In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the lin...In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the linear stability of this model is studied.It is shown that Neimark-Sacker bifurcation occurs when τ crosses certain critical values.The explicit formulae which determine the stability,direction,and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form.Finally,numerical simulations are performed to verify the analytical results.展开更多
The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation ar...The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 61272069,61272114,61073026,61170031,and 61100076)
文摘In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the linear stability of this model is studied.It is shown that Neimark-Sacker bifurcation occurs when τ crosses certain critical values.The explicit formulae which determine the stability,direction,and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form.Finally,numerical simulations are performed to verify the analytical results.
基金Supported by National Natural Science Foundation of China (No10872141)Doctoral Foundation of Ministry of Education of China (No20060056005)Natural Science Foundation of Tianjin University of Science and Technology (No20070210)
文摘The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.
