共查询到20条相似文献,搜索用时 46 毫秒
1.
Baoxian Wang Jigui Jian 《Communications in Nonlinear Science & Numerical Simulation》2010,15(2):189-204
In this paper, a four-neuron BAM neural network with distributed delays is considered, where kernels are chosen as weak kernels. Its dynamics is studied in terms of local stability analysis and Hopf bifurcation analysis. By choosing the average delay as a bifurcation parameter and analyzing the associated characteristic equation, Hopf bifurcation occurs when the bifurcation parameter passes through some exceptive values. The stability of bifurcating periodic solutions and a formula for determining the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to validate the theorem obtained. 相似文献
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《Nonlinear Analysis: Real World Applications》2007,8(3):903-921
Using the system parameter instead of the delays as the bifurcation parameter, linear stability and Hopf bifurcation including its direction and stability of a two-neuron network with three delays are investigated in this paper. The main tools to obtain our results are the normal form method and the center manifold theory introduced by Hassard. Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior. 相似文献
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This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations. 相似文献
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Urszula Fory? 《Journal of Mathematical Analysis and Applications》2009,352(2):922-203
In the paper we present known and new results concerning stability and the Hopf bifurcation for the positive steady state describing a chronic disease in Marchuk's model of an immune system. We describe conditions guaranteeing local stability or instability of this state in a general case and for very strong immune system. We compare these results with the results known in the literature. We show that the positive steady state can be stable only for very specific parameter values. Stability analysis is illustrated by Mikhailov's hodographs and numerical simulations. Conditions for the Hopf bifurcation and stability of arising periodic orbit are also studied. These conditions are checked for arbitrary chosen realistic parameter values. Numerical examples of arising due to the Hopf bifurcation periodic solutions are presented. 相似文献
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《Nonlinear Analysis: Real World Applications》2007,8(1):73-89
In this paper, we discuss the stability and bifurcation of the distributed delays Cohen–Grossberg neural networks with two neurons. By choosing the average delay as a bifurcation parameter, we prove that Hopf bifurcation occurs. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to support the theoretical predictions. 相似文献
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Gustavo Ferron Madeira Arnaldo Simal do Nascimento 《Journal of Differential Equations》2011,251(11):3228-3247
We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henry?s approach to the problem.Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided.The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments. 相似文献
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A portable bifurcation and stability analysis package, called BISTAB, is described. The package is written in FORTRAN V and can follow the connected set of equilibrium curves for a system of nonlinear ordinary differential equations in the state × parameter space by varying a bifurcation parameter. The curves are traced from an initial point with the continuation method of Kubicek, and the tangent method of Keller is used to find initial points on bifurcating curves near simple bifurcation points. Linearized stability analysis, location of Hopf bifurcation points, and sorting of points for plotting are also supported. While the package contains no new numerical methods, the lack of a requirement for any derivative information higher than the Jacobian makes BISTAB computationally efficient and useful for applied problems where nonnumerical bifurcation analysis may be difficult. 相似文献
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In this paper, a three dimensional autonomous system which is similar to the Lorenz system is considered. By choosing an appropriate bifurcation parameter, we prove that a Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is presented by applying the normal form theory. Finally, an example is given and numerical simulations are performed to illustrate the obtained results. 相似文献
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In this paper, we consider a discrete-time tabu learning single neuron model. After investigating the stability of the given system, we demonstrate that Pichfork bifurcation, Flip bifurcation and Neimark–Sacker bifurcation will occur when the bifurcation parameter exceed a critical value, respectively. A formula is given for determining the direction and stability of Neimark–Sacker bifurcation by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided. 相似文献
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An amplitude equation is considered. The linear stability of the equation with direct control is investigated, and hence a bifurcation set is provided in the appropriate parameter plane. It is found that there exist stability switches when delay varies, and the Hopf bifurcation occurs when delay passes through a sequence of critical values. Furthermore, the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, some numerical simulations are performed to illustrate the obtained results. 相似文献
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ABSTRACTWe prove a general theorem for nonlinear matrix models of the type used in structured population dynamics that describes the bifurcation that occurs when the extinction equilibrium destabilizes as a model parameter is varied. The existence of a bifurcating continuum of positive equilibria is established, and their local stability is related to the direction of bifurcation. Our theorem generalizes existing theorems found in the literature in two ways. First, it allows for a general appearance of the bifurcation parameter (existing theorems require the parameter to appear linearly). This significantly widens the applicability of the theorem to population models. Second, our theorem describes circumstances in which a backward bifurcation can produce stable positive equilibria (existing theorems allow for stability only when the bifurcation is forward). The signs of two diagnostic quantities determine the stability of the bifurcating equilibrium and the direction of bifurcation. We give examples that illustrate these features. 相似文献
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In this paper, a tumor immune model with time delay is studied. First, the stability of nonnegative equilibria is analyzed. Then the time delay τ is selected as a bifurcation parameter and the existence of Hopf bifurcation is proved. Finally, by using the canonical method and the central manifold theory, the criteria for judging the direction and stability of Hopf bifurcation are given. 相似文献
14.
