Centers of quasi-homogeneous polynomial differential equations of degree three

We characterize the centers of the quasi-homogeneous planar polynomial differential systems of degree three. Such systems do not admit isochronous centers. At most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order.     

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2

We classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 2. In other words we characterize all the global phase portraits of the quadratic polynomial vector fields having all their orbits contained in conics. For such a vector field there are exactly 25 different global phase portraits in the Poincaré disc, up to a reversal of sense.     

On the critical periods of perturbed isochronous centers

Consider a family of planar systems having a center at the origin and assume that for ε=0 they have an isochronous center. Firstly, we give an explicit formula for the first order term in ε of the derivative of the period function. We apply this formula to prove that, up to first order in ε, at most one critical period bifurcates from the periodic orbits of isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers. Moreover necessary and sufficient conditions for the existence of this critical period are explicitly given. From the tools developed in this paper we also provide a new characterization of planar isochronous centers.     

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     

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.     

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 $m$.     

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.     

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.     

Isochronicity of centers in a switching Bautin system

In this paper isochronicity of centers is discussed for a class of discontinuous differential system, simply called switching system. We give some sufficient conditions for the system to have a regular isochronous center at the origin and, on the other hand, construct a switching system with an irregular isochronous center at the origin. We give a computation method for periods of periodic orbits near the center and use the method to discuss a switching Bautin system for center conditions and isochronous center conditions. We further find all of those systems which have an irregular isochronous center.     

时间可逆系统的等时中心问题

若在中心附近的闭轨线都具有相同的周期,则此中心称为等时中心. 时间可逆多项式系统的等时中心问题是一类公开问题. 为了构造性地解决这一难题,讨论一类范围更广的时间可逆解析动力系统, 给出相应横截交换系统的递推公式,此公式可以用于等时中心条件的推导. 在递推公式的基础上,以吴特征集方法为工具,给出一类时间可逆三次系统具有横截交换系统的充要条件.     

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.     

Bifurcation of critical periods from the rigid quadratic isochronous vector field

This paper is concerned with the study of the number of critical periods of perturbed isochronous centers. More concretely, if X₀ is a vector field having an isochronous center of period T₀ at the point p and X? is an analytic perturbation of X₀ such that the point p is a center for X? then, for a suitable parameterization ξ of the periodic orbits surrounding p, their periods can be written as T(ξ,?)=T₀+T₁(ξ)?+T₂(ξ)?²+?. Firstly we give formulas for the first functions Tl(ξ) that can be used for quite general vector fields. Afterwards we apply them to study how many critical periods appear when we perturb the rigid quadratic isochronous center , inside the class of centers of the quadratic systems or of polynomial vector fields of a fixed degree.     

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.     

The center conditions and local bifurcation of critical periods for a Liénard system

In this paper we study the problems of centers and isochronous centers and the local bifurcation of critical periods for a Liénard system with forth damping. Calculating the singular point values and period constants, we find all center conditions and isochronous center conditions. Moreover, the numbers of local critical periods bifurcating from centers and isochronous centers is obtained by computing the orders of weak centers.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

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.     

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.     

Isochronous centers of Lienard type equations and applications

In this work we study Eq. (E) with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of (E) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems (LD,F). We also classify a 5-parameters family of reversible cubic systems with isochronous centers.     

Isochronous Centers and Isochronous Functions

Abstract We study isochronous centers of two classes of planar systems of ordinary differential equations.Forthe first class which is the Linard systems of the form =y-F(x),=-g(x) with a center at the origin, we provethat if g is isochronous(see Definiton 1.1),then the center is isochronous if and only if F≡0.For the secondclass which is the Hamiltonian systems of the form =-g(y),=f(x) with a center at the origin,we prove thatif f or g is isochronous,then the center is isochronous if and only if the other is also isochronous.     

Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel’nikov method

We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n + 2)-degree-of-freedom near integrable Hamiltonian with n centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).