Lower bounds for the local cyclicity for families of centers

Giné, Jaume; Gouveia, Luiz F. S.; Torregrosa, Joan

doi:10.1016/j.jde.2020.11.035

Cited by 20 publications

(30 citation statements)

References 19 publications

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“…Although the lower bound for the cyclicity of the quartic system 6.2, see Proposition 23, is not better than the current one, M (4) ≥ 21, we have added here because it is very close to that value and there are no many quartic systems with high local cyclicity. Systems exhibiting the best lower bound cyclicity can be found in [18]. We have studied only the cyclicity problem of this system because the number of small amplitude limit cycles obtained from the ones in Propositions 3 and 4 is not very high using only first-order Taylor developments of the Lyapunov constants.…”

Section: Involutionsmentioning

confidence: 99%

“…In [17] it is conjectured that M (n) = n 2 + 3n − 7. Recently, in [18], this conjecture should be updated in one, M (n) = n 2 + 3n − 6, because it is false, at least for n = 3. In the recent work [21] the local cyclicity of some Darboux cubic centers in [41], the ones with the highest codimension, is studied.…”

mentioning

confidence: 99%

“…The difficulties are related with the fact that we are almost using the total number of essential free parameters. For small degrees n, the best lower bound values for M (n) can be found in [18,21].…”

mentioning

confidence: 99%

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Orbitally symmetric systems with applications to planar centers

Bastos¹,

Buzzi²,

Torregrosa³

2021

CPAA

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We present a generalization of the most usual symmetries in differential equations known as the time-reversibility and the equivariance ones. We check that the typical properties are also valid for the new definition that unifies both. With it, we are able to present new families of planar polynomial vector fields having equilibrium points of center type. Moreover, we provide the highest lower bound for the local cyclicity of an equilibrium point of polynomial vector fields of degree 6, M (6) ≥ 48.

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Section: Involutionsmentioning

confidence: 99%

mentioning

confidence: 99%

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Orbitally symmetric systems with applications to planar centers

Bastos¹,

Buzzi²,

Torregrosa³

2021

CPAA

Self Cite

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“…From ( 15), ( 16), (17), and ( 18) we can write the first terms in the Taylor series of the displacement map (11). We will see in the next sections that we only need these four coefficients to characterize the centers and the maximum weak-focus order at infinity.…”

Section: Half-return Maps Near Infinitymentioning

confidence: 99%

“…That is, although generically the cyclicity of a family of centers remains constant, over some singular locus it can increase. We will closely follow the scheme of [17].…”

Section: Limit Cycles Bifurcating From the Centersmentioning

confidence: 99%

Limit cycles from a monodromic infinity in planar piecewise linear systems

Freire¹,

Ponce²,

Torregrosa³

et al. 2020

Preprint

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Planar piecewise linear systems with two linearity zones separated by a straight line and with a periodic orbit at infinity are considered. By using some changes of variables and parameters, a reduced canonical form with five parameters is obtained. Instead of the usual Bendixson transformation to work near infinity, a more direct approach is introduced by taking suitable coordinates for the crossing points of the possible periodic orbits with the separation straight line. The required computations to characterize the stability and bifurcations of the periodic orbit at infinity are much easier. It is shown that the Hopf bifurcation at infinity can have degeneracies of co-dimension three and, in particular, up to three limit cycles can bifurcate from the periodic orbit at infinity. This provides a new mechanism to explain the claimed maximum number of limit cycles in this family of systems. The centers at infinity classification together with the limit cycles bifurcating from them are also analyzed.

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Centers and Limit Cycles of Vector Fields Defined on Invariant Spheres

2021

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The aim of this paper is the study of the center-focus and cyclicity problems inside the class X of 3-dimensional vector fields that admit a first integral that leaves invariant any sphere centered at the origin. We classify the centers of linear, quadratic homogeneous and a family of quadratic vector fields F ⊂ X, restricted to one of these spheres. Moreover, we show the existence of at least 4 limit cycles in family F.

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Lower bounds for the local cyclicity for families of centers

Cited by 20 publications

References 19 publications

Orbitally symmetric systems with applications to planar centers

Orbitally symmetric systems with applications to planar centers

Limit cycles from a monodromic infinity in planar piecewise linear systems

Centers and Limit Cycles of Vector Fields Defined on Invariant Spheres

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