Pub Date : 2016-07-02DOI: 10.1080/1726037X.2016.1250500
K. Yamasaki, T. Yajima
Abstract We considered the differential geometric structure of non-equilibrium dynamics in non-linear interactions, such as competition and predation, based on Kosambi-Cartan-Chern (KCC) theory. The stability of a geodesic flow on a Finslerian manifold is characterized by the deviation curvature (the second invariant in the dynamical system). According to KCC theory, the value of the deviation curvature is constant around the equilibrium point. However, in the non-equilibrium region, not only the value but also the sign of the deviation curvature depend on time. Next, we reapplied KCC theory to the dynamics of the deviation curvature and determined the hierarchical structure of the geometric stability. The dynamics of the deviation curvature in the nonequilibrium region is accompanied by a complex periodic (node) pattern in the predation (competition) system.
{"title":"Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory","authors":"K. Yamasaki, T. Yajima","doi":"10.1080/1726037X.2016.1250500","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1250500","url":null,"abstract":"Abstract We considered the differential geometric structure of non-equilibrium dynamics in non-linear interactions, such as competition and predation, based on Kosambi-Cartan-Chern (KCC) theory. The stability of a geodesic flow on a Finslerian manifold is characterized by the deviation curvature (the second invariant in the dynamical system). According to KCC theory, the value of the deviation curvature is constant around the equilibrium point. However, in the non-equilibrium region, not only the value but also the sign of the deviation curvature depend on time. Next, we reapplied KCC theory to the dynamics of the deviation curvature and determined the hierarchical structure of the geometric stability. The dynamics of the deviation curvature in the nonequilibrium region is accompanied by a complex periodic (node) pattern in the predation (competition) system.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"137 - 153"},"PeriodicalIF":0.9,"publicationDate":"2016-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1250500","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349257","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.2016.1250499
Nakone Bello, J. Singh
Abstract This paper studies the motion of a third body (test particle) in the vicinity of the triangular points L4,5 by considering the more massive as a triaxial body in the frame work of the relativistic restricted three-body problem (R3BP). It is seen that the positions and stability of the triangular points are affected by both relativistic and triaxiality factors. It turns out both the coordinates of the infinitesimal mass axe affected. It is seen that for these points, the range of stability region increases or decreases according as p>0 or p<0 where p depends upon the triaxiality and relativistic factors. Furthermore we have studied the periodic orbits around the triangular points in the range 0 < µ < µc. It is found that these orbits axe elliptical; the frequencies of long and short orbits of the periodic motion,the eccentricities,semi-major and semi-minor axes, orientation and coefficients of long and short periodic terms are all affected by triaxiality and relativistic factors.
{"title":"On the stability of the triangular points in the relativistic R3BP with a bigger triaxial primary","authors":"Nakone Bello, J. Singh","doi":"10.1080/1726037X.2016.1250499","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1250499","url":null,"abstract":"Abstract This paper studies the motion of a third body (test particle) in the vicinity of the triangular points L4,5 by considering the more massive as a triaxial body in the frame work of the relativistic restricted three-body problem (R3BP). It is seen that the positions and stability of the triangular points are affected by both relativistic and triaxiality factors. It turns out both the coordinates of the infinitesimal mass axe affected. It is seen that for these points, the range of stability region increases or decreases according as p>0 or p<0 where p depends upon the triaxiality and relativistic factors. Furthermore we have studied the periodic orbits around the triangular points in the range 0 < µ < µc. It is found that these orbits axe elliptical; the frequencies of long and short orbits of the periodic motion,the eccentricities,semi-major and semi-minor axes, orientation and coefficients of long and short periodic terms are all affected by triaxiality and relativistic factors.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"119 - 136"},"PeriodicalIF":0.9,"publicationDate":"2016-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1250499","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349096","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.2016.1250498
B. Kaur, R. Aggarwal, S. Yadav
Abstract The aim of this paper is to study the effect of perturbations in the Coriolis and centrifugal forces on the location and stability of the equilibrium solutions in the Robe’s restricted problem of 2+2 bodies under the assumption that the hydrostatic equilibrium figure of the first primary is a Roche ellipsoid and the shape of the second primary is triaxial. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the ellipsoid, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m2 is describing a circle around m1. The masses m3 and m4 mutually attract each other, do not influence the motion of m1 and m2 but are influenced by them. We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii)that originating in the attraction of m2 (iii) that arising from the centrifugal force. The linear stability of this configuration is examined. It is observed that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. The equilibrium solutions of m3 and m4 lying on x1-axis are unstable for ε > 0, ε′ > 0 and ε < 0, ε′ > 0 and stable for ε > 0, ε′ < 0 and ε < 0, ε′ < 0 ,using the data of submarines in the Earth -Moon system. The equilibrium solutions of m3 and m4 respectively when the displacement is given in the direction of x2 or x3− axis are conditionally stable.We observe that the conditions of stability are influenced by the small perturbations in the Coriolis and centrifugal forces.
