Bifurcation of Critical Periods from a Quartic Isochronous Center

Peng, Linping; Feng, Zhaosheng

doi:10.1142/s0218127414500898

Cited by 6 publications

(4 citation statements)

References 19 publications

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“…In this work, we extend the results presented in Peng and Feng by calculating the part of the averaging function of system that corresponds to the expression A . In order to do that, we perturb the center of system inside the whole class of quintic polynomial differential systems.…”

Section: Proof Of Theorem 11mentioning

confidence: 71%

“…They showed that for | ε | ≠ 0 sufficiently small, systems can have at least eight limit cycles bifurcating from the periodic orbits of the period annulus of the uniform center located at the origin of system . Note that the authors improved the result of Peng and Feng by proving five additional limit cycles.…”

Section: Introductionmentioning

confidence: 77%

“…The expression B corresponds to the perturbed system studied in Itikawa and Llibre and also in Peng and Feng . The authors of Peng and Feng obtained for this system the following averaging function:

\begin{matrix} right g_{B} (R) & left = \frac{3}{4 R} [(M_{4} - \frac{3 M_{1} + 4 M_{2} + 8 M_{3}}{36}) R^{2} - \frac{M_{1} + 2 M_{2}}{81} R^{6} - \frac{2 M_{1}}{729} R^{10} & right \\ right & left + (\frac{2 M_{1}}{729} R^{12} + \frac{2 M_{2}}{81} R^{8} - \frac{2 M_{3}}{9} R^{4} - 2 (M_{1} + M_{2} + M_{3})) \frac{1}{\sqrt{R^{4} - 9}}], \end{matrix}

where

\begin{matrix} right M_{1} & left = a_{22} - a_{40} - a_{04} + b_{31} - b_{13}, & right \\ right M_{2} & left = - 2 a_{22} + a_{40} + 3 a_{04} - b_{31} + 2 b_{13}, \\ right M_{3} & left = a_{22} - 3 a_{04} - b_{13}, M_{4} = a_{10} + b_{01} . \end{matrix}

Peng and Feng proved that the function g B ( R ) has at most three zeros in

R \in false(\sqrt{3}, + \infty false)

, and using the averaging theory of first order, they showed that the maximum number of limit cycles of their considered system emerging from the period annulus of the unperturbed system is three.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

See 2 more Smart Citations

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

Bey

Badi

Fernane

et al. 2018

Math Methods in App Sciences

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Section: Proof Of Theorem 11mentioning

confidence: 71%

Section: Introductionmentioning

confidence: 77%

\begin{matrix} right g_{B} (R) & left = \frac{3}{4 R} [(M_{4} - \frac{3 M_{1} + 4 M_{2} + 8 M_{3}}{36}) R^{2} - \frac{M_{1} + 2 M_{2}}{81} R^{6} - \frac{2 M_{1}}{729} R^{10} & right \\ right & left + (\frac{2 M_{1}}{729} R^{12} + \frac{2 M_{2}}{81} R^{8} - \frac{2 M_{3}}{9} R^{4} - 2 (M_{1} + M_{2} + M_{3})) \frac{1}{\sqrt{R^{4} - 9}}], \end{matrix}

where

\begin{matrix} right M_{1} & left = a_{22} - a_{40} - a_{04} + b_{31} - b_{13}, & right \\ right M_{2} & left = - 2 a_{22} + a_{40} + 3 a_{04} - b_{31} + 2 b_{13}, \\ right M_{3} & left = a_{22} - 3 a_{04} - b_{13}, M_{4} = a_{10} + b_{01} . \end{matrix}

Peng and Feng proved that the function g B ( R ) has at most three zeros in

R \in false(\sqrt{3}, + \infty false)

Section: Proof Of Theorem 11mentioning

confidence: 99%

See 1 more Smart Citation

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

Bey

Badi

Fernane

et al. 2018

Math Methods in App Sciences

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show abstract

“…In several papers as [3,10,25] it was shown that the limits cycles model many phenomena of the real world. After these works the non-existence, existence, the maximum number and other properties of the limit cycles have been extensively studied by mathematicians and physicists, and more recently, by biologists, economist and engineers, see for instance [4,17,18,19,26].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Crossing limit cycles for a class of piecewise linear differential centers separated by a conic

Jimenez,

Llibre,

Medrado

2020

ejde

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In previous years the study of the version of Hilbert's 16th problem for piecewise linear differential systems in the plane has increased. There are many papers studying the maximum number of crossing limit cycles when the differential system is defined in two zones separated by a straight line. In particular in [11,13] it was proved that piecewise linear differential centers separated by a straight line have no crossing limit cycles. However in [14,15] it was shown that the maximum number of crossing limit cycles of piecewise linear differential centers can change depending of the shape of the discontinuity curve. In this work we study the maximum number of crossing limit cycles of piecewise linear differential centers separated by a conic.differential centers separated by a conic For more information see https://ejde.math.txstate.edu/Volumes/2020/41/abstr.html

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Bifurcation of Limit Cycles from a Quintic Center via the Second Order Averaging Method

Peng

Feng²

2015

Int. J. Bifurcation Chaos

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This paper is concerned with the bifurcation of limit cycles from a quintic system with one center. By using the averaging theory, we show that under any small quintic homogeneous perturbations, up to order 1 in ε, at most three limit cycles bifurcate from periodic orbits of the considered system, and this upper bound can be reached. Up to order 2 in ε, at most seven limit cycles emerge from periodic orbits of the unperturbed one.

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Bifurcation of Critical Periods from a Quartic Isochronous Center

Cited by 6 publications

References 19 publications

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

Crossing limit cycles for a class of piecewise linear differential centers separated by a conic

Bifurcation of Limit Cycles from a Quintic Center via the Second Order Averaging Method

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