## Hopf bifurcation analysis in a Mackey-Glass system.(English)Zbl 1159.34056

Consider the delay equation $\dot x(t)= {a\over 1+ x^n(t-\tau)}- \gamma x(t),\tag{$$*$$}$ where $$a$$ and $$\gamma$$ are positive constants. The authors determine the equilibria of $$(*)$$, investigate their stability and establish a sequence of Hopf bifurcations at the positive equilibrium as the delay increases. Moreover, they prove globl existence results on periodic solutions.

### MSC:

 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K18 Bifurcation theory of functional-differential equations 92C30 Physiology (general) 34K20 Stability theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations
Full Text:

### References:

 [1] Coppel W. A., Stability and Asymptotic Behavior of Differential Equations (1965) · Zbl 0154.09301 [2] Dieudonné J., Foundations of Modern Analysis (1960) · Zbl 0100.04201 [3] Gopalsamy K., Dyn. Stab. Syst. 4 pp 131– [4] DOI: 10.1007/BF01057415 · Zbl 0694.34057 [5] Gyori I., Oscillation Theory of Delay Differential Equations with Applications (1991) [6] DOI: 10.1007/978-1-4612-9892-2 [7] Hassard B. D., Theory and Application of Hopf Bifurcation (1981) · Zbl 0474.34002 [8] DOI: 10.1016/S0895-7177(01)00166-2 · Zbl 1069.34107 [9] DOI: 10.1006/jdeq.1993.1097 · Zbl 0786.34033 [10] DOI: 10.1126/science.267326 · Zbl 1383.92036 [11] DOI: 10.1216/rmjm/1181073047 · Zbl 0725.34049 [12] DOI: 10.1017/S0308210500031061 · Zbl 0946.34062 [13] S. Ruan and J. Wei, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Math. Anal. 10 (2003) pp. 863–874. · Zbl 1068.34072 [14] DOI: 10.1142/S0218127404011697 · Zbl 1090.37547 [15] DOI: 10.1016/j.jmaa.2004.06.056 · Zbl 1067.34076 [16] Song Y., J. Math. Anal. Appl. 316 pp 458– [17] Wei J., Acta Math. Sin. 45 pp 93– [18] DOI: 10.1016/j.na.2003.04.002 · Zbl 1144.34373 [19] DOI: 10.1016/j.jmaa.2005.03.049 · Zbl 1085.34058 [20] Wei J., J. Comput. Appl. Math. [21] DOI: 10.1016/S1468-1218(01)00049-9 · Zbl 1095.34549 [22] DOI: 10.1090/S0002-9947-98-02083-2 · Zbl 0905.34034 [23] DOI: 10.1016/S0096-3003(95)00213-8 · Zbl 0848.92018 [24] DOI: 10.1016/j.chaos.2004.03.013 · Zbl 1129.34329
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.