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Parametric Center-Focus Problem for Abel...
Briskin, M....

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Parametric Center-Focus Problem for Abel Equation by Briskin, M. ( Author )
Lemonjar Open Access
Australian National University
01-09-2023
The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with meromorphic 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 Poincar\'e Center-Focus problem for plane vector fields. Following [3,4,8,9] we say that Abel equation has a "parametric center" if for each $\varepsilon \in \mathbb C$ the equation $y'=p(x)y^3 + \varepsilon q(x) y^2$ has a center. In the present paper we use recent results of [15,6} to show show that for a polynomial Abel equation parametric center implies strong "composition" restriction on $p$ and $q$. In particular, we show that for $\deg p,q \leq 10$ parametric center is equivalent to the so-called "Composition Condition" (CC) on $p,q$. Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in [8], where certain moments of $p,q$ vanish while (CC) is violated.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1312.1609
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