The global phase portrait describes the qualitative behaviour of the solution set for all time. In general, this is as close as we can get to solving nonlinear systems. The question of particular interest is: For what...The global phase portrait describes the qualitative behaviour of the solution set for all time. In general, this is as close as we can get to solving nonlinear systems. The question of particular interest is: For what parameter values does the global phase portrait of a dynamical system change its qualitative structure? In this paper, we attempt to answer the above question specifically for the case of certain third order nonlinear differential equations of the form . The linear case where is also considered. Our phase portrait analysis shows that under certain conditions on the coefficients as well as the function , we have asymptotic stability of solutions.展开更多
Influence of recombination centers’ changes on the form of phase portraits has been studied. It has been shown that the shape of the phase portraits depends on the concentration of semiconductor materials’ recombina...Influence of recombination centers’ changes on the form of phase portraits has been studied. It has been shown that the shape of the phase portraits depends on the concentration of semiconductor materials’ recombination centers.展开更多
Theoretical investigation of generation-recombination processes in silicon, which has a lifetime of charge carriers 10-3 s and capture cross sections of 10-16 sm2. For the study uses a method of phase portraits, which...Theoretical investigation of generation-recombination processes in silicon, which has a lifetime of charge carriers 10-3 s and capture cross sections of 10-16 sm2. For the study uses a method of phase portraits, which are widely used in the theory of vibrations. It is shown that the form of phase portraits strongly depends on the frequency of exposure to the external variable deformation.展开更多
This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+...This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+a_(4)xy^(3)+a_(5)xy^(4)=xσ(y),and any zero of 1+a_(1)y+a_(2)y^(2)+a_(3)y^(3)+a_(4)y^(4)+a_(5)y^(5),y=y is an invariant straight line.At last,all global phase portraits are drawn on the Poincare disk.展开更多
The Newton diagram and, in particular, the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine the existence of characteristic orbits and separatrices of an isolated singular...The Newton diagram and, in particular, the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine the existence of characteristic orbits and separatrices of an isolated singular point. We give an easy algorithm for obtaining the local phase portrait near the origin of a bi-dimensional differential system and we provide several examples.展开更多
In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex e...In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.展开更多
This paper deals with the bifurcations and phase portraits of an asymmetric triaxial gyrostat with two rotors, which is a 3-dimensional generalized Hamiltonian system with a quadratic Hamiltonian depending on three in...This paper deals with the bifurcations and phase portraits of an asymmetric triaxial gyrostat with two rotors, which is a 3-dimensional generalized Hamiltonian system with a quadratic Hamiltonian depending on three independent parameters. The number and stability of equilibria are analyzed, and corresponding bifurcation conditions of parameters are obtained. Moreover, by Maple software, all possible phase portraits are plotted out. Except for some planar orbits under particular parametric conditions, general orbits can not be expressed in terms of elementary or elliptic functions.展开更多
In this paper,a quartic Hamiltonian system with Z5-equivariant property is considered.Using the methods of qualitative analysis,bifurcations of the above system are analyzed,the phase portraits of the system are class...In this paper,a quartic Hamiltonian system with Z5-equivariant property is considered.Using the methods of qualitative analysis,bifurcations of the above system are analyzed,the phase portraits of the system are classified and representative orbits are shown by Maple software.展开更多
In this paper, we consider the Z8-equivariant planar Hamiltonian vector field of degree 7. By using the qualitative and numerical computation, we divide the parameters space into six-parameter-space. And we obtain the...In this paper, we consider the Z8-equivariant planar Hamiltonian vector field of degree 7. By using the qualitative and numerical computation, we divide the parameters space into six-parameter-space. And we obtain the results as following : 1. There are seven cases of the number of fixed point of above vector field in finite part, that is, 1,9,l7,25,4l,49, respectively. 2. The possible phase portraits of this vector field are fifty.展开更多
By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its...By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations.展开更多
文摘The global phase portrait describes the qualitative behaviour of the solution set for all time. In general, this is as close as we can get to solving nonlinear systems. The question of particular interest is: For what parameter values does the global phase portrait of a dynamical system change its qualitative structure? In this paper, we attempt to answer the above question specifically for the case of certain third order nonlinear differential equations of the form . The linear case where is also considered. Our phase portrait analysis shows that under certain conditions on the coefficients as well as the function , we have asymptotic stability of solutions.
