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

• He, Dongping ^[1] ; Huang, Wentao ^[2] ; Wang Qinlong ^[3]
  1. [1] Sichuan University

     Sichuan University

     China

     
  2. [2] Guangxi Normal University

     Guangxi Normal University

     China

     
  3. [3] Guilin University of Electronic Technology

     Guilin University of Electronic Technology

     China

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 2, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00401-5
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• Resumen

  • 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.

• Referencias bibliográficas
  • 1. Amel’kin, V.V., Lukashevich, N.A., Sadovskii, A.P.: Nonlinear Oscillations in Second Order Systems. Belarusian State University, Minsk...
  • 2. Chen, X., Zhang, W.: Decomposition of algebraic sets and applications to weak centers of cubic systems. J. Comput. Appl. Math. 232, 565–581...
  • 3. Chen, X., Huang, W., Romanovski, V.G., Zhang, W.: Linearizability and local bifurcation of critical periods in a cubic Kolmogorov system....
  • 4. Chen, T., Huang, W., Ren, D.: Weak centers and local critical periods for a Z2-equivariant cubic system. Nonlinear Dyn. 78, 2319–2329 (2014)
  • 5. Chicone, C., Jacobs, M.: Bifurcation of critical periods for plane vector fields. Trans. Am. Math. Soc. 312, 433–486 (1989)
  • 6. Du, C., Huang, W.: Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model. Nonlinear Dyn. 72, 197–206...
  • 7. Du, C., Liu, Y., Huang, W.: Limit cycles bifurcations for a class of Kolmogorov model in symmetrical vector field. Int. J. Bifur. Chaos...
  • 8. Du, C., Liu, Y., Huang, W.: Behavior of limit cycle bifurcations for a class of quartic Kolmogorov models in a symmetrical vector field....
  • 9. Du, C., Liu, Y., Mi, H.: The bifurcation of limit cycles for a class of cubic Kolmogorov system (in Chinese). Chin. J. Eng. Math. 24, 746–752...
  • 10. Du, C., Liu, Y., Zhang, Q.: Limit cycles in a class of quartic Kolmogorov model with three positive equilibrium points. Int. J. Bifur....
  • 11. Han, M., Lin, Y., Yu, P.: A study on the exitence of limit cycles of a planar system with third-degree polynomials. Int. J. Bifur. Chaos...
  • 12. Huang, W., Liu, Y.: Bifurcations of limit cycles from infinity for a class of quintic polynomial system. Bull. Sci. Math. 128, 291–301...
  • 13. Huang, W., Chen, T., Li, J.: Isolated periodic wave trains and local critical wave lengths for a nonlinear reaction–diffusion equation....
  • 14. Lloyd, N.G., Pearson, J.M., Sáez, E., Szántó, I.: A cubic Kolmogorov system with six limit cycles. Comput. Math. Appl. 44, 445–455 (2002)
  • 15. Li, F.: Integrability and bifurcations of limit cycles in a cubic Kolmogorov system. Int. J. Bifur. Chaos 23, 1350061 (2013)
  • 16. Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifur. Chaos 13, 47–106 (2003)
  • 17. Liu, Y., Li, J.: Theory of values of singular point in complex autonomous differential systems. Sci. China Ser. A 33, 10–23 (1990)
  • 18. Liu, Y., Chen, H.: Formulas of singurlar point quantities and the first 10 saddle quantities for a class of cubic system (in Chinese)....
  • 19. Liu, Y., Huang, W.: A new method to determine isochronous center conditions for polynomial differential systems. Bull. Sci. Math. 127,...
  • 20. Liu, Y., Li, J., Huang, W.: Singular Point Values, Center Problem and Bifurcations of Limit Cycles of Two Dimensional Differential Autonomous...
  • 21. Liu, Y., Li, J.: New study on the center problem and bifurcations of limit cycles for the Lyapunov system (I ). Int. J. Bifur. Chaos 19,...
  • 22. Lin, Y., Li, J.: The canonical form of the autonomous planar system and the critical point of the closed orbit period (in Chinese). Acta...
  • 23. Mi, H., Du, C.: The central conditidn and bifurcation of limit cycles for a class of cubic Kolmogorov system. Math. Theory Appl. 25, 19–21...
  • 24. Romanovski, V.G., Han, M.: Critical period bifurcations of a cubic system. J. Phys. A: Math. Gen. 36, 5011–5022 (2003)
  • 25. Romanovski, V.G., Fernandes, W., Tang, Y., Tian, Y.: Linearizability and critical period bifurcations of a generalized Riccati system....
  • 26. Rousseau, C., Toni, B.: Local bifurcations of critical periods in vector fields with homogeneous nonliearities of the third degree. Can....
  • 27. Rousseau, C., Toni, B.: Local bifurcations of critical periods in the reduced Kukles system. Can. J. Math. 49, 338–358 (1997)
  • 28. Toni, B.: Bifurcations of critical periods: cubic vector fields in Kapteyns normal form. Quaestiones Mathematicae 22, 43–61 (1999)
  • 29. Xu, Q., Huang, W.: The center conditions and local bifurcation of critical periods for a Liénard system. Appl. Math. Comput. 217, 6637–6643...
  • 30. Ye, Y., Ye, W.: Cubic Kolmogorov differential system with two limit cycles surrounding the same focus. Ann. Diff. Eqs. 1, 201–207 (1985)
  • 31. Yu, P., Han, M.: Critical periods of planar revertible vector field with third-degree polynomial functions. Int. J. Bifur. Chaos 19, 419–433...
  • 32. Yu, P., Han, M., Zhang, J.: Critical periods of third-order planar Hamiltonian systems. Int. J. Bifur. Chaos 20, 2213–2224 (2010)
  • 33. Wu, Y., Huang, W., Suo, Y.: Weak center and bifurcation of critical periods in a cubic Z2-equivariant Hamiltonian vector field. Int. J....
  • 34. Wu, D., Huang, W., Wu, Y.: Limit cycles of a cubic kolmogorov system (in Chinese). J. Guilin Univ. Electron. Technol. 36, 160–163 (2016)
  • 35. Wu, D., Huang, W., Wu, Y.: Bifurcation of limit cycles of a class of quartic Kolmogorov system (in Chinese). J. Henan Univ. Sci. Technol....
  • 36. Zhan, J., Huang, W., He, D.: The center and limit cycles of a quartic Kolmogorov system (in Chinese). J. Guilin Univ. Electron. Technol....
  • 37. Zhang, Q., Li, F., Zhao, Y.: Limit cycles in a cubic kolmogorov system with harvest and two positive equilibrium points. Abstr. Appl....
  • 38. Zhang, W., Hou, X., Zeng, Z.: Weak centres and bifurcation of critical periods in reversible cubic systems. Comput. Math. Appl. 40, 771–782...
  • 39. Zou, L., Chen, X., Zhang, W.: Local bifurcations of critical periods for cubic Liénard equations with cubic damping. J. Comput. Appl....