{"title":"Perturbed Robe’s restricted problem of 2+2 bodies when the primaries form a Roche ellipsoid-triaxial system","authors":"B. Kaur, R. Aggarwal, S. Yadav","doi":"10.1080/1726037X.2016.1250498","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1250498","url":null,"abstract":"Abstract The aim of this paper is to study the effect of perturbations in the Coriolis and centrifugal forces on the location and stability of the equilibrium solutions in the Robe’s restricted problem of 2+2 bodies under the assumption that the hydrostatic equilibrium figure of the first primary is a Roche ellipsoid and the shape of the second primary is triaxial. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the ellipsoid, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m2 is describing a circle around m1. The masses m3 and m4 mutually attract each other, do not influence the motion of m1 and m2 but are influenced by them. We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii)that originating in the attraction of m2 (iii) that arising from the centrifugal force. The linear stability of this configuration is examined. It is observed that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. The equilibrium solutions of m3 and m4 lying on x1-axis are unstable for ε > 0, ε′ > 0 and ε < 0, ε′ > 0 and stable for ε > 0, ε′ < 0 and ε < 0, ε′ < 0 ,using the data of submarines in the Earth -Moon system. The equilibrium solutions of m3 and m4 respectively when the displacement is given in the direction of x2 or x3− axis are conditionally stable.We observe that the conditions of stability are influenced by the small perturbations in the Coriolis and centrifugal forces.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"117 - 99"},"PeriodicalIF":0.9,"publicationDate":"2016-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1250498","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349135","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.2016.1250502
Amele Taieb, Z. Dahmani
Abstract In this paper, we introduce a new class of nonlinear singular fractional differential equations. We axe interested in the singularity by using the contraction mapping principle and Schauder fixed point theorem. We present new results on the existence and uniqueness of solutions. Moreover, we investigate the Ulam-Hyers stability and the generalized Ulam-Hyers stability for this fractional equations. Some examples are provided to illustrate the application of our results.
{"title":"A new problem of singular fractional differential equations","authors":"Amele Taieb, Z. Dahmani","doi":"10.1080/1726037X.2016.1250502","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1250502","url":null,"abstract":"Abstract In this paper, we introduce a new class of nonlinear singular fractional differential equations. We axe interested in the singularity by using the contraction mapping principle and Schauder fixed point theorem. We present new results on the existence and uniqueness of solutions. Moreover, we investigate the Ulam-Hyers stability and the generalized Ulam-Hyers stability for this fractional equations. Some examples are provided to illustrate the application of our results.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"165 - 187"},"PeriodicalIF":0.9,"publicationDate":"2016-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1250502","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349502","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 : 2016-06-26DOI: 10.1080/1726037X.2017.1323418
H. El-Sayied, S. Shenawy, N. Syied
ABSTRACT The purpose of the present article is to study and characterize several types of symmetries of generalized Robertson-Walker spacetimes. Conformal vector fields, curvature and Ricci collineations are studied. Many implications for existence of these symmetries on generalized Robertson-Walker spacetimes are obtained. Finally, Ricci solitons on generalized Robertson-Walker spacetimes admitting conformal vector fields are investigated.
