Bifurcation of limit cycles from a center in two-parameter system

In this paper we consider a two-parameter perturbated system which takes the systems discussed in [1], [2], [3] as its special case. The bifurcation region of the limit cycles is given on the parameter plane. The author also studies the stability of the limit cycles. At the end of this paper the author discusses the difference of the bifurcation of the limit cycles from a center between the case of two-parameter and the case of one-parameter.     

Fourteen limit cycles in a cubic Hamiltonian system with nine-order perturbed term

We study limit cycles of the following system:

with a>c>0, ac>1, 0<1, m,l,λ are real parameters and n is a positive integer. When n=2, J.B. Li and Z.R. Liu [Publ. Math. 35 (1991) 487] showed that the system has 11 limit cycles. When n=6, H.J. Cao, Z.R. Liu and Z.J. Jing [Chaos, Solitons & Fractals 11 (2000) 2293] showed the system has 13 limit cycles. Using the same method of detection function, we first show that the system and others four systems have the same bifurcation diagrams of limit cycle. Then we demonstrate that any one of the five systems has 14 limit cycles for n=8. The positions of the 14 limit cycles are given by numerical exploration.     

Liénard方程极限环的存在性

本文在没有常设条件G(±∞)=+∞的情况下,证明了Liénard方程存在极限环的几个充分性定理,推广了文[3～6]的某些结果.这些定理给出的条件均可估计极限环的存在区域.至少在n个极限环的充分性定理3、4的条件既不要求F(x)是奇函数,也不要求F(x)"n重互相相容"或"n重互相包含".     

FOCAL QUANTITIES AND BIFURCATION OFLIMIT CYCLES FOR E~1_3 SYSTEM

1IntroductiollInthispapersweconsiderthenumberofthelimitcycleforthefollowingcubicpolynomialdifferentialsystemwiththeEdstyleFOrthepurposeofanalysingthenumberoflimitcycles,itisnecessarytocalculatethefocalquantities.FOrsystem(1))weperformfirstthefollowinghomeomorphictransformationThen,system(l)iscarriedintothefollowingsystemwhereIn[1],usingPoincaremethodandWuelimination,wedesignaspecialal-gorithmtocomputingthefocalvaluesforplanardynamicsystems.Byouralgorithm,weobtainthefollowingformularofcalcul…     

A kind of bifurcation of limit cycle from a nilpotent critical point

In this paper, an interesting and new bifurcation phenomenon that limit cycles could be bifurcated from nilpotent node (focus) by changing its stability is investigated. It is different from lowing its multiplicity in order to get limit cycles. We prove that $n^2+n-1$ limit cycles could be bifurcated by this way for $2n+1$ degree systems. Moreover, this upper bound could be reached. At last, we give two examples to show that $N(3)=1$ and $N(5)=5$ respectively. Here, $N(n)$ denotes the number of small-amplitude limit cycles around a nilpotent node (focus) with $n$ being the degree of polynomials in the vector field.     

Condition for generating limit cycles by bifurcation of the loop of a saddle-point separatrix

In paper [1] (On the stability of a saddle-point separatrix loop and analytical criterion for its bifurcation limit cycles Acta Mathematica Sinica Vol. 28. No. 1, 55–70, Bejing China 1985), we considered the problem of generating limit cycles by the bifurcation of a stable or an unstable loop of a saddle-point separatrix. We gave for the first time a criterion for the stability of the loop as following:L ₀ is stable (unstable) if \(\int_{ - \infty }^\infty {(P'_{0x} + Q'_{0y} )dt< 0(0 > 0)} \) wherex=?(t),y=?(t) then a sufficient condition for the bifurcation which generates limit cycles. This paper generalizes the result of [1] to the case where the loop contains a center or the loop tends to an infinite saddle-point, and removes the restriction that the saddle-point should be an elementary singular point. Applying the results of this paper, the author studies a two-parameter system $$\left\{ \begin{gathered} x = lx^2 + y^2 - y + 5\varepsilon xy \hfill \\ y = (3l + 5)xy + x + \varepsilon x^2 \hfill \\ \end{gathered} \right.$$ The results obtained by the author in this in real field coincides with the results given by Prof. Qin Yuanxun by means of the complex qualitative theory in complex field.     

The number of limit cycles of cubic Hamiltonian system with perturbation

In this paper, by using qualitative analysis, we investigate the number of limit cycles of perturbed cubic Hamiltonian system with perturbation in the form of (2n+2m) or (2n+2m+1)th degree polynomials . We show that the perturbed systems has at most (n+m) limit cycles, and has at most n limit cycles if m=1. If m=1, n=1 and m=1, n=2, the general conditions for the number of existing limit cycles and the stability of the limit cycles will be established, respectively. Such conditions depend on the coefficients of the perturbed terms. In order to illustrate our results, two numerical examples on the location and stability of the limit cycles are given.     

Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton systems

In this paper, we consider the bifurcation of limit cycles for system $\dot{x}=-y(x^2+a^2)^m,~\dot{y}=x(x^2+a^2)^m$ under perturbations of polynomials with degree n, where $a\neq0$, $m\in \mathbb{N}$. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from periodic orbits of the center of the unperturbed system. Particularly, if $m=2, n=5$, the sharp bound is 5.     

