Pub Date : 2017-07-03DOI: 10.1080/1726037X.2017.1390847
T. Bayrakdar, A. A. Ergin
Abstract Hamiltonian vector field, Poisson vector field and the gradient of Hamiltonian function defines Darboux frame along an integral curve of a Hamiltonian dynamical system on a surface whose normal vector field corresponds to the Poisson structure for a given Hamiltonian system. We show that the existence of compatible Poisson structures determined by the normal legs of the Darboux frame is resolved to the characteristic equation for the Weingarten map. We also show that a Hamiltonian dynamical system in three dimensions has bi-Hamiltonian representation determined by the normal legs of Frenet-Serret triad if and only if an integral curve of Hamiltonian vector field is both a geodesic and a line of curvature.
{"title":"Hamiltonian dynamical systems and geometry of surfaces in 3-D","authors":"T. Bayrakdar, A. A. Ergin","doi":"10.1080/1726037X.2017.1390847","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1390847","url":null,"abstract":"Abstract Hamiltonian vector field, Poisson vector field and the gradient of Hamiltonian function defines Darboux frame along an integral curve of a Hamiltonian dynamical system on a surface whose normal vector field corresponds to the Poisson structure for a given Hamiltonian system. We show that the existence of compatible Poisson structures determined by the normal legs of the Darboux frame is resolved to the characteristic equation for the Weingarten map. We also show that a Hamiltonian dynamical system in three dimensions has bi-Hamiltonian representation determined by the normal legs of Frenet-Serret triad if and only if an integral curve of Hamiltonian vector field is both a geodesic and a line of curvature.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"163 - 176"},"PeriodicalIF":0.9,"publicationDate":"2017-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1390847","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45094456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-03DOI: 10.1080/1726037X.2017.1411042
Mohamd Saleem Lone, Osmar Aléssio, Mohammed Jamali, M. Shahid
Abstract In this paper, we present the algorithms for calculating the differential geometric quantities {t, n, b1, b2, b3, k2, k3, k4}, geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space ℝ5. In transversal intersection, the normals of the surfaces at the intersection point are linearly independent, while as in nontransversal intersection, the normals of the surfaces at the intersection point axe linearly dependent.
{"title":"Differential geometry of transversal intersection curves of hypersurfaces in ℝ5","authors":"Mohamd Saleem Lone, Osmar Aléssio, Mohammed Jamali, M. Shahid","doi":"10.1080/1726037X.2017.1411042","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1411042","url":null,"abstract":"Abstract In this paper, we present the algorithms for calculating the differential geometric quantities {t, n, b1, b2, b3, k2, k3, k4}, geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space ℝ5. In transversal intersection, the normals of the surfaces at the intersection point are linearly independent, while as in nontransversal intersection, the normals of the surfaces at the intersection point axe linearly dependent.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"147 - 162"},"PeriodicalIF":0.9,"publicationDate":"2017-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1411042","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47792089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-02DOI: 10.1080/1726037X.2017.1327162
A. Guezane-Lakoud, R. Khaldi
ABSTRACT This paper concerns the existence of solution to an initial fractional problem of arbitrary order. The main tools for this study are the lower ad upper solutions method and Schauder’s fixed point Theorem.
{"title":"Upper and lower solutions method for higher order fractional initial value problems","authors":"A. Guezane-Lakoud, R. Khaldi","doi":"10.1080/1726037X.2017.1327162","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1327162","url":null,"abstract":"ABSTRACT This paper concerns the existence of solution to an initial fractional problem of arbitrary order. The main tools for this study are the lower ad upper solutions method and Schauder’s fixed point Theorem.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"29 - 35"},"PeriodicalIF":0.9,"publicationDate":"2017-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1327162","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46716713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-02DOI: 10.1080/1726037X.2017.1323417
A. Hasmani, R. Panchal
Abstract It is known that electric and magnetic parts of the Weyl tensor in General Relativity are analogous to that occur in the theory of electromagnetism. On the other hand, Newman-Penrose formalism provides a new dimension to the solutions and interpretations for the theory of relativity. Its significance in understanding spacetimes with some particular Petrov types, solutions with different material contents is well known. This paper provides spin coefficient form of electric and magnetic parts of the Weyl tensor, which leads to a new and efficient way for the computation of electric and magnetic parts of the Weyl tensor. It is observed that the computational time is greatly reduced when done using computer algebra system in comparison to that is done manually. This technique is elaborated in the example of Pure Radiation metric.
