LIMIT CYCLES BIFURCATION FOR A CLASS OF DEGENERATE SINGULARITY

In this paper, bifurcation of limit cycles for the degenerate equilibrium to a three- dimensional system is investigated. Firstly, we use formal series to calculate the focal values at the high-order critical point on center manifold. Then an example is studied, and the existence of 3 limit cycles on the center manifold is proved. In terms of high- order singularities in high-dimensional systems, our results are new.     

Cyclical strategies in two-dimensional optimal control models: Necessary conditions and existence

This paper derives necessary conditions such that cyclical policies may be optimal in concave, two state variable (economic) control problems. These conditions identify four different routes. One major implication is that two of these four conditions may be met by separable models. This possibility has been overlooked so far. Therefore, even separable and structurally very simple models may be characterized by optimal cyclical policies. Indeed, it will be shown that stable limit cycles exist for concave and separable control problems.     

Pathways to hopf bifurcations in dynamic continuous-time optimization problems

The purpose of this paper is to characterize pathways to Hopf bifurcation in continuous time, concave, two-dimensional optimal control models. It is shown that essentially two pathways exist: control-state interaction and growth. The knowledge of such pathways provides a criterion at the stage of modelling on the potential complexity of optimal trajectories.The author knowledges the many discussions with Professor Gustav Feichtinger.     

Automatic search for multiple limit cycles in three-dimensional Lotka-Volterra competitive systems with classes 30 and 31 in Zeeman's classification

Among the six classes of Zeeman's classification for three-dimensional Lotka-Volterra competitive systems with limit cycles, besides the classes 26, 27, 28 and 29, multiple limit cycles are found in classes 30 and 31 by an algorithmic method proposed by Hofbauer and So [J. Hofbauer, J.W. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett. 7 (1994) 65-70]. This also gives an answer to a problem proposed in [J. Hofbauer, J.W. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett. 7 (1994) 65-70].     

The Stability and Hopf Bifurcation of the Prey-predator System with Delay and Migration

In this paper we first investigate the system with the inftuence of delay and migration and give a theoretical analysis of the alternative change of the stability discovered by Stepan with computer program, then we reduce the system with the center manifold theorem and present an approximation form of Hopf bifurcation solutions. Finally we give the numerical analysis of stability for a concrete periodic solution.     

On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z3 Invariant Quintic Perturbations

A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.     

Terrorism Control in the Tourism Industry

In some countries, for instance Egypt, terrorists try to hurt thecountry income from the tourism industry by violent actions againsttourists. Another example are actions of the Kurds to bring tourism down inthe east of Turkey. This paper is a first attempt to model some relevantaspects of these prey–predator relations. The country tries tomaximize profits from the tourism industry, where profit is defined as thedifference between revenue from the tourism industry and the sum ofexpenditures on tourism industry investments and expenditures on enforcementassociated with reducing terrorism. It turns out that, for reasonableparameter values, the optimal trajectory exhibits a cyclical strategy. Theinterpretation is that, after starting out with a low number of tourists andterrorists, tourism investments are undertaken to increase tourism. Thisattracts terrorists reducing the effect of tourism investments. Therefore,investment declines and so does the number of tourists. This makes it lessattractive for terrorists to act, so we are back in the original situation,where the whole thing starts again.     

Cubic Lienard Equations with Quadratic Damping (Ⅱ)

Abstract Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienardequations with quadratic damping have at most three limit cycles. This implies that the guess in which thesystem has at most two limit cycles is false. We give the sufficient conditions for the system has at most threelimit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by usingnumerical simulation.     

Cubic Lienard equation with quadradic damping (I)

1. IntroductionLienard equationdZx dx~ f(.)g g(x) = 0 (l.0)dtZ dthas been extensively studied with particular emphasis on the ekistence and uniqueness oflimit cycles (see e.g. [l--4] and references there in). The number of limit cycles of (l.0) hasbeen also investigated by several authors (see e.g. [5--8]).In the present paper we study the general cubic Lienard equation, namelydx da~ = y ~ F(x), Z ~ ~g(x) (1.1)dt' dtwhereF(x) = ale a,x: a,x', (l.2)g(x) = blx b,x' b,x'. (1.3)Clea…     

Bifurcations and distribution of limit cycles for near-Hamiltonian polynomial systems

This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles.     

BIFURCATION OF CUBIC INTEGRABLE SYSTEM UNDER CUBIC PERTURBATION

In this paper, we consider the bifurcation for a class of cubic integrable system under cubic perturbation. Using bifurcation theory and qualitative analysis, we obtain a complete bifurcation diagram of the system in a neighbourhood of the origin for parameter plane.     

Bifurcations of limit circles and center conditions for a class of non-analytic cubic Z 2 polynomial differential systems

In this paper, bifurcations of limit cycles at three fine focuses for a class of Z ₂-equivariant non-analytic cubic planar differential systems are studied. By a transformation, we first transform nonanalytic systems into analytic systems. Then sufficient and necessary conditions for critical points of the systems being centers are obtained. The fact that there exist 12 small amplitude limit cycles created from the critical points is also proved. Henceforth we give a lower bound of cyclicity of Z ₂-equivariant non-analytic cubic differential systems.     

一类生态系统的Abelian积分的定性分析

利用Abelian积分的等价性原理,把一类生态系统化为Lienard方程,然后利用Hopf分支等定性理论,简便的证明了该系统无环和有唯一极限环的条件.