Multiple recurrent outbreak cycles in an autonomous epidemiological model due to multiple limit cycle bifurcation

Multiple recurrent outbreak cycles have been commonly observed in infectious diseases such as measles and chicken pox. This complex outbreak dynamics in epidemiologicals is rarely captured by deterministic models. In this paper, we investigate a simple 2-dimensional SI epidemiological model and propose that the coexistence of multiple attractors attributes to the complex outbreak patterns. We first determine the conditions on parameters for the existence of an isolated center, then properly perturb the model to generate Hopf bifurcation and obtain limit cycles around the center. We further analytically prove that the maximum number of the coexisting limit cycles is three, and provide a corresponding set of parameters for the existence of the three limit cycles. Simulation results demonstrate the case with the maximum coexisting attractors, which contains one stable disease free equilibrium and two stable endemic periodic solutions separated by one unstable periodic solution. Therefore, different disease outcomes can be predicted by a single nonlinear deterministic model based on different initial data.     

Bifurcation of limit cycles near equivariant compound cycles

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.     

Ⅲ类二次系统极限环的惟一性(Ⅰ)

证明了一般的Ⅲ类二次系统当参数a很小时极限环的大范围惟一性,对于一般的参数值a,在适当的条件下也证明了极限环的大范围惟一性.文中也给出了极限环随参数d变化时产生和消失的过程.     

A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems

In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in ${\mathbb{R}}^2$ and of the discontinuous piecewise differential systems in ${\mathbb{R}}^2$ formed by two linear differential systems separated by a straight line. For this class of discontinuous piecewise differential systems the study of their algebraic and explicit non-algebraic limit cycles just is starting. Here we provide the first explicit non-algebraic limit cycle for the discontinuous piecewise linear differential systems. Additionally we recall seven open questions related with these types of limit cycles.     

On the number of limit cycles in double homoclinic bifurcations

LetL be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under whichL generates at most two large limit cycles by perturbations. We also give conditions for the existence of at most five or six limit cycles which appear nearL under perturbations.     

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

This paper concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quintic homogeneous perturbations, at most 14 limit cycles birfucate from the period annulus of the considered system.     

Multiple limit cycles and global stability in predator-prey model

1.IntroductionWeconsideranextensivesystemofthefamiliarLotka-Volterrasysteminwhichthepopulationshaveself-inhibit(i.e.,withtheadditionofdampingterm)thatcanbemodeledbytheequationswherexandyrepresentthepreydensityandpredatordensityrespectively.Thespecifi...     

Bifurcation of limit cycles for the quadratic differential system (III)l=n=0

In this paper we will prove that limit cycles for the quadratic differential system (III)l=n=0 in Chinese classification are concentratedly distributed, and that the maximum number of limit cycles around O (0,0) is at least two. This project is supported by the Natural Science Foundation of Educational Committee of Jiangsu Province.     

Bifurcation of limit cycles in piecewise smooth systems via Melnikov function

In this short paper, we present some remarks on the role of the rstorder Melnikov functions in studying the number of limit cycles of piecewisesmooth near-Hamiltonian systems on the plane.     

平面折射系统极限环的存在和唯一性

本文考虑平面折射系统的极限环个数问题.根据左、右子系统的动力学性态,可以将其分为如下6种类型:焦点-焦点、焦点-鞍点、焦点-结点、鞍点-鞍点、鞍点-结点和结点-结点.利用Poincaré映射,本文证明折射系统为焦点-结点情形时最多存在1个极限环.     

The hyperelliptic limit cycles of the Liénard systems

For Liénard systems , with fm and gn real polynomials of degree m and n respectively, in [H. Zoladek, Algebraic invariant curves for the Liénard equation, Trans. Amer. Math. Soc. 350 (1998) 1681-1701] the author showed that if m?3 and m+1<n<2m there always exist Liénard systems which have a hyperelliptic limit cycle. Llibre and Zhang [J. Llibre, Xiang Zhang, On the algebraic limit cycles of Liénard systems, Nonlinearity 21 (2008) 2011-2022] proved that the Liénard systems with m=3 and n=5 have no hyperelliptic limit cycles and that there exist Liénard systems with m=4 and 5<n<8 which do have hyperelliptic limit cycles. So, it is still an open problem to characterize the Liénard systems which have an algebraic limit cycle in cases m>4 and m+1<n<2m. In this paper we will prove that there exist Liénard systems with m=5 and m+1<n<2m which have hyperelliptic limit cycles.     

Exact number of limit cycles for a family of rigid systems

For a given family of planar differential equations it is a very difficult problem to determine an upper bound for the number of its limit cycles. Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle. This note mainly deals with this second problem for a family of systems with a homogeneous nonlinear part. While the condition that allows us to separate the existence and the nonexistence of limit cycles can be described, it is very intricate.

     

Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.     

广义Lienard方程零解的全局渐近稳定性和极限环的存在性

研究了一类广义Lienard方程 x=φ（y）,y=-f（x）φ（y）-g（x）式中φ，F，g：R→R连续且保证系统初值解惟一，给出零解全局渐近稳定性条件，并讨论极限环的存在性．     

Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four

A class of degree four differential systems that have an invariant conic x ² + Cy ² = 1, C ∈ ℝ, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.     

Z 2-equivariant cubic system which yields 13 limit cycles

For the planar Z_2-equivariant cubic systems having two elementary focuses,the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved.The necessary and sufficient conditions for the existence of the bi-center are obtained.On the basis of this work,in this paper,we show that under small Z_2-equivariant cubic perturbations,this cubic system has at least 13 limit cycles with the scheme 16∪6.     

Uniqueness of limit cycles in predator-prey system: the role of weight functions

We consider a Gause type model of interactions between predator and prey populations. Using the ideas of Cheng and Liou we give a sufficient condition for uniqueness of the limit cycle, which is more general than their condition. That is, we include a kind of weight function in the condition. It was motivated by a result due to Hwang, where the prey isocline plays a role of weight function. Moreover, we show that the interval where the condition from Hwang's result is to be fulfilled can be narrowed.     

On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.