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Nine Limit Cycles Around a Weak Focus in a Class of Three-Dimensional Cubic Kukles Systems

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Abstract

In this paper, we study the bifurcation of limit cycles, centers, and isochronous centers for a class of three-dimensional Kukles systems of degree 3. Through calculating the singular point quantities, we obtain a necessary condition for the origin to be a center, then the Darboux integrability theory is used to prove that the necessary condition is also sufficient. Then, we demonstrate that the origin is an isochronous center under the obtained center condition. Finally, we determine that the highest order of the origin to be a weak focus is ten, while the maximum number of small-amplitude limit cycles bifurcating from this weak focus is nine. Moreover, we show that this maximum number of 9 can be realized. It is worthwhile to say that this number is also a new lower bound on the number of limit cycles bifurcating from the single weak focus for three-dimensional cubic polynomial systems.

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Acknowledgements

We would like to thank the referees for their useful suggestions and valuable comments that help us to improve the presentation and the proofs of our results. This work is supported by the National Natural Science Foundation of China (Grant Nos. 12061016 and 12161023) and the China Scholarship Council (No. 202206240081).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541004, People’s Republic of China

    Yuting Ouyang & Wentao Huang

  2. School of Mathematics, Sichuan University, Chengdu, 610065, People’s Republic of China

    Dongping He

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  1. Yuting Ouyang
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Correspondence to Dongping He.

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Appendix

Appendix

The polynomials of \(F_{2}(a_{3},b_{1},b_{3}), F_{3}(a_{3},b_{1},b_{3})\) and \(F_{4}(a_{3},b_{1},b_{3})\) in Theorem 3.2 are listed as follows,

$$\begin{aligned} F_{2}&=2022624740155688160107545989458895234375\\&\quad - 15316494704666857881858571415177679360000 b_{3}\\&\quad -(1553587757954031426141787091743423500000\\&\quad +7973294911946495115625284221517803136000b_{3}) a_{3} b_{3}^2\\&\quad -(9398979367830229402956769289254212300000\\&\quad + 21555997615539736407761025938041511884800b_{3}) a_{3} b_{1}^2\\&\quad +(6030163175958628391414483824019530973184 b_{1}^2\\&\quad -10261699541799483351418377575423121162240 b_{3}^2)a_{3}^2 b_{1}^2 b_{3}\\&\quad -3119765195147056350252989345072825958400 a_{3}^2 b_{3}^5,\\ F_{3}&=22416364654207270260327968396149281392811506528841600000000\\&\quad -317388102786837396350977747854653432597539597346806640625 a_{3}\\&\quad + 59925263360098270575469339726191561871682457500578867200000 a_{3} b_{1}^2\\&\quad -79931005933744432807585163543607867389908077524255637504000 a_{3}^2 b_{1}^4\\&\quad + 21014398278258981086521882855969102370094013852221670686720a_{3}^3 b_{1}^6\\&\quad -1386282411015437374549093885967618062534874600054850000000 a_{3} b_{3}\\&\quad + 8832599449419067682109266021918548216342727568525696000000 a_{3}^2b_{1}^2 b_{3}\\&\quad + 26370447653794797873748979854317806887509182027101140480000 a_{3} b_{3}^2 \\&\quad + 34097395067305410895435685620447308305257090870339987046400 a_{3}^2 b_{1}^2 b_{3}^2 \\&\quad -24137123115809812270432392465893006907271983815929953779712 a_{3}^3 b_{1}^4 b_{3}^2 \\&\quad - 256837295755460978805705283966170887820007262225280000000 a_{3}^2 b_{3}^3 \\&\quad + 6375293556050179862244785126499385493679839317176680448000 a_{3}^2 b_{3}^4 \\&\quad + 2401092400967568840669154630395102998271130541014458040320 a_{3}^3 b_{1}^2 b_{3}^4 \\&\quad - 393733998260847008673126477143792552357488768234343628800 a_{3}^3 b_{3}^6 , \\ F_{4}&=318700913579856271473070593609518530549076238682995290043187500000 \\&\quad +2999224387405884414350430410257685086273329454118482233028042500000 a_{3} b_{1}^2 \\&\quad -1385687742866903771054067262536931375303198759016114627051942400000a_{3}^2 b_{1}^4 \\&\quad + 1685983390055613100435524193708257012002922649712385245232888000000 b_{3} \\&\quad -50238444106189459172772912121666401527891362079494411602392578125 a_{3} b_{3} \\&\quad + 2559083498031404753602581717983008956145429188952805150557724160000 a_{3} b_{1}^2 b_{3} \\&\quad -4725060815843362135194556778274828754273051166953311180819077529600 a_{3}^2 b_{1}^4 b_{3} \\&\quad + 1042861809627396334218224353234053576003653939137913600973830356992 a_{3}^3 b_{1}^6 b_{3} \\&\quad + 692789413860564064017095958590465709815296654211674693243687500000 a_{3} b_{3}^2 \\&\quad + 605297132682549487157381165951281814640926697229232429536800000000 a_{3}^2 b_{1}^2 b_{3}^2 \\&\quad + 981033744707993101078330229099399322898557725481883260067104000000 a_{3} b_{3}^3 \\&\quad + 699067911658642310312531640081450476201007020636555183170191360000 a_{3}^2 b_{1}^2 b_{3}^3 \\&\quad -1360286213453169614795715487409537580376227805738638859362631680000 a_{3}^3 b_{1}^4 b_{3}^3 \\&\quad + 103754510099843208359502245493791158858905340333609783337760000000 a_{3}^2 b_{3}^4 \\&\quad + 285759539856945402698757366603602115172153240801723643184906240000 a_{3}^2 b_{3}^5 \\&\quad - 91071010835538305720889179718535521951694612303427787253245542400 a_{3}^3 b_{1}^2 b_{3}^5 \\&\quad + 73845021353075868396239141571048949484194274145922979791372288000 a_{3}^3 b_{3}^7. \end{aligned}$$

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Ouyang, Y., He, D. & Huang, W. Nine Limit Cycles Around a Weak Focus in a Class of Three-Dimensional Cubic Kukles Systems. Qual. Theory Dyn. Syst. 23, 101 (2024). https://doi.org/10.1007/s12346-024-00959-4

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