Pub Date : 2023-08-01DOI: 10.56415/basm.y2023.i1.p78
A. Gasull, A. Guillamón, Víctor Mañosa
We present several results on the determination of the number and distribution of limit cycles or centers for planar systems of differential equations. In most cases, the study of a recurrence is one of the key points of our approach. These results include the counting of the number of configurations of stabilities of nested limit cycles, the study of the number of different configurations of a given number of limit cycles, the proof of some quadratic lower bounds for Hilbert numbers and some questions about the number of centers for planar polynomial vector fields.
{"title":"Counting configurations of limit cycles and centers","authors":"A. Gasull, A. Guillamón, Víctor Mañosa","doi":"10.56415/basm.y2023.i1.p78","DOIUrl":"https://doi.org/10.56415/basm.y2023.i1.p78","url":null,"abstract":"We present several results on the determination of the number and distribution of limit cycles or centers for planar systems of differential equations. In most cases, the study of a recurrence is one of the key points of our approach. These results include the counting of the number of configurations of stabilities of nested limit cycles, the study of the number of different configurations of a given number of limit cycles, the proof of some quadratic lower bounds for Hilbert numbers and some questions about the number of centers for planar polynomial vector fields.","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132049862","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}
Pub Date : 2023-08-01DOI: 10.56415/basm.y2023.i1.p3
J. Giné, J. Llibre
In this work we summarize some well-known criteria for the nonexistence of periodic orbits in planar differential systems. Additionally we present two new criteria and illustrate with examples these criteria.
{"title":"Criteria for the nonexistence of periodic orbits in planar differential systems","authors":"J. Giné, J. Llibre","doi":"10.56415/basm.y2023.i1.p3","DOIUrl":"https://doi.org/10.56415/basm.y2023.i1.p3","url":null,"abstract":"In this work we summarize some well-known criteria for the nonexistence of periodic orbits in planar differential systems. Additionally we present two new criteria and illustrate with examples these criteria.","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131460731","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}
Pub Date : 2023-08-01DOI: 10.56415/basm.y2023.i1.p16
T. Petek, V. Romanovski
We study time-reversibility and invariants of the group of transformations $xto x, yto alpha y, z to alpha ^{-1}z$ for three-dimensional polynomial systems with $0:1:-1$ resonant singular point at the origin. An algorithm to find the Zariski closure of the set of time-reversible systems in the space of parameters is proposed. The interconnection of time-reversibility and invariants of the group mentioned above is discussed.
本文研究了具有$0:1:-1$原点共振奇点的三维多项式系统$x到x， $ y到 α y， $ z 到 α ^{-1}z$的变换群的时间可逆性和不变量。提出了一种在参数空间中求时间可逆系统集的Zariski闭包的算法。讨论了上述群的时间可逆性与不变量的相互关系。
{"title":"Time-Reversibility and Ivariants of Some 3-dim Systems","authors":"T. Petek, V. Romanovski","doi":"10.56415/basm.y2023.i1.p16","DOIUrl":"https://doi.org/10.56415/basm.y2023.i1.p16","url":null,"abstract":"We study time-reversibility and invariants of the group of transformations $xto x, yto alpha y, z to alpha ^{-1}z$ for three-dimensional polynomial systems with $0:1:-1$ resonant singular point at the origin. An algorithm to find the Zariski closure of the set of time-reversible systems in the space of parameters is proposed. The interconnection of time-reversibility and invariants of the group mentioned above is discussed.","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114446673","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}
Pub Date : 2023-08-01DOI: 10.56415/basm.y2023.i1.p29
Yantao Yang, Xiang Zhang
In this survey paper, we summarize our results and also some related ones on local integrability of analytic autonomous differential systems near an equilibrium. The results are on necessary conditions related to existence of local analytic or meromorphic first integrals, on existence of analytic normalization of local analytically integrable system, and also on some sufficient conditions for existence of local analytic first integrals. Among which the results are also on regularity of the local first integrals, including analytic and Gevrey smoothness. We also present some open questions for further investigation.
