Narrow Results

In this paper, a one-parameter Hamiltonian system under cubic perturbations is investigated and the upper bound of the number of zeros of the Abelian integral is obtained by using Horozov and Iliev's method.

Addressing the weakened Hilbert's 16th problem or the Hilbert–Arnold problem, this paper gives an upper bound B(n) ≤ 7n + 5 for the number of zeros of the Abelian integrals for a class of Liénard systems. We proved the main result using the Picard–Fuchs equations and the algebraic structure of the integrals.

The limit cycle bifurcations of a Z₂ equivariant planar Hamiltonian vector field of degree 7 under Z₂ equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.

An explicit upper bound Z(2, n) ≤ n + m - 1 is derived for the number of zeros of Abelian integrals M₁(h) = ∮_γ(h) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where γ(h) is an oval lying on the algebraic curve H_λ = (1/2)x² + (1/2)y² - (1/3)x³ - λy³ = h, P(x, y), Q(x, y) are polynomials of x and y, and max{deg P(x, y), deg Q(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M₁(h, λ) near λ = 0 and assume (∂^m/∂λ^m)M₁(h, λ)|_{λ = 0} not vanish identically.

In this paper, we study the number of limit cycles that bifurcate from the periodic orbits of a cubic reversible isochronous center under cubic perturbations. It is proved that in this situation the least upper bound for the number of zeros (taking into account the multiplicity) of the Abelian integral associated with the system is equal to four. Moreover, for each k = 0, 1, …, 4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.

In this paper, we deal with the following differential system

In this work, we study the Abelian integral $I(h)$ corresponding to the following Liénard system,

The authors investigate a kind of degenerate quadratic Hamiltonian systems with saddle-loop. Under quadratic perturbations, it is proved that the perturbed system has at most two limit cycles in the finite plane. The proof relies on a careful analysis of a related Abelian integral.