We consider number of limit cycles of perturbed quintic Hamiltonian system with perturbation in the form of (2n+2m) or (2n+2m+1) degree polynomials. We show that the perturbed system has at most n + 2m limit cycles. For m = 1 and n = 1 we showed that the perturbed system can have at most one limit cycles. If m = 1 and n = 2 we give some general conditions based on coefficients of the perturbed ...

We study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of isochronous cubic polynomial centers ẋ = y(−1+2αx+2βx), ẏ = x+α(y−x)+ 2βxy, α ∈ R, β < 0, when it is perturbed inside the classes of all continuous and discontinuous cubic polynomial differential systems. We obtain that the maximum number of limit cycles which can be obtained by the averaging...

Using the method of qualitative analysis we show that five perturbed cubic Hamiltonian systems have the same distribution of limit cycles and have 11 limit cycles for some parameters. The accurate location of each limit cycle is given by numerical exploration. In other words, we demonstrate the existence of 11 limit cycles and their distribution in five perturbed systems in two ways, the result...

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. Mor...

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In this article, we study the planar cubic polynomial differential system ẋ = −yR(x, y) ẏ = xR(x, y) where R(x, y) = 0 is a conic and R(0, 0) 6= 0. We find a bound for the number of limit cycles which bifurcate from the period annulus of the center, under piecewise smooth cubic polynomial perturbations. Our results show that the piecewise smooth cubic system can have at least 1 more limit cycle...

This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows thatH(4) ≥ 20, whereH(n) is the Hilbert number for the second part of Hilbert’s 16th problem. It is well known that ...

A Warreld group is a direct summand of a simply presented abelian group. In this paper, we describe the Pon-trjagin dual groups of Warreld groups, both for the p-local and the general case. A variety of characterizations of these dual groups is obtained. In addition, numerical invariants are given that distinguish between two such groups which are not topologically isomorphic.