Narrow Results

In this paper, we consider Liénard systems of the form

This paper is concerned with the bifurcation of limit cycles of a class of quadratic reversible Lotka–Volterra system with b = -1/3. By using the Chebyshev criterion to study the number of zeros of Abelian integrals, we prove that this system has at most two limit cycles produced from the period annulus around the center under quadratic perturbations, which provide a positive answer for a case of the conjecture proposed by S. Gautier et al.

This paper is focused on the bifurcation of critical periods from a quartic rigidly isochronous center under any small quartic homogeneous perturbations. By studying the number of zeros of the first several terms in the expansion of the period function in ε, it shows that under any small quartic homogeneous perturbations, up to orders 1 and 2 in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system respectively, and the upper bound can be reached. Up to order 3 in ε, there are at most six critical periods from the periodic orbits of the unperturbed system. Moreover, we consider a family of perturbed systems of this quartic rigidly isochronous center, and obtain that up to any order in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed one, and the upper bound is sharp.

This paper is concerned with the bifurcation of limit cycles from a quintic system with one center. By using the averaging theory, we show that under any small quintic homogeneous perturbations, up to order 1 in ε, at most three limit cycles bifurcate from periodic orbits of the considered system, and this upper bound can be reached. Up to order 2 in ε, at most seven limit cycles emerge from periodic orbits of the unperturbed one.

This paper derives explicit formulas of the qth period bifurcation function for any perturbed isochronous system with a center, which improve and generalize the corresponding results in the literature. Based on these formulas to the perturbed quadratic and quintic rigidly isochronous centers, we prove that under any small homogeneous perturbations, for ε in any order, at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system. For ε in order of 1, 2, 3, 4 and 5, at most three critical periods bifurcate from the periodic orbits of the unperturbed quintic system. Moreover, in each case, the upper bound is sharp. Finally, a family of perturbed quintic rigidly isochronous centers is shown, which has three, for ε in any order, as the exact upper bound of the number of critical periods.

This paper deals with the bifurcation of limit cycles for a quintic system with one center. Using the averaging method, we explain how limit cycles can bifurcate from the periodic annulus around the center of the considered system by adding perturbed terms which are the sum of homogeneous polynomials of degree $4k+1$ for $2\le k\le n$. We show that up to first-order averaging, at most five limit cycles can bifurcate from the period annulus of the unperturbed system for $n=2$, at most $5n-7$ limit cycles can bifurcate from the periodic annulus of the unperturbed system for any $n\ge 3$, and the upper bound is sharp for $n=2$ and for $n=3$.

This paper investigates the planar differential systems $ẋ=-yG(x,y)$, $ẏ=xG(x,y)$ under the perturbations of polynomials of $(x,y)$ with degree $n$, where $G(x,y)={(x-a)}^{m_1}{(y-b)}^{m_2}$ with $ab\ne 0$ and $m_1,m_2\in \mathbb{N}$. The upper bound for the number of zeros of Abelian integral from the period annulus around the center is obtained.