Center-focus Problem Research Articles

Hilbert Number for a Family of Piecewise Nonautonomous Equations

For family x′=(a0+a1cost+a2sint)|x|+b0+b1cost+b2sint\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x'=(a_0+a_1\\cos t+a_2 \\sin t)|x|+b_0+b_1 \\cos t+b_2 \\sin t$$\\end{document}, we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert problem 16).

Monodromy Problem and Tangential Center-Focus Problem for Products of Lines in General Position in P 2

Monodromy Problem and Tangential Center-Focus Problem for Products of Lines in General Position in P 2

The blow-up method applied to monodromic singularities

The blow-up method proves its effectiveness to characterize the integrability of the resonant saddles giving the necessary conditions to have formal integrability and the sufficiency doing the resolution of the associated recurrence differential equation using induction. In this work we apply the blow-up method to monodromic singularities in order to solve the center-focus problem. The case of nondegenerate monodromic singularities is straightforward since any real nondegenerate monodromy singularity can be embedded into a complex system with a resonant saddle. Here we apply the method to nilpotent and degenerate monodromic singularities solving the center problem when the center conditions are algebraic.

A Complete Classification on the Center-Focus Problem of a Generalized Cubic Kukles System with a Nilpotent Singular Point

A Complete Classification on the Center-Focus Problem of a Generalized Cubic Kukles System with a Nilpotent Singular Point

Qualitative Analysis of the Minimal Higgins Model of Glycolysis

Glycolysis, one of the leading metabolic pathways, involves many different periodic oscillations emerging at positive steady states of the biochemical models describing this essential process. One of the models employing the molecular diffusion of intermediates is the Higgins biochemical model to explain sustained oscillations. In this paper, we investigate the center-focus problem for the minimal Higgins model for general values of the model parameters with the help of computational algebra. We demonstrate that the model always has a stable focus point by finding a general form of the first Lyapunov number. Then, varying two of the model parameters, we obtain the first three coefficients of the period function for the stable focus point of the model and prove that the singular point is actually a bi-weak monodromic equilibrium point of type [1, 2]. Additionally, we prove that there are two (small) intervals for a chosen parameter a > 0 for which one critical period bifurcates from this singular point after small perturbations

The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor

Abstract We consider analytic families of planar vector fields depending analytically on the parameters in Λ \Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ \ Λ ∗ \Lambda \backslash {\Lambda }^{\ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.

The isochronous center for Kukles homogeneous systems of degree eight

In this paper, we investigate Kukles homogeneous systems ẋ=−y,ẏ=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n. There are two conjectures on the center-focus problem and isochronous center problem of the above systems. These two conjectures are claimed to be proven in Giné et al. (2015,2017). However, the proofs may have some gaps, hence they are still open. The gaps of Giné et al. (2015,2017) have been shown in Guo et al. (2021). In this paper, we consider isochronous centers in the family of Kukles homogeneous systems symmetric about the y-axis of degree eight. By using Period Abel constants and Gröbner basis of polynomial systems, we obtain that there is only one isochronous center excepting the linear center.

Nilpotent Center in a Continuous Piecewise Quadratic Polynomial Hamiltonian Vector Field

In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line [Formula: see text], where these kinds of systems have a nilpotent center at [Formula: see text], which comes from the combination of two cusps of both Hamiltonian systems. By the Poincaré compactification we classify the global phase portraits of these systems. We must mention that it is extremely rare to find works studying the center-focus problem in piecewise smooth systems with nonelementary singular points as we did here.

Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude.

Center problem with characteristic directions and inverse integrating factors

We consider the center-focus problem for analytic families of planar vector fields having a monodromic singularity with characteristic directions. We give a method to compute the center conditions of the family provided there is an inverse integrating factor whose expression in polar coordinates has the exceptional divisor of the first polar blow-up as isolated zero-set. Some non-trivial examples are analyzed.

The isochronous centers for Kukles homogeneous system of degree nine

In this paper, we consider Kukles homogeneous systems ẋ=−y,ẏ=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n. There are two conjectures on the center-focus problem and isochronous center problem of the above systems. These two conjectures are claimed to be proven in Giné et al. (2015) and Giné et al. (2017). However, the proofs may have some gaps, hence they are still open. In this paper, we consider isochronous centers in the family of Kukles homogeneous systems of degree nine. By using Period Abel constants and Gröbner basis of polynomial systems, we obtain that there is no isochronous center with the exception of the linear center.

On the center-focus problem for the equation dy dx+Σ i=1 n a i (x)y i =0,0≤x≤1 where a i are polynomials

We study irreducible components of the set of polynomial plane differential systems with a center, which can be seen as a modern formulation of the classical center-focus problem. The emphasis is given on the interrelation between the geometry of the center set and the Picard–lefschetz theory of the bifurcation (or Poincaré–Pontryagin–Melnikov) functions. Our main illustrative example is the center-focus problem for the Abel equation on a segment, which is compared to the related polynomial Liénard equation.

