The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.
References
[1]
Engelborghs, K., Luzyanina, T. and Samae, G. (2001) DDE-BIFTOOL, a Matlab Package for Bifurcation Analysis of Delay Differential Equations. Technical Report TW330.
[2]
Sieber, J., Engelborghs, K., Samaey, G. and Roose, D. (2015) DDE-BIFTOOL Manual—Bifurcation Analysis of Delay Differential Equations. ArXiv: 1406.7144. https://arxiv.org/abs/1406.7144
[3]
Verheyden, K., Luzyanina, T. and Roose, D. (2004) Location and Numerical Preservation of Characteristic Roots of Delay Differential Equations by LMS Methods. Technical Report TW382.
[4]
Haurie, C., Dale, D.C. and Mackey, M.C. (1998) Cyclical Neutropenia and Other Periodic Hematological Disorders: A Review of Mechanisms and Mathematical Models. Blood, 92, 2629-2640. https://doi.org/10.1182/blood.V92.8.2629.420a35_2629_2640
[5]
Bernard, S., Bélair, J. and Mackey, M.C. (2004) Bifurcation in a White-Blood-Cell Production Model. Comptes Rendus Biologies, 327, 201-210. https://doi.org/10.1016/j.crvi.2003.05.005
[6]
Pujo-Menjouet, L. and Mackey, M.C. (2003) Contribution to the Study of Periodic Chronic Myelogenous Leukemia. Comptes Rendus Biologies, 327, 235-244. https://doi.org/10.1016/j.crvi.2003.05.004
[7]
Ma, S.Q. (2021) Stability and Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model, Springer Monographs in Mathematics. International Journal of Modern Nonlinear Theory and Application, 10, 13-27. https://doi.org/10.4236/ijmnta.2021.101002
[8]
Mackey, M.C. (1978) United Hypothesis for the Origin of Aplasric Anemia and Periodic Hematopoiesis. Blood, 51, 941-956. https://doi.org/10.1182/blood.V51.5.941.bloodjournal515941
[9]
Mackey, M.C., Pujo-Menjouet, L. and Wu, J. (2006) Periodic Oscillations of Blood Cell Populations in Chronic Myelogenous Leukemia. SIAM Journal on Mathematical Analysis, 38, 166-187. https://doi.org/10.1137/04061578X
[10]
Daniel, C. and Humphries, A.R. (2017) Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model. SIAM Journal on Applied Dynamical Systems, 18, Article ID: 1165086.
[11]
Beretta, E. and Kuang, Y. (2002) Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependant Parameters. SIAM Journal on Mathematical Analysis, 33, 1144-1165. https://doi.org/10.1137/S0036141000376086
[12]
Ma, S.Q. (2019) Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays. International Journal of Bifurcation and Chaos, 29, Article ID: 1950163. https://doi.org/10.1142/S0218127419501633
[13]
Wang, Z.H. and Hu, H.Y. (2000) Stability Switches of Time Delayed Dynamic Systems with Unknown Parameters. Journal of Sound and Vibration, 233, 215-233. https://doi.org/10.1006/jsvi.1999.2817
[14]
Xu, J. and Chung, K.W. (2003) Effects of Time Delayed Position Feedback on a Van der Pol-Duffing Oscilator. Physica D: Nonlinear Phenomena, 80, 17-39. https://doi.org/10.1016/S0167-2789(03)00049-6
[15]
Hale, J.K. and Lunel, S.M.V. (1993) Introduction of Functional Differential Equations. In: Bloch, A., Charles, L., Epstein, A.G. and Greengard, L., Eds., Applied Mathematical Sciences, Vol. 99, Springer, New York, 1-10. https://doi.org/10.1007/978-1-4612-4342-7