{"title":"On symmetries of generalized Robertson-Walker spacetimes and applications","authors":"H. El-Sayied, S. Shenawy, N. Syied","doi":"10.1080/1726037X.2017.1323418","DOIUrl":"https://doi.org/10.1080/1726037X.2017.1323418","url":null,"abstract":"ABSTRACT The purpose of the present article is to study and characterize several types of symmetries of generalized Robertson-Walker spacetimes. Conformal vector fields, curvature and Ricci collineations are studied. Many implications for existence of these symmetries on generalized Robertson-Walker spacetimes are obtained. Finally, Ricci solitons on generalized Robertson-Walker spacetimes admitting conformal vector fields are investigated.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"15 1","pages":"51 - 69"},"PeriodicalIF":0.9,"publicationDate":"2016-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2017.1323418","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349795","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 : 2016-01-02DOI: 10.1080/1726037X.2016.1177922
B. Pal
Abstract In this paper, we study mixed super quasi-Einstein warped product manifolds for arbitrary dimension n ≥ 3 and we give an example of mixed super quasi-Einstein manifold (MS(QE)n) to ensure the existence of such manifold. Also in the last section we also give an example of warped product on mixed super quasi-Einstein manifold.
{"title":"On mixed super quasi-Einstein warped products","authors":"B. Pal","doi":"10.1080/1726037X.2016.1177922","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1177922","url":null,"abstract":"Abstract In this paper, we study mixed super quasi-Einstein warped product manifolds for arbitrary dimension n ≥ 3 and we give an example of mixed super quasi-Einstein manifold (MS(QE)n) to ensure the existence of such manifold. Also in the last section we also give an example of warped product on mixed super quasi-Einstein manifold.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"35 - 50"},"PeriodicalIF":0.9,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1177922","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60348766","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 : 2016-01-02DOI: 10.1080/1726037X.2016.1177930
Nakone Bello, J. Singh
Abstract In this paper, we study the effect of oblateness of the more massive primary in the relativistic R3BP. We observe that the locations of the triangular points and their stability are affected by the relativistic and oblateness factors. It is also noticed that the oblateness factor possesses destabilizing behavior. Therefore, the size of the region of stability decreases with increase in the value of the oblateness factor. Further, a numerical study on the locations of the triangular points and the critical mass for the Earth-Moon , Jupiter and its Moons, Saturn and its Moons systems is given.
{"title":"Stability of triangular points in the relativistic R3BP when the bigger primary is an oblate spheroid","authors":"Nakone Bello, J. Singh","doi":"10.1080/1726037X.2016.1177930","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1177930","url":null,"abstract":"Abstract In this paper, we study the effect of oblateness of the more massive primary in the relativistic R3BP. We observe that the locations of the triangular points and their stability are affected by the relativistic and oblateness factors. It is also noticed that the oblateness factor possesses destabilizing behavior. Therefore, the size of the region of stability decreases with increase in the value of the oblateness factor. Further, a numerical study on the locations of the triangular points and the critical mass for the Earth-Moon , Jupiter and its Moons, Saturn and its Moons systems is given.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"51 - 64"},"PeriodicalIF":0.9,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1177930","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60348869","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 : 2016-01-02DOI: 10.1080/1726037X.2016.1177935
Ö. Bektaş, Nurten (Bayrak) Gürses, S. Yüce
Abstract In this study, the osculating curves in Euclidean space E3 and E4, well known in differential geometry, are studied through the instrumentality of quaternions. We inoculate sundry delineations for quaternionic osculating curves in the Euclidean space E3, then we portray the quaternionic osculating curve in E4 as a quaternionic curve whose position vector every time reclines in the orthogonal complement N½ (or N⅓) of its first binormal vector field N2 (or N3), where {T,N1,N2,N3} be the Frenet instrumentations of the quaternionic curve in the Euclidean space E4. We feature quaternionic osculating curves from the point of view their curvature functions K, k and (r — K) and serve the necessary and the sufficient conditions for arbitrary quaternionic curve in E4 to be a quaternionic osculating. Moreover, we gain an explicit equation of a quaternionic osculating curve in E4. In the last two section, we described quaternionic osculating curves in the semi-Euclidean space and some theorems are testified.