{"title":"Electric and magnetic parts of the Weyl tensor and spin coefficients","authors":"A. Hasmani, R. Panchal","doi":"10.1080/1726037X.2017.1323417","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1323417","url":null,"abstract":"Abstract It is known that electric and magnetic parts of the Weyl tensor in General Relativity are analogous to that occur in the theory of electromagnetism. On the other hand, Newman-Penrose formalism provides a new dimension to the solutions and interpretations for the theory of relativity. Its significance in understanding spacetimes with some particular Petrov types, solutions with different material contents is well known. This paper provides spin coefficient form of electric and magnetic parts of the Weyl tensor, which leads to a new and efficient way for the computation of electric and magnetic parts of the Weyl tensor. It is observed that the computational time is greatly reduced when done using computer algebra system in comparison to that is done manually. This technique is elaborated in the example of Pure Radiation metric.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"37 - 49"},"PeriodicalIF":0.9,"publicationDate":"2017-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1323417","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43823810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-02DOI: 10.1080/1726037X.2017.1327163
D. Reddy, T. Kanakavalli, G. A. Rao
Abstract A locally rotationally symmetric (LRS) Bianchi type-II spacetime is considered in the presence of dark matter and anisotropic modified holographic Ricci dark energy in the scalar-tensor theory of Saez and Ballester (Phys. Lett. A 113: 467, 1986). To obtain exact solutions of Saez-Ballester gravitational field equations we use the following physically significant conditions: (i) the scalar expansion of the space time is proportional to its shear scalar, (ii) hybrid expansion law for the scale factor proposed by Akarsu et al. (JCAP 01: 022, 2014) and (iii) modified Ricci dark energy density given by Chen and Jing (Phys. Lett. B 679: 144, 2009). The solution describes modified holographic Ricci dark energy models, in this theory, which exhibit a smooth transition from decelerated phase to the present accelerated phase of the universe. This result is in accordance with the present day observations of cosmology. The dynamical parameters of the models are computed and their physical importance, in cosmology, is discussed.
{"title":"LRS Bianchi Type-II modified holographic Ricci dark energy models in a scalar-tensor theory of gravitation","authors":"D. Reddy, T. Kanakavalli, G. A. Rao","doi":"10.1080/1726037X.2017.1327163","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1327163","url":null,"abstract":"Abstract A locally rotationally symmetric (LRS) Bianchi type-II spacetime is considered in the presence of dark matter and anisotropic modified holographic Ricci dark energy in the scalar-tensor theory of Saez and Ballester (Phys. Lett. A 113: 467, 1986). To obtain exact solutions of Saez-Ballester gravitational field equations we use the following physically significant conditions: (i) the scalar expansion of the space time is proportional to its shear scalar, (ii) hybrid expansion law for the scale factor proposed by Akarsu et al. (JCAP 01: 022, 2014) and (iii) modified Ricci dark energy density given by Chen and Jing (Phys. Lett. B 679: 144, 2009). The solution describes modified holographic Ricci dark energy models, in this theory, which exhibit a smooth transition from decelerated phase to the present accelerated phase of the universe. This result is in accordance with the present day observations of cosmology. The dynamical parameters of the models are computed and their physical importance, in cosmology, is discussed.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"85 - 97"},"PeriodicalIF":0.9,"publicationDate":"2017-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1327163","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47525712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-02DOI: 10.1080/1726037X.2017.1323416
T. Körpınar
Abstract In this paper, we investigate inextensible flows of curves according to alternative moving frame in Euclidean space 𝔼3. Using the Prenet frame of the given curve, we present partial differential equations. The concepts with the inextensible flows are analyzed by using alternative moving frame.
{"title":"On Inextensible flows of curves according to alternative moving frame","authors":"T. Körpınar","doi":"10.1080/1726037X.2017.1323416","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1323416","url":null,"abstract":"Abstract In this paper, we investigate inextensible flows of curves according to alternative moving frame in Euclidean space 𝔼3. Using the Prenet frame of the given curve, we present partial differential equations. The concepts with the inextensible flows are analyzed by using alternative moving frame.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"15 - 27"},"PeriodicalIF":0.9,"publicationDate":"2017-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1323416","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49594017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-02DOI: 10.1080/1726037X.2017.1323415
A. Uçum, M. Sakaki, K. Ilarslan
Abstract In this paper we give the necessary and sufficient conditions for bi-null curves in ℝ63 to be osculating, normal or rectifying curves in terms of their curvature functions.
