近哈密顿系统极限环的Hopf分支与同宿分支

利用Hopf与同宿两种分支中出现的系数研究了近哈密顿系统Hopf和同宿分支产生的极限环的个数与分布,得到了全局分支产生极限环的一个新的充分条件.     

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.     

On Hopf bifurcations of piecewise planar Hamiltonian systems

In this paper we study the number of limit cycles appearing in Hopf bifurcations of piecewise planar Hamiltonian systems. For the case that the Hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively. For some systems of special form we obtain the Hopf cyclicity.     

Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems

We investigate a generalized Hopf bifurcation emerged from a corner located at the origin which is the intersection of n$n$ discontinuity boundaries in planar piecewise smooth dynamical systems with the Jacobian matrix of each smooth subsystem having either two different nonzero real eigenvalues or a pair of complex conjugate eigenvalues. We obtain a novel result that the generalized Hopf bifurcation can occur even when the Jacobian matrix of each smooth subsystem has two different nonzero real eigenvalues. According to the eigenvalues of the Jacobian matrices and the number of smooth subsystems, we provide a general method and prove some generalized Hopf bifurcation theorems by studying the associated Poincaré map.     

Analytical study of a triple Hopf bifurcation in a tritrophic food chain model

We provide an analytical proof of the existence of a stable periodic orbit contained in the region of coexistence of the three species of a tritrophic chain. The method used consists in analyzing a triple Hopf bifurcation. For some values of the parameters three limit cycles born via this bifurcation. One is contained in the plane where the top-predator is absent. Another one is not contained in the domain of interest where all variables are positive. The third one is contained where the three species coexist. The techniques for proving these results have been introduced in previous articles by the second author and are based on the averaging theory of second-order. Existence of this triple Hopf bifurcation has been previously discovered numerically by Kooij et al.     

关于一类非多项式平面微分系统的极限环及分支问题

旨在讨论一类非多项式平面微分系统.通过使用Dulac准则和Bendixson准则获得极限环不存在性的充分条件,引入广义Liénard系统理论以研究极限环的存在性及稳定性,应用Hopf分岔理论证明自原点分岔出极限环的充分条件.此外,给出一个范例以验证分析和结果的有效性.     

The 3-D Hopf bifurcation in bio-reactor when one competitor produces a toxin

In this paper, a model of competition in the bio-reactor of two competitors for a single nutrient where one of the competitors can produce toxin against its opponent is investigated. The conditions of the three dimensional Hopf bifurcation are obtained. The Hopf bifurcation implies the existence of limit cycles in the model that corresponds to the nonlinear oscillation in the reactor.     

Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models

In this paper, we discuss the special diffusive hematopoiesis model

一类非线性系统的周期扰动Hopf分支

研究了小周期扰动对一类存在Hopf分支的非线性系统的影响.特别是应用平均法讨论了扰动频率与Hopf分支固有频率在共振及二阶次调和共振的情形周期解分支的存在性.表明了在某些参数区域内,系统存在调和解分支和次调和解分支,并进一步讨论了二阶次调和分支周期解的稳定性.     

The Hopf Bifurcations in the Permanent Magnet Synchronous Motors

Based on the focus quantities and other techniques, the stability properties of equilibria and the limit cycles arising from Hopf bifurcations are investigated for two models of permanent magnet synchronous motors. The first model is of surface-magnet type and can have at most two unstable small limit cycles, which are symmetric with respect to $x$-axis. The other model is of interior-magnet type and can have at most four small limit cycles in two symmetric nests.     

Hopf bifurcation near a double singular point

The nonlinear equation f(x,λ,) = 0, f:X × R²→X, where X is a Banach space and f satisfies a Z₂-symmetry relation is considered. Interest centres on a certain type of double singular point, where the solution x is symmetric and f_x has a double zero eigenvalue, with one eigenvector symmetric and one antisymmetric.

We show that under certain nondegeneracy conditions, which are stated both algebraically and geometrically, there exists a path of Hopf bifurcations or imaginary Hopf bifurcations passing through the double singular point, and for which x is not symmetric except at the double singular point. An easy geometrical test is found to decide which type of phenomenon occurs. A biproduct of the analysis is that explicit expressions are obtained for quantities which help to provide a reliable numerical method to compute these paths. A pseudo-spectral method was used to obtain numerical results for the Brusselator equations to illustrate the theory.     

一类余维3的鞍-焦点异宿环分支

对余维3系统X_μ(x)具有包含一个双曲鞍-焦点O_1和一个非双曲鞍-焦点O_2的异宿环f进行了研究.证明了在f的邻域内有可数无穷条周期轨线和异宿轨线,当非粗糙异宿轨线Γ~0破裂时X_μ(x)会产生同宿轨分支,并给出了相应的分支曲线和两种同宿环共存的参数值.在3参数扰动下Γ~0破裂和O_2点产生Hopf分支的情况下,在f的邻域内有一条含O_1点同宿环,可数无效多条的轨线同宿于O_2点分支出的闭轨H_0,一条或无穷多条(可数或连续统的)异宿轨线等.     

Hopf bifurcation and uniqueness of limit cycle for a class of quartic system

This paper studies a class of quartic system which is more general and realistic than the quartic accompanying system. x＇=-y＋ex＋lx^2＋mxy＋ny^2,y＇=x（1-Ay）（1＋Cy^2）,（＊） where C 〉 0. Sufficient conditions are obtained for the uniqueness of limit cycle of system （＊） and some more in-depth conclusion such as Hopf bifurcation.     

Cyclicity of planar homoclinic loops and quadratic integrable systems

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.     

Stability and uniqueness of periodic orbits produced during homoclinic bifurcation

Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].Supported by the National Natural Science Foundation of China.     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

Global stability and the Hopf bifurcation for some class of delay differential equation

In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right‐hand side depending only on the past. We extend the results from paper by U. Fory? (Appl. Math. Lett. 2004; 17 (5):581–584), where the right‐hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.     

ON CLOSED ORBIT FOR A CLASS OF PLANAR SYSTEMS OF HIGH DEGREE

1.IntroductionThepolynomialdifferentialequationofdegreep qwherep2l,q21areintegersandx20,y20,a>0arerealnumber,asknownin[12],describesthedynamicalbehaviorsofmulti-molecularbiochemicalreactionHerex,yandp,qdenoterespectivelytheconcentrationsofsubstancesandnumbersofmolecules.UsuallyEq.(1.1)iscalled"TI-moleculesmodel"withn~p q.LikePrigogine'sB.....l.t.rI4],Higgins'm.del[5]andTyson-Lighe'smodel[6],itisofsignificanceinourunderstandingofthephenomenaofnonequilibriumstates[3].Thecaseofn=3isdiscussedp…