Related papers: A restricted version of the Hilbert's 16th problem…

In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach…

We prove that the number of limit cycles, which bifurcate from a two-saddle loop of a planar quadratic Hamiltonian system, under an arbitrary quadratic deformation, is less than or equal to three.

To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth…

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two…

In this article we prove in a new way that a generic polynomial vector field in $\mathbb C^2$ possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set…

A new method is presented for the analysis of limit cycle oscillations in mixed-feedback systems. The calculation of the limit cycle is reformulated as the zero finding of a mixed-monotone relation, that is, of the difference of two…

We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycles. Next we give an upper bound for the number of hyperedges in a hypergraph with…

We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are…

This paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of quadratic codimension-four centers $Q_4$. Gavrilov and Iliev set an upper bound of {\it eight} for the number of limit cycles produced from…

New criteria are established for upper bounds on the number of limit cycles of periodic Abel differential equations having two periodic invariant curves, one of them bounded. The criteria are applied to obtain upper bounds of either zero or…

Let k be a subfield of a p-adic field of odd residue characteristic, and let L be the function field of a variety of dimension n >= 1 over k. Then Hilbert's Tenth Problem for L is undecidable. In particular, Hilbert's Tenth Problem for…

This paper contains two parts. In the first part, we shall study the Abelian integrals for Zoladek's example [13], in which it is claimed the existence integrals of 11 small-amplitude limit cycles around a singular point in a particular…

We show that the number of limit cycles of Lienard vector field L is equal to the index of first order linear PDE defined by L.

It this paper we study a class of perturbed Hamiltonian systems under perturbations of thirteen order in order to detect the number of limit cycles which bifurcate from some periodic orbits of the unperturbed Hamiltonian system. The system…

We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from…

During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work…

Lienard systems are very important mathematical models describing oscillatory processes arising in applied sciences. In this paper, we study polynomial Lienard systems of arbitrary degree on the plane, and develop a new method to obtain a…

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector…

Hilbert's 14th problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the field of fractions of the algebra. It has a negative answer due to the counterexample of Nagata.…

In this paper, we consider the quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces. By using the Legendre property of quadratic forms or the compactness of operators in the presentations of…