Pub Date : 2021-12-29DOI: 10.1080/14689367.2021.2016631
Huasong Xiao
We say that a vector field has the weakly shadowing property if for any there exists such that for every d-pseudo orbit there exists an exact orbit whose -neighbourhood containing the pseudo orbit. It is proved in Li Ming and Zhongjie Liu [Weak shadowing property for flows on oriented surfaces, Proc. Amer. Math. Soc. 145(6) (2017), pp. 2591–2605.] that vector fields in the C 1-interior of the set of vector fields on an oriented smooth closed surface having the weakly shadowing property are structurally stable. In this paper, we show that the above conclusion does not hold for non-oriented surfaces. Precisely, we construct a non- -stable vector field on the Klein bottle which has the weakly shadowing property robustly.
{"title":"Weakly shadowable vector fields on non-oriented surfaces","authors":"Huasong Xiao","doi":"10.1080/14689367.2021.2016631","DOIUrl":"https://doi.org/10.1080/14689367.2021.2016631","url":null,"abstract":"We say that a vector field has the weakly shadowing property if for any there exists such that for every d-pseudo orbit there exists an exact orbit whose -neighbourhood containing the pseudo orbit. It is proved in Li Ming and Zhongjie Liu [Weak shadowing property for flows on oriented surfaces, Proc. Amer. Math. Soc. 145(6) (2017), pp. 2591–2605.] that vector fields in the C 1-interior of the set of vector fields on an oriented smooth closed surface having the weakly shadowing property are structurally stable. In this paper, we show that the above conclusion does not hold for non-oriented surfaces. Precisely, we construct a non- -stable vector field on the Klein bottle which has the weakly shadowing property robustly.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"127 - 135"},"PeriodicalIF":0.5,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45611504","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 : 2021-12-29DOI: 10.1080/14689367.2021.2012558
B. Leal, Sergio Muñoz
We consider the (three parameters) family of nonlinear mappings where a, b, c are positive real numbers. is the classical Hénon-Devaney map [1, 2, 6]. For a large open region of parameters, we exhibit invariant Cantor sets embedded in the plane with two hyperbolic fixed points of saddle type.
{"title":"Invariant Cantor sets in the parametrized Hénon-Devaney map","authors":"B. Leal, Sergio Muñoz","doi":"10.1080/14689367.2021.2012558","DOIUrl":"https://doi.org/10.1080/14689367.2021.2012558","url":null,"abstract":"We consider the (three parameters) family of nonlinear mappings where a, b, c are positive real numbers. is the classical Hénon-Devaney map [1, 2, 6]. For a large open region of parameters, we exhibit invariant Cantor sets embedded in the plane with two hyperbolic fixed points of saddle type.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"105 - 126"},"PeriodicalIF":0.5,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43108375","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 : 2021-11-18DOI: 10.1080/14689367.2021.2006150
U. Jamilov, A. Y. Khamrayev
We consider Volterra and non-Volterra cubic stochastic operators. For a Volterra cubic stochastic operator defined on the two-dimensional simplex, it is shown that the vertices and the centre of the simplex are fixed points. The trajectory of such an operator starting from any point from the boundary of the simplex is convergent, and the trajectory of such an operator starting from any point from the interior of the simplex except the centre does not converge. Moreover, therein proven the mean of any trajectory does not converge. For a non-Volterra cubic stochastic operator defined on the two-dimensional simplex, it is proved that the uniqueness of a fixed point, which is repelling and any trajectory starting from the boundary of the simplex converges to a periodic trajectory which consists of three vertices of the simplex. The set of limit points of the trajectory starting from the interior of the simplex except the centre is an infinite subset of the boundary of the simplex.
{"title":"On dynamics of Volterra and non-Volterra cubic stochastic operators","authors":"U. Jamilov, A. Y. Khamrayev","doi":"10.1080/14689367.2021.2006150","DOIUrl":"https://doi.org/10.1080/14689367.2021.2006150","url":null,"abstract":"We consider Volterra and non-Volterra cubic stochastic operators. For a Volterra cubic stochastic operator defined on the two-dimensional simplex, it is shown that the vertices and the centre of the simplex are fixed points. The trajectory of such an operator starting from any point from the boundary of the simplex is convergent, and the trajectory of such an operator starting from any point from the interior of the simplex except the centre does not converge. Moreover, therein proven the mean of any trajectory does not converge. For a non-Volterra cubic stochastic operator defined on the two-dimensional simplex, it is proved that the uniqueness of a fixed point, which is repelling and any trajectory starting from the boundary of the simplex converges to a periodic trajectory which consists of three vertices of the simplex. The set of limit points of the trajectory starting from the interior of the simplex except the centre is an infinite subset of the boundary of the simplex.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"66 - 82"},"PeriodicalIF":0.5,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44182312","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 : 2021-11-09DOI: 10.1080/14689367.2021.1999907
M. Rahimi, M. Mohammadi Anjedani
In this paper, given a sequence of positive integers, we assign a linear operator on a Hilbert space, to any compact topological dynamical system of finite entropy. Then we represent the sequence entropy of the systems in terms of the eigenvalues of the linear operator. In this way, we present a spectral approach to the sequence entropy of the dynamical systems. This spectral representation to the sequence entropy of a system is given for systems with some additional condition called admissibility condition. We also prove that, there exist a large family of dynamical systems, satisfying the admissibility condition.
