A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences begin{document}$ {{ell_{rho}^2}} $end{document}. First the existence of a pullback attractor in begin{document}$ {{ell_{rho}^2}} $end{document} is established by utilizing the dense inclusion of begin{document}$ ell^2 subset {{ell_{rho}^2}} $end{document}. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.
A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences begin{document}$ {{ell_{rho}^2}} $end{document}. First the existence of a pullback attractor in begin{document}$ {{ell_{rho}^2}} $end{document} is established by utilizing the dense inclusion of begin{document}$ ell^2 subset {{ell_{rho}^2}} $end{document}. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.
{"title":"Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces","authors":"Xiaoying Han, P. Kloeden","doi":"10.3934/dcdss.2021143","DOIUrl":"https://doi.org/10.3934/dcdss.2021143","url":null,"abstract":"<p style='text-indent:20px;'>A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences <inline-formula><tex-math id=\"M1\">begin{document}$ {{ell_{rho}^2}} $end{document}</tex-math></inline-formula>. First the existence of a pullback attractor in <inline-formula><tex-math id=\"M2\">begin{document}$ {{ell_{rho}^2}} $end{document}</tex-math></inline-formula> is established by utilizing the dense inclusion of <inline-formula><tex-math id=\"M3\">begin{document}$ ell^2 subset {{ell_{rho}^2}} $end{document}</tex-math></inline-formula>. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78403482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, ``strang splitting'' idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms begin{document}$ L_2 $end{document} and begin{document}$ L_infty $end{document} . Numerical results arising from the simulation experiments are also presented.
This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, ``strang splitting'' idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms begin{document}$ L_2 $end{document} and begin{document}$ L_infty $end{document} . Numerical results arising from the simulation experiments are also presented.
{"title":"Numerical treatment of Gray-Scott model with operator splitting method","authors":"Berat Karaagac","doi":"10.3934/dcdss.2020143","DOIUrl":"https://doi.org/10.3934/dcdss.2020143","url":null,"abstract":"This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, ``strang splitting'' idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms begin{document}$ L_2 $end{document} and begin{document}$ L_infty $end{document} . Numerical results arising from the simulation experiments are also presented.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"12 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72615126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider here a damped forced nonlinear logarithmic Schrodinger equation in begin{document}$ mathbb{R}^N $end{document} . We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrodinger equations in the framework of infinite-dimensional dynamical systems.
We consider here a damped forced nonlinear logarithmic Schrodinger equation in begin{document}$ mathbb{R}^N $end{document} . We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrodinger equations in the framework of infinite-dimensional dynamical systems.
{"title":"Global attractor for damped forced nonlinear logarithmic Schrödinger equations","authors":"O. Goubet, E. Zahrouni","doi":"10.3934/dcdss.2020393","DOIUrl":"https://doi.org/10.3934/dcdss.2020393","url":null,"abstract":"We consider here a damped forced nonlinear logarithmic Schrodinger equation in begin{document}$ mathbb{R}^N $end{document} . We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrodinger equations in the framework of infinite-dimensional dynamical systems.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"26 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72629017","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}
In this paper we study a class of singular systems with double-phase energy. The main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. In such a way, we continue the analysis introduced in [ 6 ] to the case of lack of compactness corresponding to the whole Euclidean space. After establishing a related compactness property, we establish the existence of solutions for the Baouendi-Grushin singular system.
{"title":"Singular double-phase systems with variable growth for the Baouendi-Grushin operator","authors":"Anouar Bahrouni, Vicentiu D. Rădulescu","doi":"10.3934/DCDS.2021036","DOIUrl":"https://doi.org/10.3934/DCDS.2021036","url":null,"abstract":"In this paper we study a class of singular systems with double-phase energy. The main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. In such a way, we continue the analysis introduced in [ 6 ] to the case of lack of compactness corresponding to the whole Euclidean space. After establishing a related compactness property, we establish the existence of solutions for the Baouendi-Grushin singular system.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75133943","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}
J. A. Cardoso, Patricio Cerda, Denilson S. Pereira, P. Ubilla
We prove the existence of a bounded positive solution for the following stationary Schrodinger equation begin{document}$ begin{equation*} -Delta u+V(x)u = f(x,u),,,, xinmathbb{R}^n,,, ngeq 3, end{equation*} $end{document} where begin{document}$ V $end{document} is a vanishing potential and begin{document}$ f $end{document} has a sublinear growth at the origin (for example if begin{document}$ f(x,u) $end{document} is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [ 6 ]. In addition, if begin{document}$ f $end{document} has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type begin{document}$ rho(x)f(u) $end{document} where begin{document}$ f $end{document} is a concave-convex function and begin{document}$ rho $end{document} satisfies the begin{document}$ mathrm{(H)} $end{document} property introduced in [ 6 ]. We also note that we do not impose any integrability assumptions on the function begin{document}$ rho $end{document} , which is imposed in most works.
