Pub Date : 2023-01-22DOI: 10.1080/14689367.2022.2163881
Álvaro Castañeda, Pablo Monzón, G. Robledo
Given a nonautonomous linear system of ordinary differential equations with nonuniform contractions on the half line, we study the smoothness and preserving orientation properties of a global linearization between this system and a family of nonlinear perturbations. As an application of the previous study, we improve a converse stability result for the nonlinear system in terms of density functions.
{"title":"Nonuniform contractions and converse stability results via a smooth topological equivalence","authors":"Álvaro Castañeda, Pablo Monzón, G. Robledo","doi":"10.1080/14689367.2022.2163881","DOIUrl":"https://doi.org/10.1080/14689367.2022.2163881","url":null,"abstract":"Given a nonautonomous linear system of ordinary differential equations with nonuniform contractions on the half line, we study the smoothness and preserving orientation properties of a global linearization between this system and a family of nonlinear perturbations. As an application of the previous study, we improve a converse stability result for the nonlinear system in terms of density functions.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"179 - 196"},"PeriodicalIF":0.5,"publicationDate":"2023-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41740492","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 : 2023-01-12DOI: 10.1080/14689367.2023.2210518
I. Morris, Jonah Varney
A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant -cocycle on the full shift is not exponential, then it must be either bounded or linear, with no other possibilities occurring. We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols.
{"title":"A note on the marginal instability rates of two-dimensional linear cocycles","authors":"I. Morris, Jonah Varney","doi":"10.1080/14689367.2023.2210518","DOIUrl":"https://doi.org/10.1080/14689367.2023.2210518","url":null,"abstract":"A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant -cocycle on the full shift is not exponential, then it must be either bounded or linear, with no other possibilities occurring. We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"525 - 540"},"PeriodicalIF":0.5,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43986398","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 : 2023-01-05DOI: 10.1080/14689367.2023.2221193
Amal Al Dowais
In this paper, we estimate the largest Lyapunov exponent for open billiards in the plane. We show that the largest Lyapunov exponent is differentiable with respect to a billiard deformation.
{"title":"Differentiability of the largest Lyapunov exponent for planar open billiards","authors":"Amal Al Dowais","doi":"10.1080/14689367.2023.2221193","DOIUrl":"https://doi.org/10.1080/14689367.2023.2221193","url":null,"abstract":"In this paper, we estimate the largest Lyapunov exponent for open billiards in the plane. We show that the largest Lyapunov exponent is differentiable with respect to a billiard deformation.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"556 - 575"},"PeriodicalIF":0.5,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46681249","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 : 2023-01-02DOI: 10.1080/14689367.2022.2162371
O. Podvigina
We consider a three-dimensional generalized Lotka–Volterra (GLV) system assuming that it has equilibria on each of the coordinate axes, stable along the respective directions, and heteroclinic trajectories, and , that belong to coordinate planes. For such a system we give a complete classification of possible types of dynamics, characterized by the existence or non-existence of various two-dimensional heteroclinic connections. For each of these classes, we derive inequalities satisfied by coefficients of the system. The results can be used for the construction of GLV systems possessing various heteroclinic cycles or networks.
{"title":"Two-dimensional heteroclinic connections in the generalized Lotka–Volterra system","authors":"O. Podvigina","doi":"10.1080/14689367.2022.2162371","DOIUrl":"https://doi.org/10.1080/14689367.2022.2162371","url":null,"abstract":"We consider a three-dimensional generalized Lotka–Volterra (GLV) system assuming that it has equilibria on each of the coordinate axes, stable along the respective directions, and heteroclinic trajectories, and , that belong to coordinate planes. For such a system we give a complete classification of possible types of dynamics, characterized by the existence or non-existence of various two-dimensional heteroclinic connections. For each of these classes, we derive inequalities satisfied by coefficients of the system. The results can be used for the construction of GLV systems possessing various heteroclinic cycles or networks.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"163 - 178"},"PeriodicalIF":0.5,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43489850","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 : 2022-12-17DOI: 10.1080/14689367.2022.2139224
Farhana Akond Pramy, Ben Mestel, Robert Hasson, Katrine Rogers
The Stretch-Fold-Shear (SFS) operator is a functional linear operator acting on complex-valued functions of a real variable x on some domain containing in It arises from a stylized model in kinematic dynamo theory where magnetic field growth corresponds to an eigenvalue of modulus greater than 1. When the shear parameter α is zero, the spectrum of can be determined exactly, and the eigenfunctions corresponding to non-zero eigenvalues are related to the Bernoulli polynomials. The spectrum for has not been rigorously determined although the spectrum has been approximated numerically. In this paper, a computer-assisted proof is presented to provide rigorous bounds on the leading eigenvalue for , showing inter alia that has an eigenvalue of modulus greater than 1 for all α satisfying , thereby partially confirming an outstanding conjecture on the SFS operator.
