Abstract
Bifurcation of limit cycles is discussed for three-dimensional Lotka-Volterra competitive systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. Some new results are obtained for 6 classes 26–31 in Zeeman’s classification, especially, an example with four limit cycles in class 29 is given for the first time. The algorithm applied here is effective for solving the above general cyclicity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gyllenberg, M., Yan, P.: Four limit cycles for a three-dimensional competitive Lotka-Volterra system with a heteroclinic cycle. Comput. Math. Appl. 58, 649–669 (2009)
Lu, Z., Luo, Y.: Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle. Comput. Math. Appl. 46, 231–238 (2003)
Zeeman, M.L.: Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems. Dyn. Stab. Syst. 8, 189–217 (1993)
Xiao, D., Li, W.: Limit cycles for the competitive three-dimensional Lotka-Volterra system. J. Differ. Equ. 164, 1–15 (2000)
Hofbauer, J., So, J.W.-H.: Multiple limit cycles for three dimensional Lotka-Volterra equations. Appl. Math. Lett. 7, 65–70 (1994)
Lu, Z., Luo, Y.: Two limit cycles in three-dimensional Lotka-Volterra systems. Comput. Math. Appl. 44, 51–66 (2002)
Lian, X., Lu, Z., Luo, Y.: Automatic search for multiple limit cycles in three-dimensional Lotka–Volterra competitive systems with classes 30 and 31 in Zeeman’s classification. J. Math. Anal. Appl. 348, 34–37 (2008)
Gyllenberg, M., Yan, P., Wang, Y.: A 3D competitive Lotka-Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So. Appl. Math. Lett. 19, 1–7 (2006)
Wang, Q., Liu, Y., Chen, H.: Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems. Bull. Sci. Math. 134(7), 786–798 (2010)
Carr, J.: Applications of Center Manifold Theory. Springer, New York (1981)
Liu, Y., Li, J.: Theory of values of singular point in complex autonomous differential system. Sci. China Ser. A 33, 10–24 (1990)
Liu, Y.: Theory of center-focus for a class of higher-degree critical points and infinite points. Sci. China Ser. A 44, 37–48 (2001)
Liu, Y., Huang, W.: A cubic system with twelve small amplitude limit cycles. Bull. Sci. Math. 129, 83–98 (2005)
Chen, H., Liu, Y.: Linear recursion formulas of quantities of singular point and applications. Appl. Math. Comput. 148, 163–171 (2004)
Wang, Q., Liu, Y., Du, C.: Small limit cycles bifurcating from fine focus points in quartic order z 3-equivariant vector fields. J. Math. Anal. Appl. 337, 524–536 (2008)
Butler, G., Freedman, H.I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96, 425–430 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, Q., Huang, W. & Wu, H. Bifurcation of Limit Cycles for 3D Lotka-Volterra Competitive Systems. Acta Appl Math 114, 207–218 (2011). https://doi.org/10.1007/s10440-011-9609-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-011-9609-7