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Bifurcation of Limit Cycles and Isochronous Centers on Center Manifolds for a Class of Cubic Kolmogorov Systems in \({\mathbb {R}}^{3}\)

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Abstract

Our work is concerned with the number of limit cycles and isochronous center conditions for a class of three-dimensional cubic Kolmogorov systems with an equilibrium point in the positive octant where the system has biological meaning. A formal series method of computing singular point values (equivalent to focal values) is applied to investigate the Hopf bifurcation and center problem on center manifolds for the Kolmogorov system. Using this we derive two sets of conditions for the equilibrium point to be a center on a center manifold for the system, and prove that at most seven small-amplitude limit cycles can be bifurcated from an isolated positive equilibrium point. We prove that seven limit cycles can be created in this way, obtaining a new result on the number of limit cycles in three-dimensional cubic Kolmogorov systems. Moreover, two sets of necessary conditions for the existence of an isochronous center on the center manifold for such systems are obtained by the computation of period constants. The Darboux theory of linearizability is applied to show the sufficiency of the conditions.

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Acknowledgements

This research was partially supported by the National Natural Science Foundation of China (No. 12061016), the Science and Technology Project of Guangxi (No. Guike AD21220114) and the Basic Ability Enhancement Program for Young and Middle-aged Teachers of Guangxi (No. 2022KY0904).

Author information

Authors and Affiliations

  1. School of Education, Guangxi Vocational Normal University, Nanning, 530007, People’s Republic of China

    Jieping Gu

  2. College of Mathematics and Statistics, Guangxi Normal University, Guilin, 541006, People’s Republic of China

    André Zegeling & Wentao Huang

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  1. Jieping Gu
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  2. André Zegeling
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  3. Wentao Huang
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Correspondence to Wentao Huang.

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This research was partially supported by the National Natural Science Foundation of China (No. 12061016), the Science and Technology Project of Guangxi (No. Guike AD21220114) and the Basic Ability Enhancement Program for Young and Middle-aged Teachers of Guangxi (No. 2022KY0904).

Appendix

Appendix

The polynomials of \(F_{2}, F_{3}\) and \(F_{4}\) in Theorem 3.2 are listed in this Appendix.

