In this paper, following the phase portraits analysis, we investigate the integrability of a system which physically describes the transverse oscillation of an elastic beam under end-thrust. As a result, we find that ...In this paper, following the phase portraits analysis, we investigate the integrability of a system which physically describes the transverse oscillation of an elastic beam under end-thrust. As a result, we find that this system actually comprises two families of travelling waves： the sub- and super-sonic periodic waves of positive- and negative- definite velocities, respectively, and the localized sub-sonic loop-shaped waves of positive-definite velocity. Expressing the energy-like of this system while depicting its phase portrait dynamics, we show that these multivaiued localized travelling waves appear as the boundary solutions to which the periodic travelling waves tend asymptotically展开更多
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
For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits ...For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.展开更多
Individuals with NGLY1 Deficiency, an inherited autosomal recessive disorder, exhibit hyperkinetic movements including athetoid, myoclonic, dysmetric, and dystonic movements impacting both upper and lower limb motion....Individuals with NGLY1 Deficiency, an inherited autosomal recessive disorder, exhibit hyperkinetic movements including athetoid, myoclonic, dysmetric, and dystonic movements impacting both upper and lower limb motion. This report provides the first set of laboratory-based measures characterizing the gait patterns of two individuals with NGLY1 Deficiency, using both linear and non-linear measures, during treadmill walking, and compares them to neurotypical controls. Lower limb kinematics were obtained with a camera-based motion analysis system and bilateral time normalized lower limb joint time series waveforms were developed. Linear measures of joint range of motion, stride times and peak angular velocity were obtained, and confidence intervals were used to determine if there were differences between the patients and control. Correlations between participant and control mean joint waveforms were calculated and used to evaluate the similarities between patients and controls. Non-linear measures included: joint angle-angle diagrams, phase-portrait areas, and continuous relative phase (CRP) measures. These measures were used to assess joint coordination and control features of the lower limb motion. Participants displayed high correlations with their control counterparts for the hip and knee joint waveforms, but joint motion was restricted. Peak angular velocities were also significantly less than those of the controls. Both angle-angle and phase-portrait areas were less than the controls although the general shapes of those diagrams were similar to those of the controls. The NGLY1 Deficient participants’ CRP measures displayed disrupted coordination patterns with the knee-ankle patterns displaying more disruption than the hip-knee measures. Overall, the participants displayed a functional walking pattern that differed in many quantitative ways from those of the neurotypical controls. Using both linear and non-linear measures to characterize gait provides a more comprehensive and nuanced characterization of NGLY1 gait and can be used to develop interventions targeted toward specific aspects of disordered gait.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In this paper, following the phase portraits analysis, we investigate the integrability of a system which physically describes the transverse oscillation of an elastic beam under end-thrust. As a result, we find that this system actually comprises two families of travelling waves： the sub- and super-sonic periodic waves of positive- and negative- definite velocities, respectively, and the localized sub-sonic loop-shaped waves of positive-definite velocity. Expressing the energy-like of this system while depicting its phase portrait dynamics, we show that these multivaiued localized travelling waves appear as the boundary solutions to which the periodic travelling waves tend asymptotically
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For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.
文摘Individuals with NGLY1 Deficiency, an inherited autosomal recessive disorder, exhibit hyperkinetic movements including athetoid, myoclonic, dysmetric, and dystonic movements impacting both upper and lower limb motion. This report provides the first set of laboratory-based measures characterizing the gait patterns of two individuals with NGLY1 Deficiency, using both linear and non-linear measures, during treadmill walking, and compares them to neurotypical controls. Lower limb kinematics were obtained with a camera-based motion analysis system and bilateral time normalized lower limb joint time series waveforms were developed. Linear measures of joint range of motion, stride times and peak angular velocity were obtained, and confidence intervals were used to determine if there were differences between the patients and control. Correlations between participant and control mean joint waveforms were calculated and used to evaluate the similarities between patients and controls. Non-linear measures included: joint angle-angle diagrams, phase-portrait areas, and continuous relative phase (CRP) measures. These measures were used to assess joint coordination and control features of the lower limb motion. Participants displayed high correlations with their control counterparts for the hip and knee joint waveforms, but joint motion was restricted. Peak angular velocities were also significantly less than those of the controls. Both angle-angle and phase-portrait areas were less than the controls although the general shapes of those diagrams were similar to those of the controls. The NGLY1 Deficient participants’ CRP measures displayed disrupted coordination patterns with the knee-ankle patterns displaying more disruption than the hip-knee measures. Overall, the participants displayed a functional walking pattern that differed in many quantitative ways from those of the neurotypical controls. Using both linear and non-linear measures to characterize gait provides a more comprehensive and nuanced characterization of NGLY1 gait and can be used to develop interventions targeted toward specific aspects of disordered gait.
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.
