On the approximation of the limit cycles function.
On the number of limit cycles of a generalized Abel equation
New results are proved on the maximum number of isolated -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 zeros of Melnikov functions
We provide an effective uniform upper bound for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order of the Melnikov function. The generic case was considered by Binyamini, Novikov and Yakovenko [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.
Period function's convexity for Hamiltonian centers with separable variables
A convexity theorem for the period function T of Hamiltonian systems with separable variables is proved. We are interested in systems with non-monotone T. This result is applied to proving the uniqueness of critical orbits for second order ODE's.
Periodic solutions of polynomial non-autonomous differential equations.
Polynomial systems: a lower bound for the weakened 16th Hilbert problem.
In this paper we provide the greatest lower bound about the number of (non-infinitesimal) limit cycles surrounding a unique singular point for a planar polynomial differential system of arbitrary degree.
Pseudo-abelian integrals on slow-fast Darboux systems
We study pseudo-abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.
Quadratic systems with homoclinic cycles described by quintic curves.
The number of limit cycles of a quintic Hamiltonian system with perturbation.
The period function near a polycycle with two semi-hyperbolic vertices
Uniqueness of limit cycles for a class of cubic systems with two invariant straight lines.
When singular points determine quadratic systems.