We prove the existence of a bounded positive solution for the following stationary Schrodinger equation begin{document}$ begin{equation*} -Delta u+V(x)u = f(x,u),,,, xinmathbb{R}^n,,, ngeq 3, end{equation*} $end{document} where begin{document}$ V $end{document} is a vanishing potential and begin{document}$ f $end{document} has a sublinear growth at the origin (for example if begin{document}$ f(x,u) $end{document} is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [ 6 ]. In addition, if begin{document}$ f $end{document} has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type begin{document}$ rho(x)f(u) $end{document} where begin{document}$ f $end{document} is a concave-convex function and begin{document}$ rho $end{document} satisfies the begin{document}$ mathrm{(H)} $end{document} property introduced in [ 6 ]. We also note that we do not impose any integrability assumptions on the function begin{document}$ rho $end{document} , which is imposed in most works.
{"title":"Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems","authors":"J. A. Cardoso, Patricio Cerda, Denilson S. Pereira, P. Ubilla","doi":"10.3934/dcds.2020392","DOIUrl":"https://doi.org/10.3934/dcds.2020392","url":null,"abstract":"We prove the existence of a bounded positive solution for the following stationary Schrodinger equation begin{document}$ begin{equation*} -Delta u+V(x)u = f(x,u),,,, xinmathbb{R}^n,,, ngeq 3, end{equation*} $end{document} where begin{document}$ V $end{document} is a vanishing potential and begin{document}$ f $end{document} has a sublinear growth at the origin (for example if begin{document}$ f(x,u) $end{document} is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [ 6 ]. In addition, if begin{document}$ f $end{document} has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type begin{document}$ rho(x)f(u) $end{document} where begin{document}$ f $end{document} is a concave-convex function and begin{document}$ rho $end{document} satisfies the begin{document}$ mathrm{(H)} $end{document} property introduced in [ 6 ]. We also note that we do not impose any integrability assumptions on the function begin{document}$ rho $end{document} , which is imposed in most works.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"904 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77497164","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}
We propose a simple and accurate procedure how to extract the values of model parameters in a flame/smoldering evolution equation from 2D movie images of real experiments. The procedure includes a novel method of image segmentation, which can detect an expanding smoldering front as a plane polygonal curve. The evolution equation is equivalent to the so-called Kuramoto-Sivashinsky (KS) equation in a certain scale. Our results suggest a valid range of parameters in the KS equation as well as the validity of the KS equation itself.
{"title":"A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies","authors":"M. Goto, K. Kuwana, Yasuhide Uegata, S. Yazaki","doi":"10.3934/dcdss.2020233","DOIUrl":"https://doi.org/10.3934/dcdss.2020233","url":null,"abstract":"We propose a simple and accurate procedure how to extract the values of model parameters in a flame/smoldering evolution equation from 2D movie images of real experiments. The procedure includes a novel method of image segmentation, which can detect an expanding smoldering front as a plane polygonal curve. The evolution equation is equivalent to the so-called Kuramoto-Sivashinsky (KS) equation in a certain scale. Our results suggest a valid range of parameters in the KS equation as well as the validity of the KS equation itself.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80280721","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}
In this paper, we are concerned with the phase stability testing at constant volume, temperature, and moles ( begin{document}$ VTN $end{document} -specification) of a multicomponent mixture, which is an unconstrained minimization problem. We present and compare the performance of five chosen optimization algorithms: Differential Evolution, Cuckoo Search, Harmony Search, CMA-ES, and Elephant Herding Optimization. For the comparison of the evolution strategies, we use the Wilcoxon signed-rank test. In addition, we compare the evolution strategies with the classical Newton-Raphson method based on the computation times. Moreover, we present the expanded mirroring technique, which mirrors the computed solution into a given simplex.
In this paper, we are concerned with the phase stability testing at constant volume, temperature, and moles ( begin{document}$ VTN $end{document} -specification) of a multicomponent mixture, which is an unconstrained minimization problem. We present and compare the performance of five chosen optimization algorithms: Differential Evolution, Cuckoo Search, Harmony Search, CMA-ES, and Elephant Herding Optimization. For the comparison of the evolution strategies, we use the Wilcoxon signed-rank test. In addition, we compare the evolution strategies with the classical Newton-Raphson method based on the computation times. Moreover, we present the expanded mirroring technique, which mirrors the computed solution into a given simplex.