The concept of stability on finite time interval is proposed and some stability theorems are established. The delayed bifurcation transition of Dufflng's equations with a time-dependent parameter is analyzed. Function is used to predict the bifurcation transition value. The sensitivity of the solutions to initial values and parameters is also studied. 相似文献
15.
Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay
Songtao Guo Xiaofeng Liao Chuandong Li 《Nonlinear Analysis: Real World Applications》2008,9(4):1292-1309
In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results. 相似文献
16.
《Nonlinear Analysis: Real World Applications》2008,9(1):103-113
In this paper, a discrete-time Hopfield neural network with delay is considered. We give some sufficient conditions ensuring the local stability of the equilibrium point for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Neimark–Sacker bifurcation (or Hopf bifurcation for map) would occur when the delay exceeds a critical value. A formula for determining the direction bifurcation and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided. 相似文献
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YoonMee Ham 《偏微分方程(英文版)》1996,9(3):277-288
We consider a parabolic free boundary problem which has a bifurcation parameter and double interfaces. We investigate the sign change in a real part of eigenvalues and the transversality condition as a bifurcation parameter cross the critical value in order to examine the stability of the stationary solutions. The occurence of a Hopf bifurcation will be shown at a critical value. 相似文献
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A kind of three species delayed predator-prey system with reserve area for prey and in the presence of toxicity is proposed in this paper.Local stability of the coexistence equilibrium of the system and the existence of a Hopf bifurcation is established by choosing the time delay as the bifurcation parameter.Explicit formulas to determine the direction and stability of the Hopf bifurcation are obtained by means of the normal form theory and the center manifold theorem.Finally,we give a numerical example to illustrate the obtained results. 相似文献
19.
Pengmiao Hao Dejun Fan Junjie Wei Qinghe Liu 《Communications in Nonlinear Science & Numerical Simulation》2012,17(12):4753-4766
Inspired by a simulation specific to a delayed HIV model with stage-structure, some dynamic behaviors are studied in this paper, including global stability of disease-free equilibrium and local Hopf bifurcation when taking the delay as a parameter. The corresponding characteristic equation is a transcendental equation, with the parameters delay-dependent, thus we use the conventional analysis introduced by Beretta and Kuang to obtain sufficient conditions to the existence of Hopf bifurcation. Then some properties of Hopf bifurcation such as direction, stability and period are determined, and several examples illustrate our results. 相似文献
20.
Xing He Yonglu Shu Chuandong Li Huan Jin 《Journal of Applied Mathematics and Computing》2012,39(1-2):319-332
This paper reports the nonlinear dynamics of a novel three-scroll chaotic system. The local stability of hyperbolic equilibrium and non-hyperbolic equilibrium are investigated by using center manifold theorem. Pitchfork bifurcation, degenerate pitchfork bifurcation and Hopf bifurcation are analyzed when the parameters are varied in the space of parameter. For a suitable choice of the parameters, the existence of singularly degenerate heteroclinic cycles and Hopf bifurcation without parameters are also investigated. Some numerical simulations are given to support the analytic results. 相似文献