摘要本文给出了中双零曲线的充要条件ℝ63在它们的曲率函数方面是密切曲线、法线曲线或整流曲线。
{"title":"On osculating, normal and rectifying bi-null curves in ℝ63","authors":"A. Uçum, M. Sakaki, K. Ilarslan","doi":"10.1080/1726037X.2017.1323415","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1323415","url":null,"abstract":"Abstract In this paper we give the necessary and sufficient conditions for bi-null curves in ℝ63 to be osculating, normal or rectifying curves in terms of their curvature functions.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"1 - 13"},"PeriodicalIF":0.9,"publicationDate":"2017-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1323415","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44300350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-12-19DOI: 10.1080/1726037X.2017.1324587
M. F. Nia
ABSTRACT In this paper, we focus attention on extending the topological conjugacy of adding machine maps and minimal systems to iterated function systems. We provide necessary and sufficient conditions for an iterated function system to be conjugated to an adding machine map. It is proved that every minimal iterated function system which has some non-periodic regular point is semi-conjugate to an adding machine map. Furthermore, we investigate the topological conjugacy of an infinite family of tent maps, as well as the restriction of a map to its ω—limit set with an iterated function system.
{"title":"Adding machine maps and minimal sets for iterated function systems","authors":"M. F. Nia","doi":"10.1080/1726037X.2017.1324587","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1324587","url":null,"abstract":"ABSTRACT In this paper, we focus attention on extending the topological conjugacy of adding machine maps and minimal systems to iterated function systems. We provide necessary and sufficient conditions for an iterated function system to be conjugated to an adding machine map. It is proved that every minimal iterated function system which has some non-periodic regular point is semi-conjugate to an adding machine map. Furthermore, we investigate the topological conjugacy of an infinite family of tent maps, as well as the restriction of a map to its ω—limit set with an iterated function system.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"71 - 83"},"PeriodicalIF":0.9,"publicationDate":"2016-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1324587","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-09-26DOI: 10.1080/1726037X.2021.2011110
S. Tchuiaga
Abstract This paper meticulously revisit and study the flux geometry of any compact connected oriented manifold (M, Ω). We generalize several well- known factorization results, exhibit some orbital conditions under which flux geometry can be studied, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, derive that any smooth isotopy in the group of all vanishing-flux volume-preserving diffeomorphisms is a vanishing- flux path, and show that the kernel of flux for volume-preserving diffeomorphisms is C 1−closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map: We recover several well-known results from symplectic geometry. We use the above studies to construct a right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map and study the induced geometry. In the case of a symplectic volume form, the restriction of our metric to the group Ham(N, ω), of all Hamiltonian diffeomorphisms of a closed symplectic manifold (N, ω), is controlled from above by the usual Hofer metric in general, while the Hofer-like metric control our metric in the case where the Riemannian structure is compatible with the symplectic structure (in particular, our construction implies the non-degeneracy of the Hofer and Hofer-like norms).
{"title":"Hofer-Like Geometry and Flux Theory","authors":"S. Tchuiaga","doi":"10.1080/1726037X.2021.2011110","DOIUrl":"https://doi.org/10.1080/1726037X.2021.2011110","url":null,"abstract":"Abstract This paper meticulously revisit and study the flux geometry of any compact connected oriented manifold (M, Ω). We generalize several well- known factorization results, exhibit some orbital conditions under which flux geometry can be studied, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, derive that any smooth isotopy in the group of all vanishing-flux volume-preserving diffeomorphisms is a vanishing- flux path, and show that the kernel of flux for volume-preserving diffeomorphisms is C 1−closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map: We recover several well-known results from symplectic geometry. We use the above studies to construct a right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map and study the induced geometry. In the case of a symplectic volume form, the restriction of our metric to the group Ham(N, ω), of all Hamiltonian diffeomorphisms of a closed symplectic manifold (N, ω), is controlled from above by the usual Hofer metric in general, while the Hofer-like metric control our metric in the case where the Riemannian structure is compatible with the symplectic structure (in particular, our construction implies the non-degeneracy of the Hofer and Hofer-like norms).","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"19 1","pages":"227 - 270"},"PeriodicalIF":0.9,"publicationDate":"2016-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60350292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-07-02DOI: 10.1080/1726037X.2016.1250503
T. Waezizadeh
Abstract In this paper we introduce the SEIR epidemic model with latent and infectious time delays, which are denoted by ω and τ respectively. As follows we consider two different cases, ω = 0 and ω ≠ 0 ≠ τ. Stability near disease-free and endemic equilibrium points in different cases are investigated.
{"title":"Seir epidemic model with two time delays","authors":"T. Waezizadeh","doi":"10.1080/1726037X.2016.1250503","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1250503","url":null,"abstract":"Abstract In this paper we introduce the SEIR epidemic model with latent and infectious time delays, which are denoted by ω and τ respectively. As follows we consider two different cases, ω = 0 and ω ≠ 0 ≠ τ. Stability near disease-free and endemic equilibrium points in different cases are investigated.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"189 - 200"},"PeriodicalIF":0.9,"publicationDate":"2016-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1250503","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349695","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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