Zero-Hopf bifurcations of a family of Z<inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><msub><mrow/><mrow><mn mathvariant="normal">2</mn></mrow></msub></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/DB9A84F2-A675-4432-90B2-34269845E61F-M002.jpg"><?fx-imagestate width="1.60866666" height="5.41866684"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/DB9A84F2-A675-4432-90B2-34269845E61F-M002c.jpg"><?fx-imagestate width="1.60866666" height="5.41866684"?></graphic></alternatives></inline-formula> symmetric cubic jerk systems

HU Xiaoyan; SANG Bo

doi:10.13471/j.cnki.acta.snus.2021A027

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Zero-Hopf bifurcations of a family of Z

2

symmetric cubic jerk systems

Research articles | 更新时间：2023-11-01

- Zero-Hopf bifurcations of a family of Z $2$ symmetric cubic jerk systems
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 62, Issue 3, Pages: 169-174(2023)
- 作者机构：
  
  聊城大学数学科学学院，山东聊城 252059
- 作者简介：
- 基金信息：
- DOI：10.13471/j.cnki.acta.snus.2021A027
  CLC： O175.12
- Published：25 May 2023，
  
  Published Online：27 March 2023，
  
  Received：17 April 2021，
  
  Accepted：03 September 2021
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胡小燕,桑波.一类Z2对称三次jerk系统的zero-Hopf分支[J].中山大学学报(自然科学版),2023,62(03):169-174.

HU Xiaoyan,SANG Bo.Zero-Hopf bifurcations of a family of Z2 symmetric cubic jerk systems[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2023,62(03):169-174.
胡小燕,桑波.一类Z2对称三次jerk系统的zero-Hopf分支[J].中山大学学报(自然科学版),2023,62(03):169-174. DOI： 10.13471/j.cnki.acta.snus.2021A027.

HU Xiaoyan,SANG Bo.Zero-Hopf bifurcations of a family of Z2 symmetric cubic jerk systems[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2023,62(03):169-174. DOI： 10.13471/j.cnki.acta.snus.2021A027.

Zero-Hopf bifurcations of a family of Z

2

symmetric cubic jerk systems

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摘要

基于线性空间理论，构造了一类

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49419023&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49419015&type=

2.79399991

3.21733332

对称三次jerk系统，使得当

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49419054&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49419032&type=

7.11199999

2.28600001

时系统以原点为zero-Hopf奇点，即具有一个零特征根和一对纯虚特征根的孤立奇点。在此基础上，讨论了扰动系统的小振幅极限环的个数。利用四阶平均理论，证明扰动系统从奇点至多可分支出5个小振幅极限环且此上界是可达的，从而改进了已有的一个结果。

Abstract

Based on the theory of linear space， a family of cubic jerk systems with

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49419078&type=

3.21733332

3.72533321

symmetry is constructed， which has a zero-Hopf equilibrium at the origin when

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49419115&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49419094&type=

8.38199997

2.70933342

， i.e. an isolated equilibrium with a zero eigenvalue and a pair of purely imaginary eigenvalues. Using this result， the number of small amplitude limit cycles is studied for the perturbed system. By the help of averaging theory of fourth order， it is proved that at most 5 small amplitude limit cycles can bifurcate from the equilibrium and the bound can be reached， which improves a previous result.

关键词

zero-Hopf奇点平均理论极限环

Keywords

zero-Hopf equilibriumaveraging theorylimit cycle

references

李时敏，赵育林，岑秀丽， 2015. 一类不连续平面二次微分系统的极限环［J］. 中国科学：数学， 45（1）： 43-52.

桑波， 2014. 两类一致等时系统的中心条件和极限环分支［J］. 中山大学学报（自然科学版）， 53（6）： 146-149+154.

王震，蔺小林， 2017. 具有隐藏吸引子的三维jerk系统的动力学分析与周期解［J］. 西南大学学报（自然科学版）， 39（1）： 83-91.

熊峰，黄文韬， 2017. 两类Liénard系统的小振幅极限环［J］. 中山大学学报（自然科学版）， 56（3）： 66-70.

熊峰，黄文韬， 2018. 三类Liénard系统极限环的下界［J］. 中山大学学报（自然科学版）， 57（6）： 154-161.

BARREIRA L， LLIBRE J， VALLS C， 2020. Integrability and zero-Hopf bifurcation in the Sprott A system［J］. Bull Sci Math， 162： 102874.

BRAUN F， MEREU A C， 2021. Zero-Hopf bifurcation in a 3-D jerk system［J］. Nonlinear Anal Real World Appl， 59：103245.

KRAWCEWICZ W， WU J， 1997. Theory of degrees with applications to bifurcations and differential equations［M］. New York： Wiley.

LLIBRE J， MAKHLOUF A， 2020. Zero-Hopf periodic orbits for a Rössler differential system［J］. Int J Bifurcation Chaos， 30（12）： 2050170.

LLIBRE J， MESSIAS M，de CARVALHO REINOL A， 2020. Zero-Hopf bifurcations in three-dimensional chaotic systems with one stable equilibrium［J］. Int J Bifurcation Chaos， 30（13）： 2050189.

LLIBRE J， NOVAES D D， TEIXEIRA M A， 2014. Higher order averaging theory for finding periodic solutions via Brouwer degree［J］. Nonlinearity， 27（3）： 563-583.

LLIBRE J， ZHANG X， 2009. Hopf bifurcation in higher dimensional differential systems via the averaging method［J］. Pacific J Math， 240（2）： 321-341.

SANG B， HUANG B， 2020. Zero-Hopf bifurcations of 3D quadratic jerk system［J］. Mathematics， 8（9）： 1454.

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Zero-Hopf bifurcations of a family of Z2 symmetric cubic jerk systems

DOI：10.13471/j.cnki.acta.snus.2021A027

摘要

Abstract

关键词

Keywords

references

Zero-Hopf bifurcations of a family of Z $2$ symmetric cubic jerk systems