{"title":"A computer-assisted proof of dynamo growth in the stretch-fold-shear map","authors":"Farhana Akond Pramy, Ben Mestel, Robert Hasson, Katrine Rogers","doi":"10.1080/14689367.2022.2139224","DOIUrl":"https://doi.org/10.1080/14689367.2022.2139224","url":null,"abstract":"The Stretch-Fold-Shear (SFS) operator is a functional linear operator acting on complex-valued functions of a real variable x on some domain containing in It arises from a stylized model in kinematic dynamo theory where magnetic field growth corresponds to an eigenvalue of modulus greater than 1. When the shear parameter α is zero, the spectrum of can be determined exactly, and the eigenfunctions corresponding to non-zero eigenvalues are related to the Bernoulli polynomials. The spectrum for has not been rigorously determined although the spectrum has been approximated numerically. In this paper, a computer-assisted proof is presented to provide rigorous bounds on the leading eigenvalue for , showing inter alia that has an eigenvalue of modulus greater than 1 for all α satisfying , thereby partially confirming an outstanding conjecture on the SFS operator.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"102 - 120"},"PeriodicalIF":0.5,"publicationDate":"2022-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43922160","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 : 2022-11-28DOI: 10.1080/14689367.2022.2147811
Congcong Li, Chunqiu Li, Jintao Wang
In this article, we study the statistical solution of the nonautonomous discrete Selkov model. First, we show the existence of a pullback- attractor for the system and establish the existence of a unique family of invariant Borel probability measures carried by the pullback- attractor. Then we further prove that the family of invariant Borel probability measures is a statistical solution for the discrete system and satisfies a Liouville-type theorem. Finally, we demonstrate that the invariant property of the statistical solution is indeed a particular case of the Liouville-type theorem.
{"title":"Statistical solution and Liouville-type theorem for the nonautonomous discrete Selkov model","authors":"Congcong Li, Chunqiu Li, Jintao Wang","doi":"10.1080/14689367.2022.2147811","DOIUrl":"https://doi.org/10.1080/14689367.2022.2147811","url":null,"abstract":"In this article, we study the statistical solution of the nonautonomous discrete Selkov model. First, we show the existence of a pullback- attractor for the system and establish the existence of a unique family of invariant Borel probability measures carried by the pullback- attractor. Then we further prove that the family of invariant Borel probability measures is a statistical solution for the discrete system and satisfies a Liouville-type theorem. Finally, we demonstrate that the invariant property of the statistical solution is indeed a particular case of the Liouville-type theorem.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"140 - 162"},"PeriodicalIF":0.5,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44906534","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 : 2022-11-17DOI: 10.1080/14689367.2022.2145181
M. Poulou, N. Zographopoulos
In this paper, we study the long-time behaviour of solutions of a degenerate Klein–Gordon–Schrödinger-type system which is defined in a bounded domain. First, we proved the existence, uniqueness and continuity of the solutions on the initial data, then the asymptotic compactness of the solutions and finally the existence of a global compact attractor.
{"title":"Global attractor for a degenerate Klein–Gordon–Schrödinger-type system","authors":"M. Poulou, N. Zographopoulos","doi":"10.1080/14689367.2022.2145181","DOIUrl":"https://doi.org/10.1080/14689367.2022.2145181","url":null,"abstract":"In this paper, we study the long-time behaviour of solutions of a degenerate Klein–Gordon–Schrödinger-type system which is defined in a bounded domain. First, we proved the existence, uniqueness and continuity of the solutions on the initial data, then the asymptotic compactness of the solutions and finally the existence of a global compact attractor.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"121 - 139"},"PeriodicalIF":0.5,"publicationDate":"2022-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45661645","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 : 2022-10-30DOI: 10.1080/14689367.2022.2136062
Valerie Jeong, C. Postlethwaite
A heteroclinic cycle is an invariant set in a dynamical system consisting of saddle-type equilibria and heteroclinic connections between them. It is known that deterministic perturbations (inputs) to a heteroclinic cycle generally lead to periodic solutions. Addition of noise to such a system leads to a non-intuitive result: there is a range of noise levels for which the mean residence time near the equilibria of the heteroclinic cycle increases as the noise level increases to a given threshold. We explain how the interaction between noise and inputs gives rise to this by combining analytical results from constructing a Poincaré map with a simple stochastic system. We support our results with numerical simulations.
