Pub Date : 2022-03-20DOI: 10.1080/14689367.2022.2054311
Xiaoxiao Nie, Yu Huang
In this paper, by replacing the Bowen metric with the Feldman–Katok (FK) metric and the mean metric, respectively, we introduce measure-theoretic restricted FK (mean) sensitivities and topological restricted FK (mean) sensitivities. For a topological dynamical system, we discuss the relationships among measure-theoretic asymptotic FK rate with respect to sensitivity, topological asymptotic FK rate with respect to sensitivity, Brin–Katok local entropy and topological entropy. Parallel results are obtained for the mean metric case. In addition, we characterize the measure-theoretic entropy in terms of the exponential growth rate of the n-th return time to dynamical balls with respect to FK or mean metric. We also construct conditional entropy formulae with respect to FK metrics and the mean metrics.
{"title":"Restricted sensitivity, return time and entropy in Feldman–Katok and mean metrics","authors":"Xiaoxiao Nie, Yu Huang","doi":"10.1080/14689367.2022.2054311","DOIUrl":"https://doi.org/10.1080/14689367.2022.2054311","url":null,"abstract":"In this paper, by replacing the Bowen metric with the Feldman–Katok (FK) metric and the mean metric, respectively, we introduce measure-theoretic restricted FK (mean) sensitivities and topological restricted FK (mean) sensitivities. For a topological dynamical system, we discuss the relationships among measure-theoretic asymptotic FK rate with respect to sensitivity, topological asymptotic FK rate with respect to sensitivity, Brin–Katok local entropy and topological entropy. Parallel results are obtained for the mean metric case. In addition, we characterize the measure-theoretic entropy in terms of the exponential growth rate of the n-th return time to dynamical balls with respect to FK or mean metric. We also construct conditional entropy formulae with respect to FK metrics and the mean metrics.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"357 - 381"},"PeriodicalIF":0.5,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48232224","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 : 2022-03-18DOI: 10.1080/14689367.2022.2119941
Alexander Lohse
We consider families of systems of two-dimensional ordinary differential equations with the origin 0 as a non-hyperbolic equilibrium. For any number , we show that it is possible to choose a parameter in these equations such that the stability index is precisely . In contrast to that, for a hyperbolic equilibrium x it is known that either or . Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.
{"title":"Stability indices of non-hyperbolic equilibria in two-dimensional systems of ODEs","authors":"Alexander Lohse","doi":"10.1080/14689367.2022.2119941","DOIUrl":"https://doi.org/10.1080/14689367.2022.2119941","url":null,"abstract":"We consider families of systems of two-dimensional ordinary differential equations with the origin 0 as a non-hyperbolic equilibrium. For any number , we show that it is possible to choose a parameter in these equations such that the stability index is precisely . In contrast to that, for a hyperbolic equilibrium x it is known that either or . Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"699 - 709"},"PeriodicalIF":0.5,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44344607","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 : 2022-03-09DOI: 10.1080/14689367.2022.2049707
Yuki Takahashi
In ¥cite , the authors considered analytic monotonic cocycles, and showed that the Lyapunov exponent of an analytic family of analytic monotonic cocycles is analytic. We extend the result of ¥cite , and show that a analytic family of meromorphic monotonic cocycles have analytic Lyapunov exponent. We then consider the quasiperiodic Schr¥“odinger operators that have meromorphic monotone potentials. Since the associated Schr¥“odinger cocycles are meromorphic and monotonic, by applying the result we show that the Lyapunov exponent of the associated Schr¥“odinger cocycle is analytic. For the proof we rely heavily on the techniques in ¥cite .
{"title":"Analyticity of the Lyapunov exponent of meromorphic monotonic cocycles","authors":"Yuki Takahashi","doi":"10.1080/14689367.2022.2049707","DOIUrl":"https://doi.org/10.1080/14689367.2022.2049707","url":null,"abstract":"In ¥cite , the authors considered analytic monotonic cocycles, and showed that the Lyapunov exponent of an analytic family of analytic monotonic cocycles is analytic. We extend the result of ¥cite , and show that a analytic family of meromorphic monotonic cocycles have analytic Lyapunov exponent. We then consider the quasiperiodic Schr¥“odinger operators that have meromorphic monotone potentials. Since the associated Schr¥“odinger cocycles are meromorphic and monotonic, by applying the result we show that the Lyapunov exponent of the associated Schr¥“odinger cocycle is analytic. For the proof we rely heavily on the techniques in ¥cite .","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"328 - 332"},"PeriodicalIF":0.5,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49555155","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 : 2022-03-09DOI: 10.1080/14689367.2022.2049706
Thieu Huy Nguyen, X. Bui
Motivated by a predator–prey model with cross-diffusion, we consider the evolution equation of the form where the linear operator is a sectorial operator having a gap in its spectrum. We prove the existence of an admissibly inertial manifold for such an evolution equation in the case of the spectrum of contains an isolated subset which is sufficiently far from the rest, and the nonlinear term f satisfies φ-Lipschitz condition for φ belonging to some admissible space. Next, we will study the regularity of such admissibly inertial manifolds. We then apply the obtained result to the above-mentioned predator–prey model.
