Small Amplitude Limit Cycles and Local Bifurcation of Critical Periods for a Quartic Kolmogorov System

Abstract

In this paper small amplitude limit cycles and the local bifurcation of critical periods for a quartic Kolmogrov system at the single positive equilibrium point (1, 1) are investigated. Firstly, through the computation of the singular point values, the conditions of the critical point (1, 1) to be a center and to be the highest degree fine singular point are derived respectively. Then, we prove that the maximum number of small amplitude limit cycles bifurcating from the equilibrium point (1, 1) is 7. Furthermore, through the computation of the period constants, the conditions of the critical point (1, 1) to be a weak center of finite order are obtained. Finally, we give respectively that the number of local critical periods bifurcating from the equilibrium point (1, 1) under the center conditions. It is the first example of a quartic Kolmogorov system with seven limit cycles and a quartic Kolmogorov system with four local critical periods created from a single positive equilibrium point.

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Acknowledgements

We are grateful to the Editor, Reviewers and Dr. André Zegeling for their helpful suggestions that helped to improve the paper. This research is supported by Nature Science Foundation of Guangxi (2016GXNSFDA380031).

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Correspondence to Wentao Huang.

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Appendix

Appendix

The expressions of \(T_{j}\)s \(\big (j=0,1,\ldots ,6\big )\) in Theorems 4.1 and 4.6 have the form:

