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

scholarly journals Periodic solutions of Abel differential equations

2007 ◽  
Vol 329 (2) ◽  
pp. 1161-1169 ◽  
Author(s):  
M.A.M. Alwash
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Abel Differential Equations

Generalized Weierstrass Integrability of the Abel Differential Equations

2013 ◽  
Vol 10 (4) ◽  
pp. 1749-1760 ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Differential Equations ◽  
Abel Differential Equations

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

On the integrable rational Abel differential equations

2009 ◽  
Vol 61 (1) ◽  
pp. 33-39 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Differential Equations ◽  
Abel Differential Equations

scholarly journals Abel differential equations admitting a certain first integral

2010 ◽  
Vol 370 (1) ◽  
pp. 187-199 ◽  
Author(s):  
Jaume Giné ◽  
Xavier Santallusia
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Abel Differential Equations

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 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 The Integrating Factors for Riccati and Abel Differential Equations

2015 ◽  
Vol 7 (2) ◽  
pp. 125
Author(s):  
Chein-Shan Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Type Systems ◽  
Second Order ◽  
Numerical Schemes ◽  
Riccati Differential Equation ◽  
Group Preserving Scheme ◽  
Abel Differential Equation ◽  
Abel Differential Equations

We can recast the Riccati and Abel differential equationsinto new forms in terms of introduced integrating factors.Therefore, the Lie-type systems endowing with transformation Lie-groups$SL(2,{\mathbb R})$ can be obtained.The solution of second-order linearhomogeneous differential equation is an integrating factorof the corresponding Riccati differential equation.The numerical schemes which are developed to fulfil the Lie-group property have better accuracy and stability than other schemes.We demonstrate that upon applying the group-preserving scheme (GPS) to the logistic differential equation, it is not only qualitatively correct for all values of time stepsize $h$, and is also the most accurate one among all numerical schemes compared in this paper.The group-preserving schemes derived for the Riccati differential equation, second-order linear homogeneous and non-homogeneous differential equations, the Abel differential equation and higher-order nonlinear differential equations all have accuracy better than $O(h^2)$.

Abel dynamic equations of first and second kind

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Martin Bohner ◽  
Sabrina H. Streipert
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Dynamic Equations ◽  
Specific Class ◽  
Special Cases ◽  
Abel Differential Equations ◽  
General Time

AbstractThis paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined.

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.

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