{"title":"On the existence and regularity of admissibly inertial manifolds with sectorial operators","authors":"Thieu Huy Nguyen, X. Bui","doi":"10.1080/14689367.2022.2049706","DOIUrl":"https://doi.org/10.1080/14689367.2022.2049706","url":null,"abstract":"Motivated by a predator–prey model with cross-diffusion, we consider the evolution equation of the form where the linear operator is a sectorial operator having a gap in its spectrum. We prove the existence of an admissibly inertial manifold for such an evolution equation in the case of the spectrum of contains an isolated subset which is sufficiently far from the rest, and the nonlinear term f satisfies φ-Lipschitz condition for φ belonging to some admissible space. Next, we will study the regularity of such admissibly inertial manifolds. We then apply the obtained result to the above-mentioned predator–prey model.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"295 - 327"},"PeriodicalIF":0.5,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43493434","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 : 2022-03-06DOI: 10.1080/14689367.2022.2048633
P. Cumsille, J. González–Marín, Gerardo Honorato, Diego Lugo
We investigate the Halley method of exponential maps. Our main result is that, unlike Newton's method, the Julia set of Halley's method may be disconnected when applied to entire maps of form where p and q are polynomials and q is non-constant. We also describe the nature of the fixed points and classify rational Halley's maps of entire functions.
{"title":"Disconnected Julia set of Halley's method for exponential maps","authors":"P. Cumsille, J. González–Marín, Gerardo Honorato, Diego Lugo","doi":"10.1080/14689367.2022.2048633","DOIUrl":"https://doi.org/10.1080/14689367.2022.2048633","url":null,"abstract":"We investigate the Halley method of exponential maps. Our main result is that, unlike Newton's method, the Julia set of Halley's method may be disconnected when applied to entire maps of form where p and q are polynomials and q is non-constant. We also describe the nature of the fixed points and classify rational Halley's maps of entire functions.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"280 - 294"},"PeriodicalIF":0.5,"publicationDate":"2022-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48158890","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 : 2022-02-26DOI: 10.1080/14689367.2022.2132917
Nicolas B'edaride, Jean-Franccois Bertazzon, I. Kabor'e
We consider non-bijective piecewise rotations of the plane. These maps belong to a family introduced in previous papers by Boshernitzan and Goetz. We derive in this paper some upper bounds to the size of the limit set. This improves results of [M. Boshernitzan and A. Goetz, A dichotomy for a two-parameter piecewise rotation, Ergodic Theory Dynam. Syst. 23(3) (2003), pp. 759–770.].
{"title":"Piecewise rotations: limit set for the non-bijective maps","authors":"Nicolas B'edaride, Jean-Franccois Bertazzon, I. Kabor'e","doi":"10.1080/14689367.2022.2132917","DOIUrl":"https://doi.org/10.1080/14689367.2022.2132917","url":null,"abstract":"We consider non-bijective piecewise rotations of the plane. These maps belong to a family introduced in previous papers by Boshernitzan and Goetz. We derive in this paper some upper bounds to the size of the limit set. This improves results of [M. Boshernitzan and A. Goetz, A dichotomy for a two-parameter piecewise rotation, Ergodic Theory Dynam. Syst. 23(3) (2003), pp. 759–770.].","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"52 - 78"},"PeriodicalIF":0.5,"publicationDate":"2022-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48288582","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 : 2022-02-10DOI: 10.1080/14689367.2022.2037519
J. Llibre, C. Valls
We study the continuous and discontinuous planar piecewise differential systems formed by linear centres together with linear Hamiltonian saddles separated by one or two parallel straight lines. When these piecewise differential systems are either continuous or discontinuous separated by one straight line, they have no limit cycles. When these piecewise differential systems are continuous and are separated by two parallel straight lines they do not have limit cycles. On the other hand, when these piecewise differential systems are discontinuous and separated by two parallel straight lines (either two centres and one saddle, or two saddles and one centre), we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle. If the piecewise differential systems separated by two parallel straight lines have three linear centres or three linear Hamiltonian saddles it is known that they have at most one limit cycle.