{"title":"An operator theoretical approach to the sequence entropy of dynamical systems","authors":"M. Rahimi, M. Mohammadi Anjedani","doi":"10.1080/14689367.2021.1999907","DOIUrl":"https://doi.org/10.1080/14689367.2021.1999907","url":null,"abstract":"In this paper, given a sequence of positive integers, we assign a linear operator on a Hilbert space, to any compact topological dynamical system of finite entropy. Then we represent the sequence entropy of the systems in terms of the eigenvalues of the linear operator. In this way, we present a spectral approach to the sequence entropy of the dynamical systems. This spectral representation to the sequence entropy of a system is given for systems with some additional condition called admissibility condition. We also prove that, there exist a large family of dynamical systems, satisfying the admissibility condition.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"56 - 65"},"PeriodicalIF":0.5,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42365199","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 : 2021-11-09DOI: 10.1080/14689367.2021.1999906
M. J. Dos Santos, M. Freitas, A. Ramos, D. S. Almeida Júnior
In this paper we study the long-time behaviour of a system consisting of two nonlinear wave equations under the action of three competing forces, damping forces, strong source and external force. It is of great interest to know how the relationship between these forces acts on the behaviour of the solutions of the system. In this sense, we investigate the well-posedness of system, as well as the existence of global and exponential attractors. In addition, we consider the upper semicontinuity of the global attractor when the coupling parameter of the system tends to zero. Once proved the existence of global solutions (in time), to obtain the existence of global and exponential attractors results, we prove that the dynamical system associated to solutions of the model is quasi-stable and gradient.
{"title":"Asymptotic analysis and upper semicontinuity to a system of coupled nonlinear wave equations","authors":"M. J. Dos Santos, M. Freitas, A. Ramos, D. S. Almeida Júnior","doi":"10.1080/14689367.2021.1999906","DOIUrl":"https://doi.org/10.1080/14689367.2021.1999906","url":null,"abstract":"In this paper we study the long-time behaviour of a system consisting of two nonlinear wave equations under the action of three competing forces, damping forces, strong source and external force. It is of great interest to know how the relationship between these forces acts on the behaviour of the solutions of the system. In this sense, we investigate the well-posedness of system, as well as the existence of global and exponential attractors. In addition, we consider the upper semicontinuity of the global attractor when the coupling parameter of the system tends to zero. Once proved the existence of global solutions (in time), to obtain the existence of global and exponential attractors results, we prove that the dynamical system associated to solutions of the model is quasi-stable and gradient.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"29 - 55"},"PeriodicalIF":0.5,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43697628","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 : 2021-11-03DOI: 10.1080/14689367.2021.1998378
Shintaro Suzuki
For every generalized β-map τ introduced by Góra [P. Góra, Invariant densities for generalized β-maps, Ergod. Theory Dyn. Syst. 27 (2007), pp. 1583–1598], we find an explicit formula for a basis of the (generalized) eigenspace corresponding to an isolated eigenvalue of its Perron–Frobenius operator on the space of functions of bounded variation. From this formula, we see that any (generalized) eigenfunction is a singular function related to the orbit at 1 by the map τ. In addition, as a consecutive work of the paper [S. Suzuki, Artin-Mazur zeta functions of generalized β-transformations, Kyushu J. Math. 71 (2017), pp. 85–103], the analytic continuation of its lap-counting function is given by the generating function for the coefficient sequence of the τ-expansion of 1.
{"title":"Eigenfunctions of the Perron–Frobenius operators for generalized beta-maps","authors":"Shintaro Suzuki","doi":"10.1080/14689367.2021.1998378","DOIUrl":"https://doi.org/10.1080/14689367.2021.1998378","url":null,"abstract":"For every generalized β-map τ introduced by Góra [P. Góra, Invariant densities for generalized β-maps, Ergod. Theory Dyn. Syst. 27 (2007), pp. 1583–1598], we find an explicit formula for a basis of the (generalized) eigenspace corresponding to an isolated eigenvalue of its Perron–Frobenius operator on the space of functions of bounded variation. From this formula, we see that any (generalized) eigenfunction is a singular function related to the orbit at 1 by the map τ. In addition, as a consecutive work of the paper [S. Suzuki, Artin-Mazur zeta functions of generalized β-transformations, Kyushu J. Math. 71 (2017), pp. 85–103], the analytic continuation of its lap-counting function is given by the generating function for the coefficient sequence of the τ-expansion of 1.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"9 - 28"},"PeriodicalIF":0.5,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48225718","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 : 2021-10-21DOI: 10.1080/14689367.2021.1993144
Zouhair Diab, J. L. Guirao, J. A. Vera
The main aim of the present paper is to study the existence of limit cycles (i.e. close trajectories in the phase space having the property that at least one other trajectory spirals into them either as time approaches infinity or as time approaches negative infinity) of a class of piecewise generalized Liénard differential system modulated by a two variable polynomial and a piecewise linear function respectively. The main tool that we use to obtain these results is the averaging theory of the dynamical systems worthy to detect the initial conditions of the birth of isolated orbits of a system.
