In this paper, we study the number of limit cycles of a near-Hamiltonian system having Za- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed s...In this paper, we study the number of limit cycles of a near-Hamiltonian system having Za- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.展开更多
In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limi...In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limit cycles.展开更多
This paper discuss the cusp bifurcation of codimension 2 （i.e. Bogdanov-Takens bifurcation） in a Leslie^Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bi...This paper discuss the cusp bifurcation of codimension 2 （i.e. Bogdanov-Takens bifurcation） in a Leslie^Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14（1） （2010）, 289-306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.展开更多
This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.
In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic l...In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.展开更多
Automatic gain control （AGC） has been used in many applications. The key features of AGC, including a steady state output and static/dynamic timing response, depend mainly on key parameters such as the reference and...Automatic gain control （AGC） has been used in many applications. The key features of AGC, including a steady state output and static/dynamic timing response, depend mainly on key parameters such as the reference and the filter coefficients. A simple model developed to describe AGC systems based on several simple assumptions shows that AGC always converges to the reference and that the timing constant depends on the filter coefficients. Measures are given to prevent oscillations and limit cycle effects. The simple AGC system is adapted to a multiple AGC system for a TV tuner in a much more efficient model. Simulations using the C language are 16 times faster than those with MATLAB, and 10 times faster than those with a mixed register transfer level （RTL）-simulation program with integrated circuit emphasis （SPICE） model.展开更多
In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving th...In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).展开更多
This paper concerns the number and distributions of limit cycles in a Z 2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different configurations o...This paper concerns the number and distributions of limit cycles in a Z 2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation. It can be concluded that H(5) ? 25 = 52, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to study the weakened 16th Hilbert problem.展开更多
In this paper, we investigate the Poincar bifurcation in cubic Hamiltonian systems with heteroclinic loop, under small general cubic perturbations. We prove that the system has at most two limit cycles and has at leas...In this paper, we investigate the Poincar bifurcation in cubic Hamiltonian systems with heteroclinic loop, under small general cubic perturbations. We prove that the system has at most two limit cycles and has at least two limit cycles, respectively.展开更多
In this paper, we are concerned with a cubic near-Hamiltonian system, whose unperturbed system is quadratic and has a symmetric homoclinic loop. By using the method developed in [12], we find that the system can have ...In this paper, we are concerned with a cubic near-Hamiltonian system, whose unperturbed system is quadratic and has a symmetric homoclinic loop. By using the method developed in [12], we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop. Further, we give a condition under which there exist 4 limit cycles.展开更多
In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polyn...In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polynomials.展开更多
In this paper, we investigate the limit cycle bifurcations in a cubic near-Hamiltonian system by perturbing a cuspidal loop and prove that 5 limit cycles can appear in a neighborhood of the cuspidal loop.
Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obt...Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obtain a plane system of the form展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11271261,11461001)
文摘In this paper, we study the number of limit cycles of a near-Hamiltonian system having Za- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.
基金Supported by the National Ministry of Education(No.20020248010)the National Natural Science Foundation of China(No.10371072)the Shanghai Leading Academic Discipline Project(No.T0401).
文摘In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limit cycles.
基金Supported by the National Natural Science Foundation of China(No.11101170)Research Project of the Central China Normal University(No.CCNU12A01007)the State Scholarship Fund of the China Scholarship Council(2011842509)
文摘This paper discuss the cusp bifurcation of codimension 2 （i.e. Bogdanov-Takens bifurcation） in a Leslie^Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14（1） （2010）, 289-306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.
基金Project supported by the National Natural Science Foundation of China (No.10371072)the New Century Excellent Ttdents in University (No.NCBT-04-038)the Shanghai Leading Academic Discipline (No.T0401).
文摘This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.
基金supported by the National Natural Science Foundation of China(No.11271261)the Natural Science Foundation of Anhui Province(No.1308085MA08)the Doctoral Program Foundation(2012)of Anhui Normal University
文摘In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.
基金Supported by the National Natural Science Foundation of China (No. 60572087)
文摘Automatic gain control （AGC） has been used in many applications. The key features of AGC, including a steady state output and static/dynamic timing response, depend mainly on key parameters such as the reference and the filter coefficients. A simple model developed to describe AGC systems based on several simple assumptions shows that AGC always converges to the reference and that the timing constant depends on the filter coefficients. Measures are given to prevent oscillations and limit cycle effects. The simple AGC system is adapted to a multiple AGC system for a TV tuner in a much more efficient model. Simulations using the C language are 16 times faster than those with MATLAB, and 10 times faster than those with a mixed register transfer level （RTL）-simulation program with integrated circuit emphasis （SPICE） model.
基金Surported by the Foundation of Shandong University of Technology (2006KJM01)
文摘In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and ob-tained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).
基金Supported by the Fund of Youth of Jiangsu University(Grant No.05JDG011)the National Natural Science Foundation of China(Nos.90610031,10671127)+1 种基金the Outstanding Personnel Program in Six Fields of Jiangsu Province(Grant No.6-A-029)Shanghai Shuguang Genzong Project(Grant No.04SGG05)
文摘This paper concerns the number and distributions of limit cycles in a Z 2-equivariant quintic planar vector field. 25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation. It can be concluded that H(5) ? 25 = 52, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to study the weakened 16th Hilbert problem.
文摘In this paper, we investigate the Poincar bifurcation in cubic Hamiltonian systems with heteroclinic loop, under small general cubic perturbations. We prove that the system has at most two limit cycles and has at least two limit cycles, respectively.
基金the National Natural Science Foundation of China under Grant (No.10671127)by Shanghai Shuguang Genzong Project(04SGG05)
文摘In this paper, we are concerned with a cubic near-Hamiltonian system, whose unperturbed system is quadratic and has a symmetric homoclinic loop. By using the method developed in [12], we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop. Further, we give a condition under which there exist 4 limit cycles.
文摘In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polynomials.
基金supported by Foundation of Shanghai Municipal Education Commission(10YZ74)
文摘In this paper, we investigate the limit cycle bifurcations in a cubic near-Hamiltonian system by perturbing a cuspidal loop and prove that 5 limit cycles can appear in a neighborhood of the cuspidal loop.
基金the National Natural Science Foundation of China.
文摘Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obtain a plane system of the form