文摘Influence of recombination centers’ changes on the form of phase portraits has been studied. It has been shown that the shape of the phase portraits depends on the concentration of semiconductor materials’ recombination centers.
文摘Theoretical investigation of generation-recombination processes in silicon, which has a lifetime of charge carriers 10-3 s and capture cross sections of 10-16 sm2. For the study uses a method of phase portraits, which are widely used in the theory of vibrations. It is shown that the form of phase portraits strongly depends on the frequency of exposure to the external variable deformation.
基金supported by National Natural Science Foundation of China(No.12301197)Natural Science Foundation of Henan(No.232300420343)+2 种基金Science and Technology Research Project of Henan Province(No.232102210057)Scientific Research Foundation for Doctoral Scholars of Haust(No.13480077)Natural Science Foundation of Hunan(No.2021JJ30166)。
文摘This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator.Such systems have the form x=-y+xf(x,y),y=x+yf(x,y),where f(x,y)=a_(1)x+a_(2)xy+a_(3)xy^(2)+a_(4)xy^(3)+a_(5)xy^(4)=xσ(y),and any zero of 1+a_(1)y+a_(2)y^(2)+a_(3)y^(3)+a_(4)y^(4)+a_(5)y^(5),y=y is an invariant straight line.At last,all global phase portraits are drawn on the Poincare disk.
基金Supported by Ministerio de Ciencia y Tecnología,Plan Nacional I+D+I co-financed with FEDER funds,in the frame of the pro jects MTM2010-20907-C02-02by Consejería de Educación y Ciencia de la Junta de Andalucía(Grant Nos.FQM-276 and P08-FQM-03770)
文摘The Newton diagram and, in particular, the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine the existence of characteristic orbits and separatrices of an isolated singular point. We give an easy algorithm for obtaining the local phase portrait near the origin of a bi-dimensional differential system and we provide several examples.
基金partially supported by a MINECO/FEDER grant MTM2013-40998-Pan AGAUR grant number 2014 SGR568+2 种基金the grants FP7-PEOPLE-2012-IRSES 318999 and 316338the MINECO/FEDER grant UNAB13-4E-1604partially supported by FCT/Portugal through UID/MAT/04459/2013
文摘In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.
基金supported by the NNSF of China under Grant No.10872183
文摘This paper deals with the bifurcations and phase portraits of an asymmetric triaxial gyrostat with two rotors, which is a 3-dimensional generalized Hamiltonian system with a quadratic Hamiltonian depending on three independent parameters. The number and stability of equilibria are analyzed, and corresponding bifurcation conditions of parameters are obtained. Moreover, by Maple software, all possible phase portraits are plotted out. Except for some planar orbits under particular parametric conditions, general orbits can not be expressed in terms of elementary or elliptic functions.
文摘In this paper,a quartic Hamiltonian system with Z5-equivariant property is considered.Using the methods of qualitative analysis,bifurcations of the above system are analyzed,the phase portraits of the system are classified and representative orbits are shown by Maple software.
基金National Natural Science Fundation of P.R.China (10071097).
文摘In this paper, we consider the Z8-equivariant planar Hamiltonian vector field of degree 7. By using the qualitative and numerical computation, we divide the parameters space into six-parameter-space. And we obtain the results as following : 1. There are seven cases of the number of fixed point of above vector field in finite part, that is, 1,9,l7,25,4l,49, respectively. 2. The possible phase portraits of this vector field are fifty.
文摘By using the fractional complex transform and the bifurcation theory to the generalized fractional differential mBBM equation, we first transform this fractional equation into a plane dynamic system, and then find its equilibrium points and first integral. Based on this, the phase portraits of the corresponding plane dynamic system are given. According to the phase diagram characteristics of the dynamic system, the periodic solution corresponds to the limit cycle or periodic closed orbit. Therefore, according to the phase portraits and the properties of elliptic functions, we obtain exact explicit parametric expressions of smooth periodic wave solutions. This method can also be applied to other fractional equations.