一类带毒生化反应竞争模型中的极限环

研究两个微生物竞争同一营养,而其中一个竞争者会产生毒素抑制另一竞争者且产物系数为γ1(S)=A1+B1Sn和γ2(S)=A2+B2Sm(n和m是自然数)函数时的生化反应模型.分析了平衡点的稳定性,并证明了三维系统经历Hopf分支后产生的周期解的稳定性,进一步又证明极限环存在于有关的二维稳定流形之上,并用实例验证了结果.     

Effects of Hard Limits on Bifurcation, Chaos and Stability

An SMIB model in the power systems,especially that concering the effects of hard limits onbifurcations,chaos and stability is studied.Parameter conditions for bifurcations and chaos in the absence ofhard limits are compared with those in the presence of hard limits.It has been proved that hard limits can affectsystem stability.We find that (1) hard limits can change unstable equilibrium into stable one;(2) hard limits canchange stability of limit cycles induced by Hopf bifurcation;(3) persistence of hard limits can stabilize divergenttrajectory to a stable equilibrium or limit cycle;(4) Hopf bifurcation occurs before SN bifurcation,so the systemcollapse can be controlled before Hopf bifurcation occurs.We also find that suitable limiting values of hard limitscan enlarge the feasibility region.These results are based on theoretical analysis and numerical simulations,such as condition for SNB and Hopf bifurcation,bifurcation diagram,trajectories,Lyapunov exponent,Floquetmultipliers,dimension of attractor and so on.     

Bifurcations of limit cycles in a quintic Lyapunov system with eleven parameters

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of computer algebra system MATHEMATICA, the first 12 quasi Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 12 small amplitude limit cycles created from the three order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems, the result of Jiang et al. (2009) [18] was improved.     

一类平面多项式系统极限环的存在唯一性

研究一类平面2n 1次多项式微分系统的极限环问题,利用Hopf分枝理论得到了该系统极限环存在性与稳定性的若干充分条件,利用Cherkas和Zheilevych的唯一性定理得到了极限环唯一性的若干充分条件.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis

In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach used for studying bifurcation of limit cycles around a center but simpler. Attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by ?o?a?dek (1995) [24] can indeed have eleven limit cycles under perturbations at least up to 7th order. Moreover, the pattern of numbers of limit cycles produced near the center is discussed up to 39th-order perturbations, and no more than eleven limit cycles are found.     

The influences of longitudinal surface roughness on sub-critical and super-critical limit cycles of short journal bearings

By applying the stochastic model of rough surfaces by Christensen (1969–1970, 1971)  and  together with the Hopf bifurcation theory by Hassard et al. (1981) [3], the present study is mainly concerned with the influences of longitudinal roughness patterns on the linear stability regions, Hopf bifurcation regions, sub-critical and super-critical limit cycles of short journal bearings. It is found that the longitudinal rough-surface bearings can exhibit Hopf bifurcation behaviors in the vicinity of bifurcation points. For fixed bearing parameter, the effects of longitudinal roughness structures provide an increase in the linear stability region, as well as a reduction in the size of sub-critical and super-critical limit cycles as compared to the smooth-bearing case.     

一类平面微分系统的极限环

研究一类平面微分系统的极限环,利用Hopf分支理论得到了该系统极限环存在性与稳定性的若干充分条件,利用ЧеркасЛА和ЖилевычЛИ的唯一性定理得到了极限环唯一性的若干充分条件.     

一个三维Chemostat竞争系统的Hopf分支和周期解

本文研究了一个三维Chemostat竞争系统的解的结构,分析了平衡点的稳定性和当系统的某一微生物物种处于竞争劣势趋于灭绝时另一微生物物种和养料的二维流形上极限环的存在性,以及系统的Hopf分支问题.文中用Friedrich方法得到了系统存在Hopf分支的条件,并判定了周期解的稳定性.     

Existence of multiple limit cycles in Chen system

In this paper, the existence of multiple limit cycles for Chen system are investigated. By using the method of computing the singular point quantities, the simple and explicit parametric conditions can be determined to the number and stability of multiple limit cycles from Hopf bifurcation. Especially, at least 4 limit cycles can be obtained for the Chen system as a three-dimensional perturbed system.     

关于一类平面n+2次系统极限环唯一性的一个注记

本文继续完善文[1]和[2]的工作,利用广义Lienard方程和张芷芬唯一性定 理证明了,当n≥3时一类n+2次生化反应系统极限环的唯一性.至此,该系统极 限环唯一性问题得到完整解决.     

Dynamical analysis of a new microbial pesticide model with the Monod growth rate

In this paper, a new mathematical model for the entomopathogenic nematode with the Monod growth rate is formulated. Firstly, continuous release of the entomopathogenic nematode is considered. The existence of limit cycles, the Hopf bifurcation and the stability of the periodic solution created by the bifurcation are proved. The sufficient conditions for the globally asymptotical stability of system are obtained. Secondly, impulsive release of the entomopathogenic nematode is also considered. By using the Floquet’s theorem and the small amplitude perturbations, we show that the pest-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations. Thus, we provide mathematical evidence on how to release the entomopathogenic nematode in order to control pests at acceptably low levels.