$$\begin{aligned} F_{2}={}&-3388836930773601086 + 26477242204739337823 A_{200} \\&-32355194533115342670 A_{200}^2- 222904044664915913422 A_{200}^3 \\&+733211393543661716531 A_{200}^4 - 553588970407494376014 A_{200}^5 \\&-415042722113742016388 A_{200}^6 + 520868142418465354792 A_{200}^7 \\&+188441583887032614 D_{020}+ 4486690245944981817 A_{200} D_{020} \\&-39309694140553333533 A_{200}^2 D_{020} +105839207707569964758 A_{200}^3 D_{020} \\ {}&-105853468116567901248A_{200}^4 D_{020} +23729378304659148288 A_{200}^5 D_{020}\\&+ 899818064234828334 D_{002} D_{020} +183820544588540682 A_{200} D_{002} D_{020} \\ {}&-29346682481314370190 A_{200}^2 D_{002} D_{020} +85700993239158094800 A_{200}^3 D_{002} \\&\times D_{020} -65493329897674634976 A_{200}^4 D_{002} D_{020}-10397352565055219328 \\&\times A_{200}^5 D_{002} D_{020} + 71007395923362699 D_{020}^2 -433309801774499262 A_{200} D_{020}^2 \\&+ 1439980973662393386 A_{200}^2 D_{020}^2 -2251393624863613656 A_{200}^3 D_{020}^2 \\&+ 239254233341629248 D_{002} D_{020}^2 -1642221817251481476 A_{200} D_{002} D_{020}^2 \\&+4307838203250057960 A_{200}^2 D_{002} D_{020}^2 -5215913248245329664 A_{200}^3 D_{002} D_{020}^2 \\&+179316309352856688 D_{002}^2 D_{020}^2 -1514779355568834432 A_{200} D_{002}^2 D_{020}^2 \\&+3791660033157476904 A_{200}^2 D_{002}^2 D_{020}^2 -3055947101265018960 A_{200}^3 D_{002}^2 D_{020}^2 \\&+ 1454821374906900 D_{020}^3 -4645022756053680 A_{200} D_{020}^3 \\&-6381858184266708 D_{002} D_{020}^3 +20759034533855904 A_{200} D_{002} D_{020}^3 \\&-28983852115238592 D_{002}^2 D_{020}^3 +74643376258541376 A_{200} D_{002}2^2 D_{020}^3 \\&-31687459136910720 D_{002}^3 D_{020}^3 +75079158177909600 A_{200} D_{002}^3 D_{020}^3,\\ F_{3}={}&25963139175293910160104976312 \\&- 421293108560483451988511136452 A_{200} \\&+3097503659523342587236013463337 A_{200}^2 \\&-13401731763459192369911291720487 A_{200}^3 \\&+36411451975510637725407477614948 A_{200}^4 \\&-61333867849568342398607402938540 A_{200}^5 \\&+59445641831886380795848103961936 A_{200}^6 \\&-27254838597306923080583363934512 A_{200}^7 \\&+2292499627784597988164169568384 A_{200}^8 \\&+819227036789540555709382300096 A_{200}^9 \\&-7424230943834016188469574095 D_{020} \\&+93724125289567137788849839509 A_{200} D_{020} \\&-533686229393790902042935735542 A_{200}^2 D_{020} \\&+1671637380658509707212102879356 A_{200}^3 D_{020} \\&-2862487154896147121897397246000 A_{200}^4 D_{020} \\&+2315755177951839659960638210992 A_{200}^5 D_{020}\\&-433448082562850983574421044832 A_{200}^6 D_{020} \\&-195809392116346982386334423232 A_{200}^7 D_{020}\\&-12340907555282416848210514838 D_{002} D_{020} \\&+144211211041525950971099123242 A_{200} D_{002} D_{020} \\&-738482337142101350855899464654 A_{200}^2 D_{002} D_{020}\\&+3094045081926421876830720 A_{200} D_{002}^4 D_{020}^4 \\&-2718164349555805044364998598816 A_{200}^4 D_{002} D_{020} \\&-1190672188218733199316480 D_{002}^4 D_{020}^4 \\&+1194961222168976375528985544096 A_{200}^6 D_{002} D_{020} \\&+104017294856447207678375910 D_{020}^2 \\&-789975526445724591864441205184 A_{200}^7 D_{002} D_{020} \\&-1416058705190090566069080606 A_{200} D_{020}^2 \\&+25694486492540574147577214112 A_{200}^3 D_{002} D_{020}^2 \\&-12189561070061234374697991024 A_{200}^3 D_{020}^2 \\&+74421962751573035052515603712 A_{200}^5 