{"title":"A survey on local integrability and its regularity","authors":"Yantao Yang, Xiang Zhang","doi":"10.56415/basm.y2023.i1.p29","DOIUrl":"https://doi.org/10.56415/basm.y2023.i1.p29","url":null,"abstract":"In this survey paper, we summarize our results and also some related ones on local integrability of analytic autonomous differential systems near an equilibrium. The results are on necessary conditions related to existence of local analytic or meromorphic first integrals, on existence of analytic normalization of local analytically integrable system, and also on some sufficient conditions for existence of local analytic first integrals. Among which the results are also on regularity of the local first integrals, including analytic and Gevrey smoothness. We also present some open questions for further investigation.","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126452981","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}
Pub Date : 2023-08-01DOI: 10.56415/basm.y2023.i1.p42
Cristina Bujac, D. Schlomiuk, N. Vulpe
We denote by ${mbox{boldmath $QSL$}}_3$ the family of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition three more families of quadratic systems possessing invariant lines of total multiplicity at least three were also studied, among them the Lotka-Volterra family. However there were still systems in ${mbox{boldmath ${mbox{boldmath $QSL$}}$}}_3$ missing from all these studies. The goals of this article are: to complete the study of the geometric configurations of invariant lines of ${mbox{boldmath ${mbox{boldmath $QSL$}}$}}_3$ by studying all the remaining cases and to give the full classification of this family modulo their configurations of invariant lines together with their bifurcation diagram. The family ${mbox{boldmath ${mbox{boldmath $QSL$}}$}}_3$ has a total of 81 distinct configurations of invariant lines. This classification is done in affine invariant terms and we also present the bifurcation diagram of these configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to ${mbox{boldmath $QSL$}}_3$ and in case it does, by producing its configuration of invariant straight lines.
{"title":"The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields","authors":"Cristina Bujac, D. Schlomiuk, N. Vulpe","doi":"10.56415/basm.y2023.i1.p42","DOIUrl":"https://doi.org/10.56415/basm.y2023.i1.p42","url":null,"abstract":"We denote by ${mbox{boldmath $QSL$}}_3$ the family of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition three more families of quadratic systems possessing invariant lines of total multiplicity at least three were also studied, among them the Lotka-Volterra family. However there were still systems in ${mbox{boldmath ${mbox{boldmath $QSL$}}$}}_3$ missing from all these studies. The goals of this article are: to complete the study of the geometric configurations of invariant lines of ${mbox{boldmath ${mbox{boldmath $QSL$}}$}}_3$ by studying all the remaining cases and to give the full classification of this family modulo their configurations of invariant lines together with their bifurcation diagram. The family ${mbox{boldmath ${mbox{boldmath $QSL$}}$}}_3$ has a total of 81 distinct configurations of invariant lines. This classification is done in affine invariant terms and we also present the bifurcation diagram of these configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to ${mbox{boldmath $QSL$}}_3$ and in case it does, by producing its configuration of invariant straight lines.","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131473701","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}
Pub Date : 2023-08-01DOI: 10.56415/basm.y2023.i1.p8
J. Llibre
The work of Chicone and Shafer published in 1982 together with the work of Bamon published in 1986 proved that any polynomial differential system of degree two has finitely many limit cycles. But the problem remains open of providing a uniform upper bound for the maximum number of limit cycles that a polynomial differential system of degree two can have, i.e. the second part of the 16th Hilbert problem restricted to the polynomial differential systems of degree two remains open. Here we present six subclasses of polynomial differential systems of degree two for which we can prove that an upper bound for their maximum number of limit cycles is one.
{"title":"Some families of quadratic systems with at most one limit cycle","authors":"J. Llibre","doi":"10.56415/basm.y2023.i1.p8","DOIUrl":"https://doi.org/10.56415/basm.y2023.i1.p8","url":null,"abstract":"The work of Chicone and Shafer published in 1982 together with the work of Bamon published in 1986 proved that any polynomial differential system of degree two has finitely many limit cycles. But the problem remains open of providing a uniform upper bound for the maximum number of limit cycles that a polynomial differential system of degree two can have, i.e. the second part of the 16th Hilbert problem restricted to the polynomial differential systems of degree two remains open. Here we present six subclasses of polynomial differential systems of degree two for which we can prove that an upper bound for their maximum number of limit cycles is one.","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129620332","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}
Pub Date : 2023-06-01DOI: 10.56415/basm.y2022.i3.p3
J. Ettayb
In this paper, we extend the Volkenborn integral and Shnirelman integral for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces over $mathbb{Q}_{p}$ and $mathbb{C}_{p}$ respectively. When the ground field is a complete non-Archimedean valued field, which is also algebraically closed, we give some functional calculus for groups of infinitesimal generator $A$ such that $A$ is a nilpotent operator on finite-dimensional non-Archimedean Banach spaces.