{"title":"On the limit cycles for a class of generalized Liénard differential systems","authors":"Zouhair Diab, J. L. Guirao, J. A. Vera","doi":"10.1080/14689367.2021.1993144","DOIUrl":"https://doi.org/10.1080/14689367.2021.1993144","url":null,"abstract":"The main aim of the present paper is to study the existence of limit cycles (i.e. close trajectories in the phase space having the property that at least one other trajectory spirals into them either as time approaches infinity or as time approaches negative infinity) of a class of piecewise generalized Liénard differential system modulated by a two variable polynomial and a piecewise linear function respectively. The main tool that we use to obtain these results is the averaging theory of the dynamical systems worthy to detect the initial conditions of the birth of isolated orbits of a system.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"1 - 8"},"PeriodicalIF":0.5,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41317303","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 : 2021-10-11DOI: 10.1080/14689367.2023.2178388
Katsukuni Nakagawa
For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all 2-locally constant functions f on the shift such that all those isomorphisms defined on the system associated with f are induced from automorphisms of the shift. We prove that this set contains a full-measure open set of the space of all 2-locally constant functions on the shift. We apply this result to rigidity problems of entropy spectra and show that the strong non-rigidity occurs if and only if so does the weak non-rigidity.
{"title":"Continuity of isomorphisms applied to rigidity problems of entropy spectra","authors":"Katsukuni Nakagawa","doi":"10.1080/14689367.2023.2178388","DOIUrl":"https://doi.org/10.1080/14689367.2023.2178388","url":null,"abstract":"For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all 2-locally constant functions f on the shift such that all those isomorphisms defined on the system associated with f are induced from automorphisms of the shift. We prove that this set contains a full-measure open set of the space of all 2-locally constant functions on the shift. We apply this result to rigidity problems of entropy spectra and show that the strong non-rigidity occurs if and only if so does the weak non-rigidity.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"301 - 313"},"PeriodicalIF":0.5,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49114474","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 : 2021-10-02DOI: 10.1080/14689367.2021.1994925
E. Abdalaoui, I. Naghmouchi
It is shown that the restriction of the action of any group with finite orbit on the minimal sets of local dendrites is equicontinuous. Consequently, we obtain that the action of any amenable group and Thompson group restricted to any minimal sets of dendrite is equicontinuous. We further provide a class of non-amenable groups whose action on the minimal sets of local dendrites is equicontinuous. Moreover, we extend some of our results to dendron. We further give a characterization of the set of invariant probability measures and its extreme points.
{"title":"Group action with finite orbits on local dendrites","authors":"E. Abdalaoui, I. Naghmouchi","doi":"10.1080/14689367.2021.1994925","DOIUrl":"https://doi.org/10.1080/14689367.2021.1994925","url":null,"abstract":"It is shown that the restriction of the action of any group with finite orbit on the minimal sets of local dendrites is equicontinuous. Consequently, we obtain that the action of any amenable group and Thompson group restricted to any minimal sets of dendrite is equicontinuous. We further provide a class of non-amenable groups whose action on the minimal sets of local dendrites is equicontinuous. Moreover, we extend some of our results to dendron. We further give a characterization of the set of invariant probability measures and its extreme points.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"36 1","pages":"714 - 730"},"PeriodicalIF":0.5,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43970778","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 : 2021-10-02DOI: 10.1080/14689367.2021.1993145
A. Alves, B. P. Silva e Silva, M. Salarinoghabi
For a positive odd integer d, we study the connectedness of the Julia set of the one-parameter family of rational maps given by with . Also, when we show that the geometric limit of the Julia set and filled Julia set of the family exists and is the unit circle.
{"title":"Geometric limit of Julia set of a family of rational functions with odd degree","authors":"A. Alves, B. P. Silva e Silva, M. Salarinoghabi","doi":"10.1080/14689367.2021.1993145","DOIUrl":"https://doi.org/10.1080/14689367.2021.1993145","url":null,"abstract":"For a positive odd integer d, we study the connectedness of the Julia set of the one-parameter family of rational maps given by with . Also, when we show that the geometric limit of the Julia set and filled Julia set of the family exists and is the unit circle.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"36 1","pages":"699 - 713"},"PeriodicalIF":0.5,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45097537","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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