Li, Yanqiu; Jiang, Weihua Global existence of periodic solutions in the linearly coupled Mackey-Glass system. (English) Zbl 1215.34080 Int. J. Bifurcation Chaos Appl. Sci. Eng. 21, No. 3, 711-724 (2011). Summary: The dynamics of a linearly coupled Mackey-Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result. Cited in 1 Document MSC: 34K13 Periodic solutions to functional-differential equations 34K18 Bifurcation theory of functional-differential equations 34K20 Stability theory of functional-differential equations 34K17 Transformation and reduction of functional-differential equations and systems, normal forms Keywords:Mackey-Glass system; time delay; Hopf bifurcation; periodic solutions PDFBibTeX XMLCite \textit{Y. Li} and \textit{W. Jiang}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 21, No. 3, 711--724 (2011; Zbl 1215.34080) Full Text: DOI References: [1] DOI: 10.1016/j.camwa.2005.09.001 · Zbl 1101.34062 · doi:10.1016/j.camwa.2005.09.001 [2] DOI: 10.1007/b79666 · doi:10.1007/b79666 [3] DOI: 10.1364/OL.25.000625 · doi:10.1364/OL.25.000625 [4] Hassard B. D., Theory and Application of Hopf Bifurcation (1981) · Zbl 0474.34002 [5] DOI: 10.1007/BF00275162 · Zbl 0523.93038 · doi:10.1007/BF00275162 [6] DOI: 10.1016/j.physd.2004.02.005 · Zbl 1059.93118 · doi:10.1016/j.physd.2004.02.005 [7] DOI: 10.1006/jdeq.1993.1097 · Zbl 0786.34033 · doi:10.1006/jdeq.1993.1097 [8] Liz E., Diff. Integ. Eq. 15 pp 875– [9] DOI: 10.1016/S0022-247X(02)00416-X · Zbl 1022.34071 · doi:10.1016/S0022-247X(02)00416-X [10] DOI: 10.1126/science.267326 · Zbl 1383.92036 · doi:10.1126/science.267326 [11] DOI: 10.1016/0375-9601(95)00555-H · Zbl 1020.34510 · doi:10.1016/0375-9601(95)00555-H [12] DOI: 10.1016/0375-9601(95)00480-Q · doi:10.1016/0375-9601(95)00480-Q [13] DOI: 10.1103/PhysRevLett.64.821 · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821 [14] DOI: 10.1103/PhysRevE.58.3067 · doi:10.1103/PhysRevE.58.3067 [15] DOI: 10.1016/j.amc.2007.01.103 · Zbl 1131.34327 · doi:10.1016/j.amc.2007.01.103 [16] DOI: 10.1098/rspa.2007.1890 · Zbl 1137.34038 · doi:10.1098/rspa.2007.1890 [17] DOI: 10.1016/S0096-3003(02)00315-6 · Zbl 1048.34114 · doi:10.1016/S0096-3003(02)00315-6 [18] Sano S., Phys. Rev. E 75 pp 016207: 1– [19] DOI: 10.1016/j.chaos.2005.08.128 · Zbl 1142.37330 · doi:10.1016/j.chaos.2005.08.128 [20] DOI: 10.1016/j.chaos.2004.05.043 · Zbl 1061.93510 · doi:10.1016/j.chaos.2004.05.043 [21] DOI: 10.1016/j.jmaa.2005.04.051 · Zbl 1105.34047 · doi:10.1016/j.jmaa.2005.04.051 [22] DOI: 10.1016/j.jmaa.2008.03.044 · Zbl 1153.65074 · doi:10.1016/j.jmaa.2008.03.044 [23] DOI: 10.1016/j.chaos.2007.09.030 · Zbl 1197.93107 · doi:10.1016/j.chaos.2007.09.030 [24] DOI: 10.1007/s11071-008-9422-7 · Zbl 1176.78020 · doi:10.1007/s11071-008-9422-7 [25] DOI: 10.1016/S0167-2789(99)00009-3 · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3 [26] DOI: 10.1016/j.physd.2004.08.023 · Zbl 1062.34077 · doi:10.1016/j.physd.2004.08.023 [27] DOI: 10.1016/j.cam.2005.10.037 · Zbl 1098.92055 · doi:10.1016/j.cam.2005.10.037 [28] DOI: 10.1088/0951-7715/20/11/002 · Zbl 1141.34045 · doi:10.1088/0951-7715/20/11/002 [29] DOI: 10.1142/S0218127407018282 · Zbl 1159.34056 · doi:10.1142/S0218127407018282 [30] DOI: 10.1090/S0002-9947-98-02083-2 · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.