In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstr...In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.展开更多
The mitotic model of frog eggs established by Borisuk and Tyson is qualitatively analyzed.The existence and stability of its steady states are further discussed.Furthermore,the bifurcation of above model is further in...The mitotic model of frog eggs established by Borisuk and Tyson is qualitatively analyzed.The existence and stability of its steady states are further discussed.Furthermore,the bifurcation of above model is further investigated by using theoretical analysis and numerical simulations. At the same time, the numerical results of Tyson are verified by theoretical analysis.展开更多
A unidirectional, weakly dispersive nonlinear model is proposed to describe the supercritical bifurcation arising from hydroelastic waves in deep water. This model equation, including quadratic, cubic, and quartic non...A unidirectional, weakly dispersive nonlinear model is proposed to describe the supercritical bifurcation arising from hydroelastic waves in deep water. This model equation, including quadratic, cubic, and quartic nonlinearities, is an extension of the famous Whitham equation. The coefficients of the nonlinear terms are chosen to match with the key properties of the full Euler equations, precisely, the associated cubic nonlinear Schrödinger equation and the amplitude of the solitary wave at the bifurcation point. It is shown that the supercritical bifurcation, rich with Stokes, solitary, generalized solitary, and dark solitary waves in the vicinity of the phase speed minimum, is a universal bifurcation mechanism. The newly developed model can capture the essential features near the bifurcation point and easily be generalized to other nonlinear wave problems in hydrodynamics.展开更多
This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,th...This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.展开更多
A nonlinear mathematical model for hydro turbine governing system with saturation nonlinearity in small perturbation has been proposed with all the essential components,i.e. turbine,PID type governor with saturation p...A nonlinear mathematical model for hydro turbine governing system with saturation nonlinearity in small perturbation has been proposed with all the essential components,i.e. turbine,PID type governor with saturation part and generator included in the model. Existence,stability and direction of Hopf bifurcation of an example HTGS are investigated in detail and presented in forms of bifurcation diagrams and time waveforms. The analysis show that a supercritical Hopf bifurcation may exist in hydraulic turbine systems in some certain conditions. Moreover,the dynamic behavior of system with different parameters such as Tw,Tab,Tyand K are studied extensively. An example with numerical simulations is presented to illustrate the theoretical results. The researches provide a reasonable explanation for the Hopf phenomenon happened in operation of hydroelectric generating unit.展开更多
In this paper,we study the complicated dynamics of general Morris-Lecar model with the impact of Cl<sup>-</sup> fluctuations on firing patterns of this neuron model. After adding Cl<sup>-</sup>...In this paper,we study the complicated dynamics of general Morris-Lecar model with the impact of Cl<sup>-</sup> fluctuations on firing patterns of this neuron model. After adding Cl<sup>-</sup> channel in the original Morris-Lecar model, the dynamics of the original model such as its bifurcations of equilibrium points would be changed and they occurred at different values compared to the primary model. We discover these qualitative changes in the point of dynamical systems and neuroscience. We will conduct the co-dimension two bifurcations analysis with respect to different control parameters to explore the complicated behaviors for this new neuron model.展开更多
文摘In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.
文摘The mitotic model of frog eggs established by Borisuk and Tyson is qualitatively analyzed.The existence and stability of its steady states are further discussed.Furthermore,the bifurcation of above model is further investigated by using theoretical analysis and numerical simulations. At the same time, the numerical results of Tyson are verified by theoretical analysis.
基金supported by the National Natural Science Foundation of China under Grant No.11772341the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No.XDB22040203。
文摘A unidirectional, weakly dispersive nonlinear model is proposed to describe the supercritical bifurcation arising from hydroelastic waves in deep water. This model equation, including quadratic, cubic, and quartic nonlinearities, is an extension of the famous Whitham equation. The coefficients of the nonlinear terms are chosen to match with the key properties of the full Euler equations, precisely, the associated cubic nonlinear Schrödinger equation and the amplitude of the solitary wave at the bifurcation point. It is shown that the supercritical bifurcation, rich with Stokes, solitary, generalized solitary, and dark solitary waves in the vicinity of the phase speed minimum, is a universal bifurcation mechanism. The newly developed model can capture the essential features near the bifurcation point and easily be generalized to other nonlinear wave problems in hydrodynamics.
基金The project supported by the National Natural Science Foundation of China (19972025)
文摘This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms,and with linear delayed velocity feedback.The analysis indicates that for a sufficiently large velocity feedback gain,the equilibrium of the system may undergo a number of stability switches with an increase of time delay,and then becomes unstable forever.At each critical value of time delay for which the system changes its stability,a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay.The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability.It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.
文摘A nonlinear mathematical model for hydro turbine governing system with saturation nonlinearity in small perturbation has been proposed with all the essential components,i.e. turbine,PID type governor with saturation part and generator included in the model. Existence,stability and direction of Hopf bifurcation of an example HTGS are investigated in detail and presented in forms of bifurcation diagrams and time waveforms. The analysis show that a supercritical Hopf bifurcation may exist in hydraulic turbine systems in some certain conditions. Moreover,the dynamic behavior of system with different parameters such as Tw,Tab,Tyand K are studied extensively. An example with numerical simulations is presented to illustrate the theoretical results. The researches provide a reasonable explanation for the Hopf phenomenon happened in operation of hydroelectric generating unit.
文摘In this paper,we study the complicated dynamics of general Morris-Lecar model with the impact of Cl<sup>-</sup> fluctuations on firing patterns of this neuron model. After adding Cl<sup>-</sup> channel in the original Morris-Lecar model, the dynamics of the original model such as its bifurcations of equilibrium points would be changed and they occurred at different values compared to the primary model. We discover these qualitative changes in the point of dynamical systems and neuroscience. We will conduct the co-dimension two bifurcations analysis with respect to different control parameters to explore the complicated behaviors for this new neuron model.
