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1.

Maoan Han 《中国科学A辑(英文版)》1997,40(12):1247-1258

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.

Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations

**总被引：3，自引：0，他引：3**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.

Bifurcations of Invariant Tori and Subharmonic Solutions of Singularly Perturbed System

*下载免费PDF全文***总被引：1，自引：1，他引：0**This paper deals with bifurcations of subharmonic solutions and invariant tori generated from limit cycles in the fast dynamics for a nonautonomous singularly perturbed system. Based on Poincare map, a series of blow-up transformations and the theory of integral manifold, the conditions for the existence of invariant tori are obtained, and the saddle-node bifurcations of subharmonic solutions are studied. 相似文献

4.

Dongmei Xiao Wenxia Li Maoan Han 《Journal of Mathematical Analysis and Applications》2006,324(1):14-29

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

5.

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

6.

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

7.

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

8.

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

9.

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

10.

Maoan Han 《Journal of Differential Equations》2003,189(2):396-411

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times. 相似文献