{"title":"Limit cycles of piecewise differential systems with linear Hamiltonian saddles and linear centres","authors":"J. Llibre, C. Valls","doi":"10.1080/14689367.2022.2037519","DOIUrl":"https://doi.org/10.1080/14689367.2022.2037519","url":null,"abstract":"We study the continuous and discontinuous planar piecewise differential systems formed by linear centres together with linear Hamiltonian saddles separated by one or two parallel straight lines. When these piecewise differential systems are either continuous or discontinuous separated by one straight line, they have no limit cycles. When these piecewise differential systems are continuous and are separated by two parallel straight lines they do not have limit cycles. On the other hand, when these piecewise differential systems are discontinuous and separated by two parallel straight lines (either two centres and one saddle, or two saddles and one centre), we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle. If the piecewise differential systems separated by two parallel straight lines have three linear centres or three linear Hamiltonian saddles it is known that they have at most one limit cycle.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"262 - 279"},"PeriodicalIF":0.5,"publicationDate":"2022-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45481384","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 : 2022-02-06DOI: 10.1080/14689367.2022.2033166
Aimee S. A. Johnson, D. McClendon
We study minimal -Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal -odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original. Furthermore, we give examples of speedups of -odometers which show the significant role played by a choice of ‘cone’ associated to the speedup.
{"title":"Topological speedups of ℤd-actions","authors":"Aimee S. A. Johnson, D. McClendon","doi":"10.1080/14689367.2022.2033166","DOIUrl":"https://doi.org/10.1080/14689367.2022.2033166","url":null,"abstract":"We study minimal -Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal -odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original. Furthermore, we give examples of speedups of -odometers which show the significant role played by a choice of ‘cone’ associated to the speedup.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"222 - 261"},"PeriodicalIF":0.5,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46598320","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 : 2022-01-02DOI: 10.1080/14689367.2021.2008877
José Luis Daíz Palencia
Travelling waves (TWs) instabilities to a degenerate diffusion problem with heterogeneous Fisher-KPP problem have not been previously analysed. The intention along this paper is to study existence, uniqueness and TW instability for a high order diffusion heterogeneous reaction Fisher-KPP problem with advection. The TW profiles are obtained analytically in the proximity of the stationary points, making use of the geometric perturbation theory. In addition, we examine a characterization of a local in time positive inner region where the TW behaves monotonically in contrast with an outer region of instabilities. Furthermore, a numerical exercise determines an accurate estimation of a local time to ensure the existence of the positive inner region, given a certain TW propagation speed.
{"title":"Analysis and instabilities of travelling waves solutions for a free boundary problem with non-homogeneous KPP reaction, with degenerate diffusion and with non-linear advection","authors":"José Luis Daíz Palencia","doi":"10.1080/14689367.2021.2008877","DOIUrl":"https://doi.org/10.1080/14689367.2021.2008877","url":null,"abstract":"Travelling waves (TWs) instabilities to a degenerate diffusion problem with heterogeneous Fisher-KPP problem have not been previously analysed. The intention along this paper is to study existence, uniqueness and TW instability for a high order diffusion heterogeneous reaction Fisher-KPP problem with advection. The TW profiles are obtained analytically in the proximity of the stationary points, making use of the geometric perturbation theory. In addition, we examine a characterization of a local in time positive inner region where the TW behaves monotonically in contrast with an outer region of instabilities. Furthermore, a numerical exercise determines an accurate estimation of a local time to ensure the existence of the positive inner region, given a certain TW propagation speed.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"83 - 104"},"PeriodicalIF":0.5,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60089253","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 : 2022-01-02DOI: 10.1080/14689367.2021.2020215
Xiujuan Wang, Jintao Wang, Chunqiu Li
This paper investigates a two-dimensional nonautonomous nonlocal Swift–Hohenberg equation with two kinds of kernels and studies the existence of invariant measures and statistical solutions, which are important research objects in the area of turbulence for fluid systems. The existence of weak solutions guarantees a norm-to-weak continuous process associated with the nonautonomous equation. We first prove the existence of the pullback attractor for the process via the pullback flattening. Then the unique existence of invariant measures is obtained by appropriate construction, so that the invariant measure is supported by this pullback attractor. This invariant measure is turned out to be exactly a statistical solution of the original nonlocal Swift–Hohenberg equation.
{"title":"Invariant measures and statistical solutions for a nonautonomous nonlocal Swift–Hohenberg equation","authors":"Xiujuan Wang, Jintao Wang, Chunqiu Li","doi":"10.1080/14689367.2021.2020215","DOIUrl":"https://doi.org/10.1080/14689367.2021.2020215","url":null,"abstract":"This paper investigates a two-dimensional nonautonomous nonlocal Swift–Hohenberg equation with two kinds of kernels and studies the existence of invariant measures and statistical solutions, which are important research objects in the area of turbulence for fluid systems. The existence of weak solutions guarantees a norm-to-weak continuous process associated with the nonautonomous equation. We first prove the existence of the pullback attractor for the process via the pullback flattening. Then the unique existence of invariant measures is obtained by appropriate construction, so that the invariant measure is supported by this pullback attractor. This invariant measure is turned out to be exactly a statistical solution of the original nonlocal Swift–Hohenberg equation.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"136 - 158"},"PeriodicalIF":0.5,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43452363","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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