{"title":"Comparison of modern heuristics on solving the phase stability testing problem","authors":"T. Smejkal, J. Mikyška, J. Kukal","doi":"10.3934/dcdss.2020227","DOIUrl":"https://doi.org/10.3934/dcdss.2020227","url":null,"abstract":"In this paper, we are concerned with the phase stability testing at constant volume, temperature, and moles ( begin{document}$ VTN $end{document} -specification) of a multicomponent mixture, which is an unconstrained minimization problem. We present and compare the performance of five chosen optimization algorithms: Differential Evolution, Cuckoo Search, Harmony Search, CMA-ES, and Elephant Herding Optimization. For the comparison of the evolution strategies, we use the Wilcoxon signed-rank test. In addition, we compare the evolution strategies with the classical Newton-Raphson method based on the computation times. Moreover, we present the expanded mirroring technique, which mirrors the computed solution into a given simplex.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"03 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85936041","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}
In this paper, we propose a time-delayed HIV/AIDS-PrEP model which takes into account the delay on pre-exposure prophylaxis (PrEP) distribution and adherence by uninfected persons that are in high risk of HIV infection, and analyze the impact of this delay on the number of individuals with HIV infection. We prove the existence and stability of two equilibrium points, for any positive time delay. After, an optimal control problem with state and control delays is proposed and analyzed, where the aim is to find the optimal strategy for PrEP implementation that minimizes the number of individuals with HIV infection, with minimal costs. Different scenarios are studied, for which the solutions derived from the Minimum Principle for Multiple Delayed Optimal Control Problems change depending on the values of the time delays and the weights constants associated with the number of HIV infected individuals and PrEP. We observe that changes on the weights constants can lead to a passage from bang-singular-bang to bang-bang extremal controls.
{"title":"Stability and optimal control of a delayed HIV/AIDS-PrEP model","authors":"Cristiana J. Silva","doi":"10.3934/dcdss.2021156","DOIUrl":"https://doi.org/10.3934/dcdss.2021156","url":null,"abstract":"In this paper, we propose a time-delayed HIV/AIDS-PrEP model which takes into account the delay on pre-exposure prophylaxis (PrEP) distribution and adherence by uninfected persons that are in high risk of HIV infection, and analyze the impact of this delay on the number of individuals with HIV infection. We prove the existence and stability of two equilibrium points, for any positive time delay. After, an optimal control problem with state and control delays is proposed and analyzed, where the aim is to find the optimal strategy for PrEP implementation that minimizes the number of individuals with HIV infection, with minimal costs. Different scenarios are studied, for which the solutions derived from the Minimum Principle for Multiple Delayed Optimal Control Problems change depending on the values of the time delays and the weights constants associated with the number of HIV infected individuals and PrEP. We observe that changes on the weights constants can lead to a passage from bang-singular-bang to bang-bang extremal controls.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76819254","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}
A new method for some advection equations is derived and analyzed, where the finite element method is constructed by using spline. A proper spline subspace is discussed for satisfying boundary conditions. Meanwhile, in order to get more accuracy solutions, spline method is connected with finite element method. Furthermore, the stability and convergence are discussed rigorously. Two numerical experiments are also presented to verify the theoretical analysis.
{"title":"Finite element method for two-dimensional linear advection equations based on spline method","authors":"Kai Qu, Qiannan Dong, Chanjie Li, Feiyu Zhang","doi":"10.3934/DCDSS.2021056","DOIUrl":"https://doi.org/10.3934/DCDSS.2021056","url":null,"abstract":"A new method for some advection equations is derived and analyzed, where the finite element method is constructed by using spline. A proper spline subspace is discussed for satisfying boundary conditions. Meanwhile, in order to get more accuracy solutions, spline method is connected with finite element method. Furthermore, the stability and convergence are discussed rigorously. Two numerical experiments are also presented to verify the theoretical analysis.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77155521","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}
T. Oberhuber, T. Dytrych, K. Launey, D. Langr, J. Draayer
Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the same. We find that the CUDA implementation delivers speedups of 2.27x – 5.93x.
{"title":"Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs","authors":"T. Oberhuber, T. Dytrych, K. Launey, D. Langr, J. Draayer","doi":"10.3934/dcdss.2020383","DOIUrl":"https://doi.org/10.3934/dcdss.2020383","url":null,"abstract":"Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the same. We find that the CUDA implementation delivers speedups of 2.27x – 5.93x.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"134 6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85517580","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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