$$\begin{aligned} T_{0}&= -8 + 42 A_{1} + 30 B_{2}+ 6 A_{1}^{2} - 36 A_{1} B_{2} - 27 B_{2}^{2} + 144 A_{1}^{3} + 189 A_{1}^{4},\\ T_{1}&= 655762 - 1442763 A_{1} - 3375285 B_{2} + 3209565 A_{1}^{2} + 2507472 A_{1} B_{2} + 5822892 B_{2}^{2} \\&\quad -\, 4615731 A_{1}^{3} - 9573543 A_{1}^{2} B_{2} + 596106 A_{1} B_{2}^{2} - 3468906 B_{2}^{3} + 3320730 A_{1}^{3} B_{2} \\&\quad +\, 8993754 A_{1}^{2} B_{2}^{2},\\ T_{2}&= 87857756934782242 - 520165861656483 A_{1} - 620084461304634225 B_{2}\\&\quad + 435797226603279690 A_{1}^{2} + 110914421374725012 A_{1} B_{2} + 1663066230601915872 B_{2}^{2} \\&\quad +\, 163275678202872654 A_{1}^{3} - 2104245905334988338 A_{1}^{2} B_{2}+ 358924917500107056 A_{1} B_{2}^{2} \\&\quad -\, 1976777728930344186 B_{2}^{3} + 3683170932955168599 A_{1}^{2} B_{2}^{2} - 599133080312319420 A_{1} B_{2}^{3} \\&\quad +\, 881592817586546520 B_{2}^{4} - 2262275093697293430 A_{1}^{2} B_{2}^{3},\\ T_{3}&= -5 - 48 A_{4} - 36 A_{1}^2 - 288 A_{1} A_{4} - 72 A_{3}^2 - 288 A_{4}^2 - 144 A_{1}^3 -720 A_{1}^2 A_{4} - 216 A_{1}^4 ,\\ T_{4}&= -79 + 250 A_{1} - 530 A_{4} - 1125 A_{1}^2 + 2160 A_{1} A_{3} + 5760 A_{1} A_{4} + 1920 A_{3}^2 - 1920 A_{4}^2 \\&\quad +\, 3840 A_{1}^3 + 12960 A_{1}^3 A_{3} - 14400 A_{1}^3 A_{4} + 1200 A_{1}^2 A_{4} - 2880 A_{1} A_{3}^2 + 12960 A_{1} A_{3} A_{4} \\&\quad - 17280 A_{1} A_{4}^2 - 8640 A_{1}^2 A_{3}^2 + 15840 A_{1}^2 A_{4}^2 + 5040 A_{3}^2 A_{4} + 20160 A_{4}^3,\\ T_{5}&=-986677 + 7733968 A_{1} + 1029888 A_{3} + 4394752 A_{4} -6760008 A_{1}^2 + 725760 A_{1} A_{3} \\&\quad - 18552480 A_{1} A_{4} - 8843280 A_{3}^2 +27077760 A_{3} A_{4} + 15543360 A_{4}^2+32021520 A_{1}^3 \\&\quad - 35536320 A_{1}^2 A_{3} - 9176640 A_{1}^2 A_{4} -9331200 A_{1} A_{3} A_{4} - 5368320 A_{1} A_{3}^2 \\&\quad - 312284160 A_{1} A_{4}^2 +3317760 A_{3}^3 - 77276160 A_{3}^2 A_{4} + 164643840 A_{3} A_{4}^2 \\&\quad - 162063360 A_{4}^3- 216979200 A_{1}^3 A_{4} + 421977600 A_{1}^2 A_{3} A_{4} +691200 A_{1}^2 A_{3}^2 \\&\quad -\, 34732800 A_{1}^2 A_{4}^2 - 4976640 A_{1} A_{3}^3 -51252480 A_{1} A_{3}^2 A_{4} - 307307520 A_{1} A_{3} A_{4}^2 \\&\quad - 522132480 A_{1} A_{4}^3 + 27993600 A_{3}^4 + 8709120 A_{3}^3 A_{4}+ 28339200 A_{3}^2 A_{4}^2 \\&\quad +\, 34836480 A_{3} A_{4}^3 -334540800 A_{4}^4 +33937920 A_{1}^3 A_{4}^2 - 14929920 A_{1}^2 A_{3}^3\\&\quad +\, 324933120 A_{1}^2 A_{3}^2 A_{4}+27371520 A_{1}^2 A_{3} A_{4}^2 - 656363520 A_{1}^2 A_{4}^3,\\ T_{6}&=16331381714875267 - 172534267554571734 A_{1} - 39245212356880320 A_{3} \\&\quad -\,134018559354405852 A_{4}+104036107340587464 A_{1}^2 + 44304725070910080 A_{1} A_{3} \\&\quad +\, 253069833391633440 A_{1} A_{4} + 177044466761179920 A_{3}^2 - 770590007914638240 A_{3} A_{4} \\&\quad -\, 481435066704352320 A_{4}^2-816618738644563680 A_{1}^3 + 766559668927933440 A_{1}^2 A_{3} \\&\quad +\, 67663822700980800 A_{1}^2 A_{4} + 205124717855020320 A_{1} A_{3}^2 - 126011450034388800 A_{1} A_{3} A_{4}\\&\quad +\, 7704536075457338880 A_{1} A_{4}^2 - 215179802254448640 A_{3}^3 + 2071684124520736320 A_{3}^2 A_{4} \\&\quad -\, 4691910815443373760 A_{3} A_{4}^2 + 5229216547878234240 A_{4}^3 +5247962565609415680 A_{1}^3 A_{4} \\&\quad -\, 376140674082983040 A_{1}^2 A_{3}^2 - 10907937668271299040 A_{1}^2 A_{3} A_{4} \\&\quad +\, 2457324523620227520 A_{1}^2 A_{4}^2+ 101322437534115840 A_{1} A_{3}^3 \\&\quad +\, 1375047849590472960 A_{1} A_{3}^2 A_{4} + 6880196680078095360 A_{1} A_{3} A_{4}^2 \\&\quad +\, 14555905284501826560 A_{1} A_{4}^3 - 707139917109158400 A_{3}^4\\&\quad -\, 593711768016944640 A_{3}^2 A_{4}^2 - 710649240879513600 A_{3} A_{4}^3 \\&\quad +\, 8278574356837693440 A_{4}^4+231518399106355200 A_{1}^2 A_{3}^3 \\&\quad -\, 8314906711244958720 A_{1}^2 A_{3}^2 A_{4} - 1074811230056793600 A_{1}^2 A_{3} A_{4}^2 \\&\quad +\, 17110824441857233920 A_{1}^2 A_{4}^3 -57098713340390400 A_{1} A_{3}^4 \\&\quad -\, 161258133067315200 A_{1} A_{3}^3 A_{4} +47892230563968000 A_{1} A_{3}^2 A_{4}^2\\&\quad + 685 348 337 699 020 800 A_{1} A_{4}^{4} - 121 243 113 992 448 000 A_{3}^4 A_{4}\\&\quad +\, 112889718020620800 A_{3}^3 A_{4}^2 - 22350459120998400 A_{3}^3 A_{4} \\&\quad -196 325 208 051 840 000 A_{3}^{2} A_{4}^{3} + 451 558 872 082 483 200 A_{3} A_{4}^{4}\\&\quad +\, 1154588991671808000 A_{4}^5+63114076963123200 A_{1}^2 A_{3}^4 \\&\quad - 193525230892492800 A_{1}^2 A_{3}^3 A_{4} - 1184557301770752000 A_{1}^2 A_{3}^2 A_{4}^2 \\&\quad +\, 354796256636236800 A_{1}^2 A_{3} A_{4}^3 + 402514361235302400 A_{1} A_{3} A_{4}^3\\&\quad +\, 1988142600384460800 A_{1}^2 A_{4}^4. \end{aligned}$$

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He, D., Huang, W. & Wang, Q. Small Amplitude Limit Cycles and Local Bifurcation of Critical Periods for a Quartic Kolmogorov System. Qual. Theory Dyn. Syst. 19, 68 (2020). https://doi.org/10.1007/s12346-020-00401-5

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Keywords

  • Kolmogrov system
  • Singular point value
  • Period constant
  • Limit cycle
  • Local critical period

Mathematics Subject Classification

  • 34C05
  • 34C07