scholarly journals The center problem and composition condition for Abel differential equations

2016 ◽  
Vol 34 (2) ◽  
pp. 210-222 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau ◽  
Xavier Santallusia
Keyword(s):  
Differential Equations ◽  
Center Problem ◽  
Composition Condition ◽  
Abel Differential Equations

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

2018 ◽  
Vol 39 (12) ◽  
pp. 3347-3352 ◽  
Author(s):  
JAUME GINÉ ◽  
MAITE GRAU ◽  
XAVIER SANTALLUSIA
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Model Problem ◽  
Polynomial Systems ◽  
Composition Condition ◽  
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.

scholarly journals Universal curves in the center problem for Abel differential equations

2014 ◽  
Vol 36 (5) ◽  
pp. 1379-1395 ◽  
Author(s):  
ALEXANDER BRUDNYI
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fundamental Group ◽  
Universal Curve ◽  
Center Problem ◽  
Iterated Integrals ◽  
Rectifiable Curve ◽  
Basic Properties ◽  
Algebraic Description ◽  
Abel Differential Equations

We study the center problem for the class ${\mathcal{E}}_{{\rm\Gamma}}$ of Abel differential equations $dv/dt=a_{1}v^{2}+a_{2}v^{3}$, $a_{1},a_{2}\in L^{\infty }([0,T])$, such that images of Lipschitz paths $\tilde{A}:=(\int _{0}^{\cdot }a_{1}(s)\,ds,\int _{0}^{\cdot }a_{2}(s)\,ds):[0,T]\rightarrow \mathbb{R}^{2}$ belong to a fixed compact rectifiable curve ${\rm\Gamma}$. Such a curve is said to be universal if whenever an equation in ${\mathcal{E}}_{{\rm\Gamma}}$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_{1},a_{2}$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with the product being the composition of series, explicit examples of universal curves, and approximation of Lipschitz triangulable curves by universal ones.

scholarly journals Center conditions at infinity for Abel differential equations

2010 ◽  
Vol 172 (1) ◽  
pp. 437-483 ◽  
Author(s):  
Miriam Briskin ◽  
Nina Roytvarf ◽  
Yosef Yomdin
Keyword(s):  
Differential Equations ◽  
Center Conditions ◽  
Abel Differential Equations ◽  
Conditions At Infinity

THE CENTER PROBLEM FOR DISCONTINUOUS LIÉNARD DIFFERENTIAL EQUATION

1999 ◽  
Vol 09 (09) ◽  
pp. 1751-1761 ◽  
Author(s):  
B. COLL ◽  
R. PROHENS ◽  
A. GASULL
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Differential Equations ◽  
General Expression ◽  
The Other ◽  
Center Problem ◽  
Homogeneous Polynomials ◽  
Lyapunov Constants

We prove that the Lyapunov constants for differential equations given by a vector field with a line of discontinuities are quasi-homogeneous polynomials. This property is strongly used to derive the general expression of the Lyapunov constants for two families of discontinuous Liénard differential equations, modulus some unknown coefficients. In one of the families these coefficients are determined and the center problem is solved. In the other family the center problem is just solved by assuming that the coefficients which appear in these expressions are nonzero. This assumption on the coefficients is supported by their computation (analytical and numerical) for several cases.

scholarly journals Algebraic geometry of the center-focus problem for Abel differential equations

2014 ◽  
Vol 36 (3) ◽  
pp. 714-744 ◽  
Author(s):  
M. BRISKIN ◽  
F. PAKOVICH ◽  
Y. YOMDIN
Keyword(s):  
Algebraic Geometry ◽  
Differential Equations ◽  
Moment Problem ◽  
Explicit Solution ◽  
Order Approximation ◽  
Abel Equation ◽  
Composition Algebra ◽  
Center Conditions ◽  
Abel Differential Equations ◽  
Compositio Math

The Abel differential equation $y^{\prime }=p(x)y^{3}+q(x)y^{2}$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincaré center-focus problem for plane vector fields. Center conditions are provided by an infinite system of ‘center equations’. During the last two decades, important new information on these equations has been obtained via a detailed analysis of two related structures: composition algebra and moment equations (first-order approximation of the center ones). Recently, one of the basic open questions in this direction—the ‘polynomial moments problem’—has been completely settled in Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728]. In this paper, we present a progress in the following two main directions: first, we translate the results of Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728] into the language of algebraic geometry of the center equations. Applying these new tools, we show that the center conditions can be described in terms of composition algebra, up to a ‘small’ correction. In particular, we significantly extend the results of Briskin, Roytvarf and Yomdin [Center conditions at infinity for Abel differential equations. Ann. of Math. (2)172(1) (2010), 437–483]. Second, applying these tools in combination with explicit computations, we start in this paper the study of the ‘second Melnikov coefficients’ (second-order approximation of the center equations), showing that in many cases vanishing of the moments and of these coefficients is sufficient in order to completely characterize centers.

scholarly journals On the Abel differential equations of third kind

2020 ◽  
Vol 25 (5) ◽  
pp. 1821-1834
Author(s):  
Regilene Oliveira ◽  
◽  
Cláudia Valls ◽  
Keyword(s):  
Differential Equations ◽  
Abel Differential Equations
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