This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback.Based on a detailed analysis of the stability switches of the t...This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback.Based on a detailed analysis of the stability switches of the trivial equilibrium of the system,the stability charts are given in a parameter space consisting of the time delay and the feedback gains.The center manifold reduc-tion and the normal form method are used to study Hopf bifurcations with respect to the time delay.To gain an insight into the persistence of a Hopf bifurcation as the time delay varies farther away from its critical value,the method of multiple scales is used to obtain the global view of Hopf bifurcations with respect to the time delay.Both the analytical results of Hopf bifurca-tions and global view of those bifurcations are validated via a collocation scheme implemented on DDE-Biftool.The most important discovery in this paper is the well-structured global view of Hopf bifurcations for the system of concern,showing the generality of the persistence of Hopf bifurcations.展开更多
Two-phase, incompressible, immiscible flow in porous media is governed by a coupled system of nonlinear partial differential equations. The pressure equation is elliptic, whereas the concentration equation is paraboli...Two-phase, incompressible, immiscible flow in porous media is governed by a coupled system of nonlinear partial differential equations. The pressure equation is elliptic, whereas the concentration equation is parabolic, and both are treated by the collocation scheme. Existence and uniqueness of solutions of the algorithm are proved. A optimal convergence analysis is given for the method.展开更多
This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback. Based on a detailed analysis of the stability switches of the ...This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback. Based on a detailed analysis of the stability switches of the trivial equilibrium of the system, the stability charts are given in a parameter space consisting of the time delay and the feedback gains. The center manifold reduction and the normal form method are used to study Hopf bifurcations with respect to the time delay. To gain an insight into the persistence of a Hopf bifurcation as the time delay varies farther away from its critical value, the method of multiple scales is used to obtain the global view of Hopf bifurcations with respect to the time delay. Both the analytical results of Hopf bifurcations and global view of those bifurcations are validated via a collocation scheme implemented on DDE-Biftool. The most important discovery in this paper is the well-structured global view of Hopf bifurcations for the system of concern, showing the generality of the persistence of Hopf bifurcations.展开更多
In this paper, a new kind of alternating direction implicit （ADI） Crank-Nicolson-type orthogonal spline collocation （OSC） method is formulated for the two-dimensional frac-tional evolution equation with a weakly s...In this paper, a new kind of alternating direction implicit （ADI） Crank-Nicolson-type orthogonal spline collocation （OSC） method is formulated for the two-dimensional frac-tional evolution equation with a weakly singular kernel arising in the theory of linear viscoelas-ticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for thetemporal component. The stability of proposed scheme is rigourously established, and nearlyoptimal order error estimate is also derived. Numerical experiments are conducted to supportthe predicted convergence rates and also exhibit expected super-convergence phenomena.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.10532050,10702024)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20070287029)
文摘This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback.Based on a detailed analysis of the stability switches of the trivial equilibrium of the system,the stability charts are given in a parameter space consisting of the time delay and the feedback gains.The center manifold reduc-tion and the normal form method are used to study Hopf bifurcations with respect to the time delay.To gain an insight into the persistence of a Hopf bifurcation as the time delay varies farther away from its critical value,the method of multiple scales is used to obtain the global view of Hopf bifurcations with respect to the time delay.Both the analytical results of Hopf bifurca-tions and global view of those bifurcations are validated via a collocation scheme implemented on DDE-Biftool.The most important discovery in this paper is the well-structured global view of Hopf bifurcations for the system of concern,showing the generality of the persistence of Hopf bifurcations.
基金Supported by NNSF of China(0441005)Research Fund for Doctoral Program of High Education by China State Education Ministry
文摘Two-phase, incompressible, immiscible flow in porous media is governed by a coupled system of nonlinear partial differential equations. The pressure equation is elliptic, whereas the concentration equation is parabolic, and both are treated by the collocation scheme. Existence and uniqueness of solutions of the algorithm are proved. A optimal convergence analysis is given for the method.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10532050)Supported by the National Natural Science Foundation of China (Grant Nos.10702024)the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20070287029)
文摘This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback. Based on a detailed analysis of the stability switches of the trivial equilibrium of the system, the stability charts are given in a parameter space consisting of the time delay and the feedback gains. The center manifold reduction and the normal form method are used to study Hopf bifurcations with respect to the time delay. To gain an insight into the persistence of a Hopf bifurcation as the time delay varies farther away from its critical value, the method of multiple scales is used to obtain the global view of Hopf bifurcations with respect to the time delay. Both the analytical results of Hopf bifurcations and global view of those bifurcations are validated via a collocation scheme implemented on DDE-Biftool. The most important discovery in this paper is the well-structured global view of Hopf bifurcations for the system of concern, showing the generality of the persistence of Hopf bifurcations.
基金supported by National Nature Science Foundation of China(11701168,11601144 and 11626096)Hunan Provincial Natural Science Foundation of China(2018JJ3108,2018JJ3109 and 2018JJ4062)+1 种基金Scientific Research Fund of Hunan Provincial Education Department(16K026 and YB2016B033)China Postdoctoral Science Foundation(2018M631403)
文摘In this paper, a new kind of alternating direction implicit （ADI） Crank-Nicolson-type orthogonal spline collocation （OSC） method is formulated for the two-dimensional frac-tional evolution equation with a weakly singular kernel arising in the theory of linear viscoelas-ticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for thetemporal component. The stability of proposed scheme is rigourously established, and nearlyoptimal order error estimate is also derived. Numerical experiments are conducted to supportthe predicted convergence rates and also exhibit expected super-convergence phenomena.