{"title":"Quaternionic osculating curves in Euclidean and semi-Euclidean space","authors":"Ö. Bektaş, Nurten (Bayrak) Gürses, S. Yüce","doi":"10.1080/1726037X.2016.1177935","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1177935","url":null,"abstract":"Abstract In this study, the osculating curves in Euclidean space E3 and E4, well known in differential geometry, are studied through the instrumentality of quaternions. We inoculate sundry delineations for quaternionic osculating curves in the Euclidean space E3, then we portray the quaternionic osculating curve in E4 as a quaternionic curve whose position vector every time reclines in the orthogonal complement N½ (or N⅓) of its first binormal vector field N2 (or N3), where {T,N1,N2,N3} be the Frenet instrumentations of the quaternionic curve in the Euclidean space E4. We feature quaternionic osculating curves from the point of view their curvature functions K, k and (r — K) and serve the necessary and the sufficient conditions for arbitrary quaternionic curve in E4 to be a quaternionic osculating. Moreover, we gain an explicit equation of a quaternionic osculating curve in E4. In the last two section, we described quaternionic osculating curves in the semi-Euclidean space and some theorems are testified.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"65 - 84"},"PeriodicalIF":0.9,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1177935","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60348620","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 : 2016-01-02DOI: 10.1080/1726037X.2016.1177920
O. Bahadır
Abstract The object of the present paper is to study quarter symmetric nonmetric connection on a LP-Sasakian manifold. In this paper, we consider some properties of the curvature tensor, projective curvature tensor, concircular curvature tensor, conformai curvature tensor with respect to quarter symmetric non-metric connection in a LP-Sasakian manifolds. Finally we consider submanifolds with respect to quarter symmetric non-metric connection.
{"title":"Lorentzian para-Sasakian manifold with quarter-symmetric non-metric connection","authors":"O. Bahadır","doi":"10.1080/1726037X.2016.1177920","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1177920","url":null,"abstract":"Abstract The object of the present paper is to study quarter symmetric nonmetric connection on a LP-Sasakian manifold. In this paper, we consider some properties of the curvature tensor, projective curvature tensor, concircular curvature tensor, conformai curvature tensor with respect to quarter symmetric non-metric connection in a LP-Sasakian manifolds. Finally we consider submanifolds with respect to quarter symmetric non-metric connection.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"17 - 33"},"PeriodicalIF":0.9,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1177920","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60348689","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 : 2016-01-02DOI: 10.1080/1726037X.2016.1177936
S. Dey, A. Bhattacharyya
Abstract The object of the present paper is to study the pseudo-projective φ-recurrent and generalized projective recurrent Lorentzian α-Sasakian manifolds. Here we show that pseudo-projective φ-recurrent Lorentzian α-Sasakian Manifold is an Einstein manifold and in the case of generalized projective φ- recurrent Lorentzian α-Sasakian manifold, we find a relation between the associated 1-forms A and B. We have also proved that the characteristic vector field ξ and vector field ρ associated to the 1-forms A and B are co-directional. We also study quasi-projectively flat Lorentzian α-Sasakian manifolds.
{"title":"Some curvature properties of Lorentzian α-Sasakian manifolds","authors":"S. Dey, A. Bhattacharyya","doi":"10.1080/1726037X.2016.1177936","DOIUrl":"https://doi.org/10.1080/1726037X.2016.1177936","url":null,"abstract":"Abstract The object of the present paper is to study the pseudo-projective φ-recurrent and generalized projective recurrent Lorentzian α-Sasakian manifolds. Here we show that pseudo-projective φ-recurrent Lorentzian α-Sasakian Manifold is an Einstein manifold and in the case of generalized projective φ- recurrent Lorentzian α-Sasakian manifold, we find a relation between the associated 1-forms A and B. We have also proved that the characteristic vector field ξ and vector field ρ associated to the 1-forms A and B are co-directional. We also study quasi-projectively flat Lorentzian α-Sasakian manifolds.","PeriodicalId":42788,"journal":{"name":"Journal of Dynamical Systems and Geometric Theories","volume":"14 1","pages":"85 - 98"},"PeriodicalIF":0.9,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/1726037X.2016.1177936","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60349005","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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