{"title":"Effect of noise on residence times of a heteroclinic cycle","authors":"Valerie Jeong, C. Postlethwaite","doi":"10.1080/14689367.2022.2136062","DOIUrl":"https://doi.org/10.1080/14689367.2022.2136062","url":null,"abstract":"A heteroclinic cycle is an invariant set in a dynamical system consisting of saddle-type equilibria and heteroclinic connections between them. It is known that deterministic perturbations (inputs) to a heteroclinic cycle generally lead to periodic solutions. Addition of noise to such a system leads to a non-intuitive result: there is a range of noise levels for which the mean residence time near the equilibria of the heteroclinic cycle increases as the noise level increases to a given threshold. We explain how the interaction between noise and inputs gives rise to this by combining analytical results from constructing a Poincaré map with a simple stochastic system. We support our results with numerical simulations.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"79 - 101"},"PeriodicalIF":0.5,"publicationDate":"2022-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48297318","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 : 2022-10-16DOI: 10.1080/14689367.2022.2128991
Shang Wu, Zhiming Liu, Jianhua Huang
In this paper, we investigate the stochastic Boussinesq equations driven by Gaussian noise on a two-dimensional domain . By the regularity estimation on the high-order Sobolev space, we prove the existence of weak solutions in non-separable Banach space. We also show that the Markov semigroup generated by the solution of stochastic Boussinesq equations is also weak Feller. Endowed with the weak-★ topology on , we prove the existence of the invariant measure by Krylov–Bogoliubov theorem.
{"title":"Invariant measure of stochastic Boussinesq equation with zero viscosity in Banach space","authors":"Shang Wu, Zhiming Liu, Jianhua Huang","doi":"10.1080/14689367.2022.2128991","DOIUrl":"https://doi.org/10.1080/14689367.2022.2128991","url":null,"abstract":"In this paper, we investigate the stochastic Boussinesq equations driven by Gaussian noise on a two-dimensional domain . By the regularity estimation on the high-order Sobolev space, we prove the existence of weak solutions in non-separable Banach space. We also show that the Markov semigroup generated by the solution of stochastic Boussinesq equations is also weak Feller. Endowed with the weak-★ topology on , we prove the existence of the invariant measure by Krylov–Bogoliubov theorem.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"1 - 19"},"PeriodicalIF":0.5,"publicationDate":"2022-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45069896","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 : 2022-10-10DOI: 10.1080/14689367.2022.2132136
Babette de Wolff
A characteristic matrix function captures the spectral information of a bounded linear operator in a matrix-valued function. In this article, we consider a delay differential equation with one discrete time delay and assume this equation is equivariant with respect to a compact symmetry group. Under this assumption, the delay differential equation can have discrete wave solutions, i.e. periodic solutions that have a discrete group of spatio-temporal symmetries. We show that if a discrete wave solution has a period that is rationally related to the time delay, then we can determine its stability using a characteristic matrix function. The proof relies on equivariant Floquet theory and results by Kaashoek and Verduyn Lunel on characteristic matrix functions for classes of compact operators. We discuss applications of our result in the context of delayed feedback stabilization of periodic orbits.
{"title":"Characteristic matrix functions for delay differential equations with symmetry","authors":"Babette de Wolff","doi":"10.1080/14689367.2022.2132136","DOIUrl":"https://doi.org/10.1080/14689367.2022.2132136","url":null,"abstract":"A characteristic matrix function captures the spectral information of a bounded linear operator in a matrix-valued function. In this article, we consider a delay differential equation with one discrete time delay and assume this equation is equivariant with respect to a compact symmetry group. Under this assumption, the delay differential equation can have discrete wave solutions, i.e. periodic solutions that have a discrete group of spatio-temporal symmetries. We show that if a discrete wave solution has a period that is rationally related to the time delay, then we can determine its stability using a characteristic matrix function. The proof relies on equivariant Floquet theory and results by Kaashoek and Verduyn Lunel on characteristic matrix functions for classes of compact operators. We discuss applications of our result in the context of delayed feedback stabilization of periodic orbits.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"30 - 51"},"PeriodicalIF":0.5,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49347262","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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