Weak and strong composition conditions for the Abel differential equation

Pakovich, Fedor

doi:10.1016/j.bulsci.2014.06.001

Cited by 7 publications

(4 citation statements)

References 13 publications

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“…Meanwhile, some solutions in closed form appeared in [39] and periodic solutions have been studied in [2]. Abel equations are also relevant for other mathematical problems like the centre problem for planar polynomial systems of differential equations [4,7,26,27,40]. Moreover, Abel equations are broadly used in Physics, where they describe the properties of a number of interesting physical systems [19,23,24,25,29,34,41,44,51].…”

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confidence: 99%

Quasi-Lie families, schemes, invariants and their applications to Abel equations

Cariñena

Lucas

2015

Journal of Mathematical Analysis and Applications

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This work analyses types of group actions on families of t-dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to a deep geometrical understanding of quasi-Lie schemes and quasi-Lie systems giving rise to several new results. Our achievements are illustrated by studying Abel and Riccati equations. We retrieve the Liouville invariant and study other new quasi-Lie invariants of Abel equations. Several Abel equations with a superposition rule are described and we characterise Abel equations via quasi-Lie schemes.

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confidence: 99%

Quasi-Lie families, schemes, invariants and their applications to Abel equations

Cariñena

Lucas

2015

Journal of Mathematical Analysis and Applications

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“…Notice however that the proofs given in [7], [5] are much more complicated than the proof given below. The idea to relate decompositions in R t [θ] with decompositions in L[z] was proposed in the paper [15], and the proof given below essentially coincides with the proof of Lemma 2.2 in [15].…”

Section: Decompositions Inmentioning

confidence: 99%

“…where A ∈ R[x], and deg A ≤ 2k, since (15) implies that the function µ −1 • W sends 0 and ∞ to i and −i. Alternatively, we can observe that tan(dθ/2) considered as a function of complex variable takes all the values in CP 1 distinct from ±i.…”

Section: Decompositions Inmentioning

confidence: 99%

On decompositions of trigonometric polynomials

Pakovich

2017

Isr. J. Math.

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“…Other parametric settings of the Center-Focus problem for Abel equation have been recently considered in [11,8,9] (see also [18]). In particular, the following result has been obtained in [9]: if the equation y ′ = [αp(x) + βq(x)]y 3 + q(x)y 2 (1.…”

Section: Introductionmentioning

confidence: 99%

Parametric Center-Focus Problem for Abel Equation

Briskin

Pakovich

Yomdin

2014

Qual. Theory Dyn. Syst.

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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 Poincaré Center-Focus problem for plane vector fields.Following [3,4,8,9] we say that Abel equation has a "parametric center" if for each ǫ ∈ C the equation y ′ = p(x)y 3 + ǫq(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 ≤ 10 parametric center is equivalent to the so-called "Composition Condition" (CC) ([2, 3]) 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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Weak and strong composition conditions for the Abel differential equation

Abstract: We establish an equivalence between two forms of the composition condition for the Abel differential equation with trigonometric coefficients.

Cited by 7 publications

References 13 publications

Quasi-Lie families, schemes, invariants and their applications to Abel equations

Quasi-Lie families, schemes, invariants and their applications to Abel equations

On decompositions of trigonometric polynomials

Parametric Center-Focus Problem for Abel Equation

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