In this paper we consider a kind of predator-prey model named Holling-Tanner model.Firstly,we prove all solutions of this model to be bounded from above.Secondly,we find a positive invariant set of the model,and prove...In this paper we consider a kind of predator-prey model named Holling-Tanner model.Firstly,we prove all solutions of this model to be bounded from above.Secondly,we find a positive invariant set of the model,and prove the existence of stable limit cycle in this invariant set by Poincaré-Bendixson theorem for the unstable equilibrium.Thirdly,we get the region of parameters in which the corresponding stable equilibrium are also globally asymptotically stable.Lastly,we give a bifurcation diagram and illustration with two limit cycles for special parameters through numerical simulation.By our knowledge,the invariant set constructed in this paper is better than that in the book written by Murray.展开更多
The nonlinear oscillatory phenomenon has been observed in the system of immune response, which corresponds to the limit cycles in the mathematical models. We prove that the system simulating an immune response studied...The nonlinear oscillatory phenomenon has been observed in the system of immune response, which corresponds to the limit cycles in the mathematical models. We prove that the system simulating an immune response studied by Huang has at least three limit cycles in the system. The conditions for the multiple limit cycles are useful in analyzing the nonlinear oscillation in immune response.展开更多
In this paper, we obtain the exact computation formulae to determine the stability of a multiple limit cycle with the third or fourth order degenerations. We employ the method of computing the expansion of the Poincar...In this paper, we obtain the exact computation formulae to determine the stability of a multiple limit cycle with the third or fourth order degenerations. We employ the method of computing the expansion of the Poincar′e map around the closed orbit using 'normal bundle' coordinates parameterized by time variable in a neighborhood of the closed orbit. An example is given to show the feasibility of our results.展开更多
In this paper, the influence of sampling intervals on the chattering in sliding mode （SM） control systems is considered. The describing function （DF） approach is employed to analyze the chattering characteristics ...In this paper, the influence of sampling intervals on the chattering in sliding mode （SM） control systems is considered. The describing function （DF） approach is employed to analyze the chattering characteristics in the sampling SM control. By the DF calculations and limit cycle existence conditions, an unstable limit cycle and two stable limit cycles are found in the SM control system. The frequencies and amplitudes of the two limit cycles can also be estimated by graphical calculations. The estimation accuracy of chattering parameters is evaluated by the simulations. The results of simulations show that the system could converge to a large and a small limit cycle from different initial conditions.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.62227810)。
文摘In this paper we consider a kind of predator-prey model named Holling-Tanner model.Firstly,we prove all solutions of this model to be bounded from above.Secondly,we find a positive invariant set of the model,and prove the existence of stable limit cycle in this invariant set by Poincaré-Bendixson theorem for the unstable equilibrium.Thirdly,we get the region of parameters in which the corresponding stable equilibrium are also globally asymptotically stable.Lastly,we give a bifurcation diagram and illustration with two limit cycles for special parameters through numerical simulation.By our knowledge,the invariant set constructed in this paper is better than that in the book written by Murray.
文摘The nonlinear oscillatory phenomenon has been observed in the system of immune response, which corresponds to the limit cycles in the mathematical models. We prove that the system simulating an immune response studied by Huang has at least three limit cycles in the system. The conditions for the multiple limit cycles are useful in analyzing the nonlinear oscillation in immune response.
基金Supported by the Fundamental Research Funds for the Central Universities
文摘In this paper, we obtain the exact computation formulae to determine the stability of a multiple limit cycle with the third or fourth order degenerations. We employ the method of computing the expansion of the Poincar′e map around the closed orbit using 'normal bundle' coordinates parameterized by time variable in a neighborhood of the closed orbit. An example is given to show the feasibility of our results.
基金supported by Industrial Research Projects in department of education of Shaanxi province(2014K05-29)Science Research Projects in department of education of Shaanxi province(14JK1669,14JF028)
文摘In this paper, the influence of sampling intervals on the chattering in sliding mode （SM） control systems is considered. The describing function （DF） approach is employed to analyze the chattering characteristics in the sampling SM control. By the DF calculations and limit cycle existence conditions, an unstable limit cycle and two stable limit cycles are found in the SM control system. The frequencies and amplitudes of the two limit cycles can also be estimated by graphical calculations. The estimation accuracy of chattering parameters is evaluated by the simulations. The results of simulations show that the system could converge to a large and a small limit cycle from different initial conditions.
