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Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka-Volterra systems. (English) Zbl 1334.34072
Subject of the paper are three-dimensional competitive Lotka-Volterra systems and the possible number of limit cycles around the unique equilibrium in the positive octant. As shown earlier, among the 33 stable equivalence classes of Zeeman only six may have limit cycles, namely at least two. The authors present a detailed review of the relevant literature and also point out mistakes in some papers on the subject. Then, as the main result of this paper, they prove the existence of four small-amplitude limit cycles for two examples of each of the two Zeeman classes 27 and 26. Moreover, they prove two conjectures established by Gyllenberg and Yan, referring to the case where a heteroclinic cycle exists on the boundary of the carrying simplex. The proofs use center manifold theory to reduce the system to a two-dimensional one. The calculation of center manifolds, normal forms, and focal values up to order four, requires extensive use of computer algebra systems such as various Maple programs.

MSC:
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
92D25 Population dynamics (general)
Software:
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[1] Androno, A. A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. G., Qualitative theory of second-order dynamic systems, (1973), Wiley New York · Zbl 0282.34022
[2] Brauer, F.; Castillo-Chávez, C., Mathematical models in population biology and epidemiology, (2000), Springer · Zbl 0967.92015
[3] Carr, J., Applications of center manifold theory, (1981), Springer New York
[4] Chow, S. N.; Hale, J. K., Methods of bifurcation theory, (1982), Springer New York
[5] Chow, S. N.; Li, C. C.; Wang, D., Normal forms and bifurcation of planar vector fields, (1994), Cambridge University Press Cambridge
[6] Gardini, L.; Lupini, R.; Messia, M. G., Hopf bifurcation and transition to chaos in Lotka-Volterra equation, J. Math. Biol., 27, 259-272, (1989) · Zbl 0715.92020
[7] Gazor, M.; Yu, P., Spectral sequences and parametric normal forms, J. Differential Equations, 252, 1003-1031, (2012) · Zbl 1242.34065
[8] Guckenhermer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1993), Springer New York
[9] Gyllenberg, M.; Yan, P., On the number of limit cycles for the three-dimensional Lotka-Volterra systems, Discrete Contin. Dyn. Syst. Ser. S, 11, 347-352, (2009) · Zbl 1163.34337
[10] 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) · Zbl 1189.34080
[11] 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) · Zbl 1085.34025
[12] Han, M.; Yang, J.; Yu, P., Hopf bifurcations for near-Hamiltonian systems, Internat. J. Bifur. Chaos, 19, 4117-4130, (2009) · Zbl 1183.34054
[13] Han, M.; Yu, P., Normal forms, Melnikov functions, and bifurcations of limit cycles, (2012), Springer New York · Zbl 1252.37002
[14] Hirsch, M. W., Systems of differential equations with competitive or cooperative III: competing species, Nonlinearity, 1, 51-71, (1988) · Zbl 0658.34024
[15] Hofbauer, J.; Sigmund, K., The theory of evolution and dynamical systems, (1988), Cambridge University Press Cambridge
[16] Hofbauer, J.; So, J. W.-H., Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7, 65-70, (1994) · Zbl 0816.34021
[17] Kuznetsov, Yuri A., Elements of applied bifurcation theory, (1998), Springer New York · Zbl 0914.58025
[18] Liu, Y.; Li, J., Theory of values of singular points in complex autonomous differential system, Sci. China Ser. A, 33, 10-24, (1990)
[19] Lu, Z.; Luo, Y., Two limit cycles in three-dimensional Lotka-Volterra systems, Comput. Math. Appl., 44, 51-66, (2002) · Zbl 1014.34034
[20] 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) · Zbl 1053.34030
[21] Tian, Y.; Yu, P., An explicit recursive formula for computing the normal form and center manifold of n-dimensional differential systems associated with Hopf bifurcation, Internat. J. Bifur. Chaos, 23, (2013), 18 pp
[22] Tian, Y.; Yu, P., An explicit recursive formula for computing the normal forms associated with semisimple cases, Commun. Nonlinear Sci. Numer. Simul., 19, 2294-2308, (2014)
[23] Y. Tian, P. Yu, An explicit recursive algorithm for computing the normal form and center manifold of 3-dimensional differential systems associated with Hopf bifurcation, private communication.
[24] Xiao, D.; Li, W., Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations, 164, 1-15, (2000) · Zbl 0960.34022
[25] Yu, P., Computation of normal forms via a perturbation technique, J. Sound Vib., 211, 19-38, (1998) · Zbl 1235.34126
[26] Yu, P.; Han, M., Small limit cycles bifurcating from fine focus points in cubic-order \(Z_2\)-equivariant vector fields, Chaos Solitons Fractals, 24, 329-348, (2005) · Zbl 1067.34033
[27] Yu, P.; Leung, A. Y.T., The simplest normal form of Hopf bifurcation, Nonlinearity, 16, 277-300, (2003) · Zbl 1086.34033
[28] Zeeman, M. L., Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stab. Syst., 8, 189-217, (1993) · Zbl 0797.92025
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