Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Abstract

Abstract. We study degree n polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré-Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is n − 1. In the present paper we prove that if the first Poincaré-Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is 2(n − 1). In the case when the perturbation is quadratic (n = 2) we obtain a complete result-there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

Cauchy-type integrals of algebraic functions

“…In turn, the study of the distribution of zeroes of H(y) (which has been recently started in [18]) corresponds to the study of zeroes of the Abelian integrals. This last problem is one of the most active research areas in the Analytic Theory of Differential Equations in the last two decades (see [6,7,[26][27][28][29][32][33][34][35]43,46,56,57]). …”

“…Let us return for a moment to the comparison between the functions I(t) and the Abelian integrals. One of the most important analytic properties of the the Abelian integrals is the fact that they satisfy certain Fuchsian linear differential equations with rational coefficients (see [6,7,[26][27][28][29][32][33][34][35]43,46,56,57]). Theorem 4.4 allows us to show that the same is true for the Cauchy type integrals of algebraic functions.…”

Periodic orbits from second order perturbation via rational trigonometric integrals

“…Concerning the infinitesimal Hilbert's 16th problem this case is a more interesting one since the Poincaré-Pontryagin-Melnikov function can have more zeros, i.e. we can produce (in general) more limit cycles than at the ones at first order, see [15,18,19]. More details about higher order Poincaré-Pontryagin-Melnikov theory can be found in [7].…”

Center conditions at infinity for Abel differential equations

“…Let us mention just a small sample of related results [21]- [25], [39], [40], [41], [20], [37], [59], [61], [73]. One can hope that a combination of different approaches will bring a better understanding of this subject.…”