Estimating the number of limit cycles in polynomials systems

We describe a method based on algorithms of computational algebra for obtaining an upper bound for the number of limit cycles bifurcating from a center or a focus of polynomial vector field. We apply it to a cubic system depending on six parameters and prove that in the generic case at most six limit cycles can bifurcate from any center or focus at the origin of the system.     

Quadratic systems with maximum number of limit cycles

We suggest a method for obtaining quadratic systems with a given distribution of limit cycles. We use it to obtain a set of quadratic systems with the distributions (3, 1), (3, 0), and 3 of limit cycles and with different configurations of singular points. The distributions are justified with the use of a modified Dulac function in a natural domain of existence 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.     

On the number of limit cycles of planar quadratic vector fields

Doklady Mathematics -     

On the number of limit cycles of a generalized Abel equation

New results are proved on the maximum number of isolated T-periodic solutions (limit cycles) of a first order polynomial differential equation with periodic coefficients. The exponents of the polynomial may be negative. The results are compared with the available literature and applied to a class of polynomial systems on the cylinder.     

On the number of limit cycles of some systems on the cylinder

In this article we give two criteria for bounding the number of non-contractible limit cycles of a family of differential systems on the cylinder. This family includes Abel equations as well as the polar expression of several types of planar polynomial systems given by the sum of three homogeneous vector fields.     

Lower bounds for the number of limit cycles of trigonometric Abel equations

We consider the Abel equation , where A(t) and B(t) are trigonometric polynomials of degree n and m, respectively, and we give lower bounds for its number of isolated periodic orbits for some values of n and m. These lower bounds are obtained by two different methods: the study of the perturbations of some Abel equations having a continuum of periodic orbits and the Hopf-type bifurcation of periodic orbits from the solution x=0.     

On the number of limit cycles bifurcating from a non-global degenerated center

We give an upper bound for the number of zeros of an Abelian integral. This integral controls the number of limit cycles that bifurcate, by a polynomial perturbation of arbitrary degree n, from the periodic orbits of the integrable system , where H is the quasi-homogeneous Hamiltonian H(x,y)=x2k/(2k)+y²/2. The tools used in our proofs are the Argument Principle applied to a suitable complex extension of the Abelian integral and some techniques in real analysis.     

On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields

We prove that the number of limit cycles which bifurcate from a two-saddle loop of an analytic planar vector field X ₀ under an arbitrary finite-parameter analytic deformation X _λ , λ ∈ (? ^N , 0), is uniformly bounded with respect to λ.     

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.

     

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.     

On the number of limit cycles which appear by perturbation of Hamiltonian two-saddle cycles of planar vector fields

We find an upper bound to the maximal number of limit cycles, which bifurcate from a hamiltonian two-saddle loop of an analytic vector field, under an analytic deformation.     

Estimate of the number of limit cycles of a second-order autonomous system and their localization

By using the Dulac-Cherkas function, we study the limit cycles of an autonomous two-dimensional system of ordinary differential equations depending on a parameter.     

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.     

Linear estimate of the number of limit cycles for a class of non-linear systems

A dynamic system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for general non-linear dynamical systems. In this paper, we investigated a class of non-linear systems under perturbations. We proved that the upper bound of the number of zeros of the related elliptic integrals of the given system is 7n + 5 including multiple zeros, which also gives the upper bound of the number of limit cycles for the given system.     

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.