D_{002} D_{020}^2 \\&+9915236348894887404366228384 A_{200}^5 D_{020}^2 \\&+449185534731771296911274034 D_{002} D_{020}^2 \\&-3111938580878696611860117336 A_{200} D_{002} D_{020}^2 \\&+2737518077153921222608089264 A_{200}^2 D_{002} D_{020}^2 \\&+6002516473363525055863417776 A_{200}^2 D_{020}^2 \\&-73764388769460477474624180192 A_{200}^4 D_{002} D_{020}^2 \\&+8650400918831621350342549920 A_{200}^4 D_{020}^2 \\&+474231388560043137905696436 D_{002}^2 D_{020}^2 \\&-1500525331333698925172096208 A_{200} D_{002}^2 D_{020}^2\\&-9789296658655327922409200640 A_{200}^2 D_{002}^2 D_{020}^2 \\&+799598619986834947411465632 A_{200}^2 D_{020}^3 \\&-134688790107991701031311785088 A_{200}^4 D_{002}^2 D_{020}^2 \\&+62117493425649286831297728 D_{020}^3 \\&+66006670790595482077055795712 A_{200}^3 D_{002}^2 D_{020}^2 \\&-408938099590575094455077760 A_{200} D_{020}^3 \\&+93534834445757249673066413184 A_{200}^5 D_{002}^2 D_{020}^2 \\&-381787286241197884598224128 A_{200}^3 D_{020}^3 \\&+272936091272111496219079440 D_{002} D_{020}^3 \\&-1707237196608148317207070560 A_{200} D_{002} D_{020}^3 \\&+3308223731031691632846560160 A_{200}^2 D_{002} D_{020}^3 \\&+8587633047424204842731520 A_{200} D_{002}^3 D_{020}^4 \\&+438408747681133305970535616 D_{002}^2 D_{020}^3 \\&-2681614184754896021160541440 A_{200} D_{002}^2 D_{020}^3 \\&+5555854266892600324591139904 A_{200}^2 D_{002}^2 D_{020}^3 \\&-937086864059502875553120 D_{002} D_{020}^4 \\&+242864315781467341132768896 D_{002}^3 D_{020}^3 \\&-1414929098130705020422241280 A_{200} D_{002}^3 D_{020}^3 \\&+3230067949763389123503988224 A_{200}^2 D_{002}^3 D_{020}^3 \\&-3283745345985598745771520 D_{002}^2 D_{020}^4 \\&-1972939111479506936190335040 A_{200}^3 D_{002} D_{020}^3 \\&+10358648389076062870903680 A_{200} D_{002}^2 D_{020}^4 \\&+2333140335794087223243840 A_{200} D_{002} D_{020}^4 \\&-4097778468137788674865712256 A_{200}^3 D_{002}^2 D_{020}^3 \\&-2945385203495641742912334336 A_{200}^3 D_{002}^3 D_{020}^3 \\&-422748978115910654688480 A_{200} D_{020}^4 \\&-2491360192974406497194880 D_{002}^3 D_{020}^4 \\&+995988900717054300615932151072 A_{200}^5 D_{002} D_{020} \\&+161275329674957209551840 D_{020}^4 \\&+2010199255132768567931961855956 A_{200}^3 D_{002} D_{020},\\ F_{4}={}&23580441053070175857314583880077428028\\&-173287862903231298561094524593989877470 A_{200}\\&-820263922542216334410913605401495199143 A_{200}^2\\&+15605413399701204204437224696613888471553 A_{200}^3 \\&-88736993268762052708158454620141081549354 A_{200}^4\\&+277708291753939144098000423906091041410340 A_{200}^5\\&-529609741464089406228364807842588188906688 A_{200}^6\\&+629913063902051244635103866217349267114056 A_{200}^7\\&-482474190195407766839292290011368802695216 A_{200}^8\\&+274248360955775273329585628625058181145888 A_{200}^9\\&-132113856268471872153191477376614508515392 A_{200}^{10}\\&+34926662125316136649949361272753140105216 A_{200}^{11}\\&-2272256410250149356671833561711380855 D_{020}\\&-66220226898206160910685993548045424829 A_{200} D_{020}\\&+1114355659575251847759161973667789140828 A_{200}^2 D_{020}\\&-6980647837458625632253324626693814847340 A_{200}^3 D_{020}\\&+23465954173347618108270172322690698992432 A_{200}^4 D_{020}\\&-45934220490917195819226055593947587868736 A_{200}^5 D_{020}\\&+53025507639536598872431573503254956758000 A_{200}^6 