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1.
Cyclicity of planar homoclinic loops and quadratic integrable systems   总被引:8,自引:0,他引:8  
A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations is established. Certain conditions are given under which the cyclicity of a homoclinic loop equals 1 or 2. As an application to quadratic systems, it is proved that the cyclicity of homoclinic loops of quadratic integrable and non-Hamiltonian systems equals 2 except for one case. Project supported by the National Natural Science Foundation of China.  相似文献
2.
This paper is concerned with the number of limit cycles of a cubic system with quartic perturbations. Fifteen limit cycles are found and their distributions are studied by using the methods of bifurcation theory and qualitative analysis. It gives rise to the conclusion: H(4)15, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem.  相似文献
3.
Dynamics in a ratio-dependent predator-prey model with predator harvesting   总被引:1,自引:0,他引:1  
The objective of this paper is to study systematically the dynamical properties of a ratio-dependent predator-prey model with nonzero constant rate predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation), the subcritical and supercritical Hopf bifurcations. These results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in [D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737-753]. Biologically, it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction, and remove the singularity of the origin, which was regarded as “pathological behavior” for a ratio-dependent predator prey model in [P. Yodzis, Predator-prey theory and management of multispecies fisheries, Ecological Applications 4 (2004) 51-58].  相似文献
4.
New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations. Project supported by the National Natural Science Foundation of China (Grant Nos. 19531070 and 19771037).  相似文献
5.
This paper deals with the oscillation problems of delay hyperbolic systems with impulses. Some sufficient conditions for oscillations of impulsive delay hyperbolic systems with Robin boundary conditions are obtained and the criteria of oscillation of the systems are established.  相似文献
6.
It has been already known that the maximum number of limit cycles near a homoclinic loop of a quadratic Hamiltonian system under quadratic perturbations is two. However, the problem of finding the maximum number of limit cycles in the 2-polycycle case is still open. This paper addresses the problem in some detail and solves it partially.  相似文献
7.
Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-type model with two distributed delays. For this model, He et al. [SIAM J. Math. Anal. 29 (1998) 681] showed that the positive equilibrium can be globally asymptotically stable if the mean delays are sufficiently small. In this paper, using the average time delay as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations. Computer simulations illustrate the result. The mistakes in [Chaos, Solitons & Fractals 17 (2003) 879] are pointed out and corrected.  相似文献
8.
We have considered a chemostat model with two distributed delays in a recent paper [Chaos, Solitons & Fractals 2004;20:995–1004], where, using the average time delay corresponding to the growth response as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations for two weak kernels. This article is a sequel to the previous work. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. The results are consistent with the numerical results in [Chaos, Solitons & Fractals 2004;20:995–1004].  相似文献
9.
We consider a Lotka–Volterra competition system with two delays. We first investigate the stability of the positive equilibrium and the existence of Hopf bifurcations, and then using the normal form theory and center manifold argument, derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions.  相似文献
10.
In this paper, we develop Kaplan–Yorke’s method and consider the existence and bifurcation of -periodic solutions for the high-dimensional delay differential systems. We also study the periodic solution and its bifurcation for this system with parameters and present some application examples.  相似文献
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