Center Conditions for a Class of Rigid Quintic Systems

The problem of determining necessary and sufficient conditions on P and Q for system.
 x = - y+P(x+y),y = x+Q(x+y) to have a center at the origin is known as the Poincaré center-focus problem. So far, people has tried many ways to solve the problem of central focus. However, it is difficult to solve the center focus problem of higher order polynomial system. In this paper, we use the Poincaré and Alwash-Lloyd methods to study the center focus problem and derive the center conditions of the five periodic differential equation.

On the Composition Center for a Class of Rigid System

In this paper, we have proved that for a class of rigid system the composition conjecture is true. We give the sufficient and necessary conditions for this rigid system to have a center at origin point and these conditions are succinct and beautiful. The ideas and methods in this paper can be used to solve the center-focus problem of more higher order polynomial differential systems.

On the Center-Focus Problem for a Family of High-Order Polynomial Differential Systems

In this paper, we have solved the center-focus problem for a family of high-order polynomial differential systems. We give the sufficient and necessary conditions for the corresponding periodic differential equation to have a center and also show that this center is a CC-center.

An inverse approach to the center problem

We consider analytic or polynomial vector fields of the form $${\mathcal {X}}=\left( -y+X\right) \dfrac{\partial }{\partial x}+\left( x+Y\right) \dfrac{\partial }{\partial y},$$ where $$X=X(x,y))$$ and $$Y=Y(x,y))$$ start at least with terms of second order. It is well-known that $${\mathcal {X}}$$ has a center at the origin if and only if $${\mathcal {X}}$$ has a Liapunov–Poincare local analytic first integral of the form $$H=\dfrac{1}{2}(x^2+y^2)+\sum _{j=3}^ {\infty } H_j$$ , where $$H_j=H_j(x,y)$$ is a homogenous polynomial of degree j. The classical center-focus problem already studied by Poincare consists in distinguishing when the origin of $${\mathcal {X}}$$ is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which H is a first integral. Moreover, given an analytic function $$V=1+\sum _{j=1}^ {\infty } V_j$$ in a neighborhood of the origin, where $$V_j$$ is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which V is a Reeb inverse integrating factor. We study the particular case of centers which have a local analytic first integral of the form $$ H=\dfrac{1}{2}(x^2+y^2)\,\left( 1+ \sum _{j=1}^{\infty } \Upsilon _j\right) , $$ in a neighborhood of the origin, where $$\Upsilon _j$$ is a homogenous polynomial of degree j for $$j\ge 1.$$ These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin We extended to analytic or polynomial differential systems the weak conditions of a center given by Alwash and Lloyd for linear centers with homogeneous polynomial nonlinearities. Furthermore the centers satisfying these weak conditions are weak centers.

A counterexample to the composition condition conjecture for polynomial Abel differential equations

Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works pointed out that all centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.

Solution of the parametric center problem for the Abel differential equation

The Abel differential equation y'=p(x)y^2+q(x)y^3 with p,q\in \mathbb R[x] is said to have a center on a segment [a,b] if all its solutions, with the initial value y(a) small enough, satisfy the condition y(b)=y(a) . The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical Center-Focus problem of Poincaré. The Abel equation is said to have a “parametric center” if for each \epsilon \in \mathbb R the equation y'=p(x)y^2+\epsilon q(x)y^3 has a center. In this paper we show that the Abel equation has a parametric center if and only if the antiderivatives P=\int p(x) dx, Q=\int q(x) dx satisfy the equalities P=\widetilde P \circ W , Q=\widetilde Q\circ W for some polynomials \widetilde P, \widetilde Q, and W such that W(a)=W(b) . We also show that the last condition is necessary and sufficient for the “generalized moments” \int_a^b P^id Q and \int_a^b Q^id P to vanish for all i\ geq 0.

Research on the Poincar\xe9 center-focus problem of some cubic differential systems by using a new method

In this article, we discuss the Poincaré center-focus problem of some cubic differential systems by using a new method (Mironenko’s method) and obtain some new sufficient conditions for a critical point to be a center. By using this method we not only solve a center-focus problem, but also at the same time, we open a class of differential systems, which do not have to be polynomial differential systems, with the same qualitative behavior at the critical point.

A hybrid symbolic-numerical approach to the center-focus problem

We propose a new hybrid symbolic-numerical approach to the center-focus problem. The method allowed us to obtain center conditions for a three-dimensional system of differential equations, which was previously not possible using traditional, purely symbolic computational techniques.