree n. It is known that these systems can have global centres only if n is odd and the global centres in the cases n = 1 and n = 3 are known. Here we work with the case n = 5 and we classify the global centres of a four parameter family of systems (1). In particular we illustrate how to study the local phase portraits of the singular points whose linear part is identically zero using only vertical blow ups.
{"title":"Reversible global centres with quintic homogeneous nonlinearities","authors":"J. Llibre, C. Valls","doi":"10.1080/14689367.2023.2228737","DOIUrl":"https://doi.org/10.1080/14689367.2023.2228737","url":null,"abstract":"A centre of a differential system in the plane is a singular point having a neighbourhood U such that is filled of periodic orbits. A global centre is a centre such that is filled of periodic orbits. To determine if a given differential system has a centre is in general a difficult problem, but it is even harder to know if it has a global centre. In the present paper we deal with the class of polynomial differential systems of the form (1) where P and Q are homogeneous polynomials of degree n. It is known that these systems can have global centres only if n is odd and the global centres in the cases n = 1 and n = 3 are known. Here we work with the case n = 5 and we classify the global centres of a four parameter family of systems (1). In particular we illustrate how to study the local phase portraits of the singular points whose linear part is identically zero using only vertical blow ups.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"632 - 653"},"PeriodicalIF":0.5,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48530577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-26DOI: 10.1080/14689367.2023.2228219
Thieu Huy Nguyen, T. N. H. Vu, Thi Kim Oanh Tran
ABSTRACT Motivated by the smoothing properties of heat semigroups on unbounded domains and the conditional stability of hyperbolic semigroups, we develop a unified approach toward the problems on the existence of periodic solutions to the evolution equation . Our method is based on the analysis of -stability of the semigroup generated by A, i.e. , t>0, for certain couple of Banach spaces and real-valued function φ satisfying . Our theory covers both cases corresponding to smoothing properties and the conditional stability of hyperbolic semigroups as well as some other important situations relating to the polynomial or exponential stability of semigroups. As illustrations for our theory, we give applications to the existence and uniqueness of periodic solutions to Navier–Stokes and damped wave equations.
{"title":"( X , Y , φ ) -Stable semigroups, periodic solutions, and applications","authors":"Thieu Huy Nguyen, T. N. H. Vu, Thi Kim Oanh Tran","doi":"10.1080/14689367.2023.2228219","DOIUrl":"https://doi.org/10.1080/14689367.2023.2228219","url":null,"abstract":"ABSTRACT Motivated by the smoothing properties of heat semigroups on unbounded domains and the conditional stability of hyperbolic semigroups, we develop a unified approach toward the problems on the existence of periodic solutions to the evolution equation . Our method is based on the analysis of -stability of the semigroup generated by A, i.e. , t>0, for certain couple of Banach spaces and real-valued function φ satisfying . Our theory covers both cases corresponding to smoothing properties and the conditional stability of hyperbolic semigroups as well as some other important situations relating to the polynomial or exponential stability of semigroups. As illustrations for our theory, we give applications to the existence and uniqueness of periodic solutions to Navier–Stokes and damped wave equations.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"612 - 631"},"PeriodicalIF":0.5,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41340666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-23DOI: 10.1080/14689367.2023.2226384
Shuzhen Hua, Jiandong Yin
In the paper, we introduce the concepts of topological stability, expansivity and pseudo-orbit tracing property of Borel measures with respect to a given continuous map on a compact topological space from the viewpoint of open covers and prove that every expansive Borel measure with the pseudo-orbit tracing property is topologically stable. Furthermore, we obtain some properties of topologically stable measures.
{"title":"Topological stability and pseudo-orbit tracing property of Borel measures from the viewpoint of open covers","authors":"Shuzhen Hua, Jiandong Yin","doi":"10.1080/14689367.2023.2226384","DOIUrl":"https://doi.org/10.1080/14689367.2023.2226384","url":null,"abstract":"In the paper, we introduce the concepts of topological stability, expansivity and pseudo-orbit tracing property of Borel measures with respect to a given continuous map on a compact topological space from the viewpoint of open covers and prove that every expansive Borel measure with the pseudo-orbit tracing property is topologically stable. Furthermore, we obtain some properties of topologically stable measures.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"597 - 611"},"PeriodicalIF":0.5,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42036587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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