D_{020}\\&-36866446340277138000645995913057529696704 A_{200}^7 D_{020}\\&+17353569419170761651908849708838143569920 A_{200}^8 D_{020}\\&-4983593761535294037865652222906796342912 A_{200}^9 D_{020}\\&-10447101259198545414043098666086337522 D_{002} D_{020}\\&+4902512274712309774114687871927999286 A_{200} D_{002} D_{020}\\&+880253736383819050456387544918652775014 A_{200}^2 D_{002} D_{020}\\&-6491039479966921238411054024845859182740 A_{200}^3 D_{002} D_{020}\\&+21637646555563947962847062763215848006056 A_{200}^4 D_{002} D_{020}\\&-38813733306144028482882479652320619842880 A_{200}^5 D_{002} D_{020}\\&+38855392721348027906795469614443526525136 A_{200}^6 D_{002} D_{020}\\&-25163072015656721473439588916042615086400 A_{200}^7 D_{002} D_{020}\\&+16145648746979875046404647869894725103040 A_{200}^8 D_{002} D_{020}\\&-6798164493891880184589047559562475019264 A_{200}^9 D_{002} D_{020}\\&-919654163765550183161810414829097260 D_{020}^2\\&+18004058330190449148889651168098971886 A_{200} D_{020}^2\\&-136991341288823236424186528376297758340 A_{200}^2 D_{020}^2\\&+527310428991319455239662043483570879952 A_{200}^3 D_{020}^2\\&-1128748016756917031712072532898349568320 A_{200}^4 D_{020}^2\\&+1336926608337651718922647303772863999152 A_{200}^5 D_{020}^2\\&-784549176692517186957542356801124757984 A_{200}^6 D_{020}^2\\&+244405210514720642167896819240222841344 A_{200}^7 D_{020}^2\\&-1908652286816563394311210787166327654 D_{002} D_{020}^2\\&+43919784158765952820239475351854172272 A_{200} D_{002} D_{020}^2\\&-327922108005287536975261458939063900024 A_{200}^2 D_{002} D_{020}^2\\&+1169882277366494646075603760946425336608 A_{200}^3 D_{002} D_{020}^2\\&-2243341372434419630461094463144370055616 A_{200}^4 D_{002} D_{020}^2\\&+2424408568430481125420155273177960198752 A_{200}^5 D_{002} D_{020}^2\\&-1497129644474062987913348607335432104704 A_{200}^6 D_{002} D_{020}^2\\&+632334954171125228308433810277279852672 A_{200}^7 D_{002} D_{020}^2\\&-551442442574039365910303197625217276 D_{002}^2 D_{020}^2\\&+23665452118543297080219670611905697552 A_{200} D_{002}^2 D_{020}^2\\&-179474934268080115460148256258900631616 A_{200}^2 D_{002}^2 D_{020}^2\\&+568783239361229655684184240812215829600 A_{200}^3 D_{002}^2 D_{020}^2\\&-859776870779144400678895230188144760864 A_{200}^4 D_{002}^2 D_{020}^2\\&+687762901348543330561422074563185260736 A_{200}^5 D_{002}^2 D_{020}^2\\&-552447006549345538652477649578211165312 A_{200}^6 D_{002}^2 D_{020}^2\\&+441929220511821120416560147894882715904 A_{200}^7 D_{002}^2 D_{020}^2\\&+98073901857296192468037792239479752 D_{020}^3\\&-666475830512727278696420976244078848 A_{200} D_{020}^3\\&+2275194532565057044864690319619025296 A_{200}^2 D_{020}^3\\&-6426454612149640747110222987111558864 A_{200}^3 D_{020}^3\\&+10446211467055499555581703970267921120 A_{200}^4 D_{020}^3\\&-4470216881863866300426905216321370048 A_{200}^5 D_{020}^3\\&+567387694151128519158136249001538360 D_{002} D_{020}^3\\&-4125404488573521979908824348389963440 A_{200} D_{002} D_{020}^3\\&+13937585542454402756993035558983723120 A_{200}^2 D_{002} D_{020}^3\\&-32330536850241307591415408924255833920 A_{200}^3 D_{002} D_{020}^3\\&+42911257143232809247436328812599739520 A_{200}^4 D_{002} D_{020}^3\\&-18891911938802274301417544280903120000 A_{200}^5 D_{002} D_{020}^3\\&+984439181378134443263022724381526304 