{"title":"Some integrals for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces","authors":"J. Ettayb","doi":"10.56415/basm.y2022.i3.p3","DOIUrl":"https://doi.org/10.56415/basm.y2022.i3.p3","url":null,"abstract":"In this paper, we extend the Volkenborn integral and Shnirelman integral for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces over $mathbb{Q}_{p}$ and $mathbb{C}_{p}$ respectively. When the ground field is a complete non-Archimedean valued field, which is also algebraically closed, we give some functional calculus for groups of infinitesimal generator $A$ such that $A$ is a nilpotent operator on finite-dimensional non-Archimedean Banach spaces.\u0000","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124595483","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}
Pub Date : 2023-06-01DOI: 10.56415/basm.y2022.i3.p30
S. Dukhnovsky
A self-similar solution of the Broadwell system is found. Here the solution is sought using a reduction that transforms the given system into a system of differential equations. Further, the solution is constructed using the Painlev'e series. Here the system already passes the Painlev'e test and it is possible to find the solution if the equations in resonance satisfy the solution of the two-dimensional Bateman equation. Exact solution of the Bateman equation is established, allowing to find new explicit solution for the original system. In the process of calculations, we use the Wolfram Mathematica program. The proof of these results is carried out at a rigorous mathematical level.
{"title":"A self-similar solution for the two-dimensional Broadwell system via the Bateman equation","authors":"S. Dukhnovsky","doi":"10.56415/basm.y2022.i3.p30","DOIUrl":"https://doi.org/10.56415/basm.y2022.i3.p30","url":null,"abstract":"A self-similar solution of the Broadwell system is found. Here the solution is sought using a reduction that transforms the given system into a system of differential equations. Further, the solution is constructed using the Painlev'e series. Here the system already passes the Painlev'e test and it is possible to find the solution if the equations in resonance satisfy the solution of the two-dimensional Bateman equation. Exact solution of the Bateman equation is established, allowing to find new explicit solution for the original system. In the process of calculations, we use the Wolfram Mathematica program. The proof of these results is carried out at a rigorous mathematical level.\u0000","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"130 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134070579","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}
Pub Date : 2023-06-01DOI: 10.56415/basm.y2022.i3.p41
I. Fryz, F. Sokhatsky
Algorithms for checking if a medial ternary quasigroup has a set of six triple-wise orthogonal principal parastrophes and a set of six triple-wise strongly orthogonal principal parastrophes are found. It is proved that $n$-ary strongly self-orthogonal linear (including medial) quasigroups do not exist when $n>3$.
{"title":"Construction of medial ternary self-orthogonal quasigroupsm","authors":"I. Fryz, F. Sokhatsky","doi":"10.56415/basm.y2022.i3.p41","DOIUrl":"https://doi.org/10.56415/basm.y2022.i3.p41","url":null,"abstract":"Algorithms for checking if a medial ternary quasigroup has a set of six triple-wise orthogonal principal parastrophes and a set of six triple-wise strongly orthogonal principal parastrophes are found. It is proved that $n$-ary strongly self-orthogonal linear (including medial) quasigroups do not exist when $n>3$.\u0000","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128318585","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}
Pub Date : 2023-06-01DOI: 10.56415/basm.y2022.i3.p56
D. Cheban
This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo-periodic, pseudo-recurrent and Poisson stable) motions of symmetric monotone non-autonomous dynamical systems (NDS). It is proved that every precompact motion of such system is asymptotically Poisson stable. We give also the description of the structure of compact global attractor for monotone NDS with symmetry. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We apply our general results to the study of the problem of existence of different classes of Poisson stable solutions and global attractors for a chemical reaction network and nonautonomous translation-invariant difference equations.
{"title":"Poisson Stable Motions and Global Attractors of Symmetric Monotone Nonautonomous Dynamical Systems","authors":"D. Cheban","doi":"10.56415/basm.y2022.i3.p56","DOIUrl":"https://doi.org/10.56415/basm.y2022.i3.p56","url":null,"abstract":"This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo-periodic, pseudo-recurrent and Poisson stable) motions of symmetric monotone non-autonomous dynamical systems (NDS). It is proved that every precompact motion of such system is asymptotically Poisson stable. We give also the description of the structure of compact global attractor for monotone NDS with symmetry. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We apply our general results to the study of the problem of existence of different classes of Poisson stable solutions and global attractors for a chemical reaction network and nonautonomous translation-invariant difference equations.\u0000","PeriodicalId":102242,"journal":{"name":"Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131407899","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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