Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

This paper consists of two parts. In the first part we study the relationship between conic centers (all orbits near a singular point of center type are conics) and isochronous centers of polynomial systems. In the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics, when we perturb such systems inside the class of all polynomial systems of degree n.     

Limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers

In this article, we study the maximum number of limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers. Using the first-order averaging method, we analyze how many limit cycles can bifurcate from the period solutions surrounding the centers of the considered systems when they are perturbed inside the class of homogeneous polynomial differential systems of the same degree. We show that the maximum number of limit cycles, $m$ and $m+1$, that can bifurcate from the period solutions surrounding the centers for the two classes of differential systems of degree $2m$ and degree $2m+1$, respectively. Both of the bounds can be reached for all $m$.     

Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics

We study the number of zeros of Abelian integrals for the quadratic centers having almost all their orbits formed by cubics, when we perturb such systems inside the class of all polynomial systems of degreen     

Criteria to bound the number of critical periods

In the present paper we study the period function of centers of potential systems. We obtain criteria to bound the number of critical periods. In case that the system is polynomial, our result enables to tackle the problem from a purely algebraic point of view, since it allows to bound the number of critical periods by counting the zeros of a polynomial. To illustrate its applicability some new and old results are proved.     

Non-isochronicity of the center at the origin in polynomial Hamiltonian systems with even degree nonlinearities

In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin.     

The Center Problem for a Linear Center Perturbed by Homogeneous Polynomials

The centers of the polynomial differential systems with homogeneous polynomials have been studied for the degrees s = 2, 3, 4, 5. for s = 2, 3, and partially classified for s = 4, 5. In this paper we recall and we give new centers for s = 6, 7 a linear center perturbed by They are completely classified these results for s = 2, 3, 4, 5,     

Centers of quasi-homogeneous polynomial planar systems

In this paper we determine the centers of quasi-homogeneous polynomial planar vector fields of degree 0, 1, 2, 3 and 4. In addition, in every case we make a study of the reversibility and the analytical integrability of each one of the above centers. We find polynomial centers which are neither orbitally reversible nor analytically integrable, this is a new scenario in respect to the one of non-degenerate and nilpotent centers.     

Conditions for polynomial Liénard centers

In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coupled functional equations are solvable, in this paper we give an algorithm to compute a Gr¨obner basis for irreducible decomposition of algebraic varieties so as to find algebraic relations among coefficients of the damping function and the restoring function. We demonstrate the algorithm for polynomial Li′enard systems of degree 5, which are divided into 25 cases. We find all conditions of those coefficients for the polynomial Li′enard center in 13 cases and prove that the origin is not a center in the other 12 cases.     

A new method to determine isochronous center conditions for polynomial differential systems

The computation of period constants is a way to study isochronous center for polynomial differential systems. In this article, a new method to compute period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As an application, we discuss the center conditions and isochronous centers for a class of high-degree system.     

NEW FAMILIES OF CENTERS AND LIMIT CYCLES FOR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH HOMOGENEOUS NONLINEARITIES

We consider the class of polynomial differential equations x = -y+Pn(x,y), y = x + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. Inside this class we identify a new subclass of systems having a center at the origin. We show that this subclass contains at least two subfamilies of isochro-nous centers. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centers when we perturb them inside the class of all polynomial differential systems of the above form. In particular, we present a function whose simple zeros correspond to the limit cycles vvhich bifurcate from the periodic orbits of Hamiltonian systems.     

Non-isochronicity of the center in polynomial Hamiltonian systems

In 2002 Jarque and Villadelprat proved that planar polynomial Hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial Hamiltonian systems of even degree. Recently, it was proved that a planar polynomial Hamiltonian system is non-isochronous if a quantity, denoted by M_2m−2, can be computed such that M_2m−2≤0. As a corollary of this criterion, the open question was answered for those systems with only even degree nonlinearities. In this paper we consider the case of M_2m−2>0 and give a new criterion for non-isochronicity. Applying the new criterion, we also answer the open question for some cases in which some terms of odd degree are included.     

Complexity reduction of C-Algorithm

The C-Algorithm introduced in [5] is designed to determine isochronous centers for Lienard-type differential systems, in the general real analytic case. However, it has a large complexity that prevents computations, even in the quartic polynomial case.The main result of this paper is an efficient algorithmic implementation of C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of it is proposed in the rational case. It is called RCA (Rational C-Algorithm) and is widely used in [1] and [2] to find many new examples of isochronous centers for the Liénard type equation.     

A PTAS for the disk cover problem of geometric objects

We present PTASs for the disk cover problem: given geometric objects and a finite set of disk centers, minimize the total cost for covering those objects with disks under a polynomial cost function on the disks’ radii. We describe the first FPTAS for covering a line segment when the disk centers form a discrete set, and a PTAS for when a set of geometric objects, described by polynomial algebraic inequalities, must be covered. The latter result holds for any dimension.     

\( {\small \bf Communication Letter:}\) Four limit cycles in quadratic near-integrable system

In this note, we report of obtaining 4 limit cycles in quadratic near-integrable polynomial systems. It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least 4 limit cycles with (3.1) distribution. This result provides a positive answer to an open problem in this area.     

Centers with degenerate infinity and their commutators

We study polynomial systems with degeneracy at infinity and a center-focus equilibrium at the origin. We give some general properties related to the existence of polynomial commutators and use these properties in order to characterize uniformly isochronous polynomial centers with polynomial commutator and, also, we show that the commutator of the centers of the analytic systems whose angular speed is constant can be chosen of radial form. Finally, we characterize the systems (−y+P_s+∑_j=kⁿ⁻¹xH_j,x+Q_s+∑_j=kⁿ⁻¹yH_j)^t with polynomial commutator, with P_j,Q_j,H_j and K_j homogeneous polynomials.     

Quantized Weyl algebras at roots of unity

We classify the centers of the quantized Weyl algebras that are polynomial identity algebras and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this family of algebras and their tensor products.     

On the centers of the weight-homogeneous polynomial vector fields on the plane

We classify all centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.     

Nonexistence of Isochronous Centers in Planar Polynomial Hamiltonian Systems of Degree Four

In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree four (that is, with its homogeneous part of degree four not identically zero) is nonisochronous. The proof uses the geometric properties of the period annulus and it requires the study of the Hamiltonian systems associated to a Hamiltonian function of the form H(x, y)=A(x)+B(x) y+C(x) y²+D(x) y³.     

A Rodrigues-type formula for Gegenbauer matrix polynomials

This paper centers on the derivation of a Rodrigues-type formula for the Gegenbauer matrix polynomial. A connection between Gegenbauer and Jacobi matrix polynomials is given.     

切换多项式非线性系统的输入-状态稳定性

利用耗散不等式研究了切换多项式非线性系统的输入-状态稳定性分析问题,在任意切换信号下,给出了使得切换多项式非线性系统输入-状态稳定的充分条件.采用平方和分解方法来寻找切换多项式非线性系统的输入-状态稳定共同Lyapunov函数.数值算例验证了所提方法的可行性.