D_{002}^2 D_{020}^3\\&-6755301297188972654880141026800383456 A_{200} D_{002}^2 D_{020}^3\\&+19775640625616901024803148954805658592 A_{200}^2 D_{002}^2 D_{020}^3\\&-36432402884262681423117244708857331968 A_{200}^3 D_{002}^2 D_{020}^3\\&+42984799930857620932520922644125224960 A_{200}^4 D_{002}^2 D_{020}^3\\&-23594210781693964524966217448724546816 A_{200}^5 D_{002}^2 D_{020}^3\\&+621109912030727350969878135157603584 D_{002}^3 D_{020}^3\\&-4102678748198749213813841388664508736 A_{200} D_{002}^3 D_{020}^3\\&+10636964358659845605220355886025544832 A_{200}^2 D_{002}^3 D_{020}^3\\&-12650421059258637611531997438190057728 A_{200}^3 D_{002}^3 D_{020}^3\\&+9364333449837162755823364454125017600 A_{200}^4 D_{002}^3 D_{020}^3\\&-9877282905665141913657872897830688256 A_{200}^5 D_{002}^3 D_{020}^3\\&-1057551798318818422765952510402400 D_{020}^4\\&+9042554891398163079221005325070240 A_{200} D_{020}^4\\&-23135626066696964889254716642463520 A_{200}^2 D_{020}^4\\&+17655073220592825590292686866298880 A_{200}^3 D_{020}^4\\&-16814614801085392730438113837594560 D_{002} D_{020}^4\\&+109919534422383943937091047707662240 A_{200} D_{002} D_{020}^4\\&-241895038871329136548016371402029120 A_{200}^2 D_{002} D_{020}^4\\&+176828797367169550045943625391743360 A_{200}^3 D_{002} D_{020}^4\\&-61146125383895113238182238927688000 D_{002}^2 D_{020}^4\\&+358072623552130894663410923356573440 A_{200} D_{002}^2 D_{020}^4\\&-650952964067190187569643227392553600 A_{200}^2 D_{002}^2 D_{020}^4\\&+393310248760855302846797349858435840 A_{200}^3 D_{002}^2 D_{020}^4\\&-92518609553773369066184385281086080 D_{002}^3 D_{020}^4\\&+494520114984306700265664632064731520 A_{200} D_{002}^3 D_{020}^4\\&-760359893395868719364605559119073280 A_{200}^2 D_{002}^3 D_{020}^4\\&+313369430034235464862191019752606720 A_{200}^3 D_{002}^3 D_{020}^4\\&-55490770586047614395496987493301760 D_{002}^4 D_{020}^4\\&+259217065783271616056699675178216960 A_{200} D_{002}^4 D_{020}^4\\&-323788026842114830979799526923962880 A_{200}^2 D_{002}^4 D_{020}^4\\&+55936111692353677534871841214310400 A_{200}^3 D_{002}^4 D_{020}^4\\&+6897818521508789801305378771200 D_{020}^5\\&-20465481755106669945247045862400 A_{200} D_{020}^5\\&-70284122485281776899050426403200 D_{002} D_{020}^5\\&+213257168422092186408461889100800 A_{200} D_{002} D_{020}^5\\&+8277764002305188093408171980800 D_{002}^2 D_{020}^5\\&-104793224794935471477528600268800 A_{200} D_{002}^2 D_{020}^5\\&+863087076538572515439410048448000 D_{002}^3 D_{020}^5\\&-2347818644191646439420325541222400 A_{200} D_{002}^3 D_{020}^5\\&+1745473679951539956125246395084800 D_{002}^4 D_{020}^5\\&-4239890543495653787648300314521600 A_{200} D_{002}^4 D_{020}^5\\&+743589257305092196247489230848000 D_{002}^5 D_{020}^5\\&-1732052200150920908539716778598400 A_{200} D_{002}^5 D_{020}^5. \end{aligned}$$

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Gu, J., Zegeling, A. & Huang, W. Bifurcation of Limit Cycles and Isochronous Centers on Center Manifolds for a Class of Cubic Kolmogorov Systems in \({\mathbb {R}}^{3}\). Qual. Theory Dyn. Syst. 22, 42 (2023). https://doi.org/10.1007/s12346-023-00745-8

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  • DOI: https://doi.org/10.1007/s12346-023-00745-8

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