Suqi Ma

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Department of Mathematics, China Agricultural University, Beijing, China.
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Abstract: The observed dynamical property illustrates that state feedback control may stabilize invariant attractor to stable state in a simple version of hematopoietic stem cell model. The stability character of the positive steady state is analyzed by the computation of the rightmost characteristic roots in complex plane. Hopf bifurcation points are tracked as the roots curve crossing imaginary axis from the left half plane to the right half plane continuously. The bifurcation direction and stability of the bifurcating periodical solution are discussed by norm form computation combined with the center manifold theory. Furthermore, the numerical simulation verifies that instead of chaos, system is stabilized to period-1, 2, 3, 4 and period-7 periodical solutions in some delay windows, and the continuous of periodical solutions is also numerical simulated with varying free parameters continuously.

Keywords: State Feedback Control, Time Delay, Hopf Bifurcation

Cite this paper: Ma, S. (2021) Stability and Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model. International Journal of Modern Nonlinear Theory and Application, 10, 13-27. doi: 10.4236/ijmnta.2021.101002.

References

[1]   Smith, H. (2011) An Introduction to Delay Differential Equations with Applications to the Life Sciences. Vol. 57, Springer, New York.
https://doi.org/10.1007/978-1-4419-7646-8

[2]   Kuang, Y. (1992) Delay Differential Equations with Application in Population Dynamics. Springer, New York.

[3]   Davis, L.C. (2003) Modifications of the Optimal Velocity Traffic Model to Include Delay Due to Driver Reaction Time. Physica A: Statistical Mechanics and Its Applications, 319, 557-567. https://doi.org/10.1016/S0378-4371(02)01457-7

[4]   Beddington, J.R. and Ray, R.M. (1975) Time Delays Are Not Necessarily Destabilizing. Mathematical Biosciences, 27, 109-117.
https://doi.org/10.1016/0025-5564(75)90028-0

[5]   Bengea, S.C., Li, X.Q. and DeCarlo, R.A. (2004) Combined Controller-Observer Design for Uncertain Time Delay Systems with Application to Engine Idle Speed Control. Journal of Dynamic Systems Measurement and Control, 126, 772-780.
https://doi.org/10.1115/1.1849239

[6]   Franklin, G.F., Powell, J.D. and Emami-Naeini, A. (2005) Feedback Control of Dynamic Systems. Pearson Prentice Hall, Upper Saddle River.

[7]   Frost, M.G. (1982) Controllability, Observability and the Transfer Function Matrix for a Delay-Differential System. International Journal of Control, 35, 175-182.
https://doi.org/10.1080/00207178208922610

[8]   Belair, J. and Campbell, S.A. (1994) Stability and Bifurcations of Equilibria in a Multiple-Delayed Differential Equation. SIAM Journal on Applied Mathematics, 54, 1402-1424. https://doi.org/10.1137/S0036139993248853

[9]   Huang, K.L. and Lu, Q.S. (1995) Some Theorems for a Class of Dynamical System with Delay and Their Applications. Acta Mathematic Application Sinica, 18, 422-428. (in Chinese).

[10]   Shi, M. and Wang, Z.H. (2011) An Effective Analytical Criterion for Stability Testing of Fractional-Delay Systems. Automatica, 47, 2001-2005.
https://doi.org/10.1016/j.automatica.2011.05.018

[11]   Wang, Z.H. and Hu, H.Y. (1999) Delay-Independent Stability of Retarded Dynamic Systems of Multiple Degrees of Freedom. Journal of Sound and Vibration, 226, 57-81. https://doi.org/10.1006/jsvi.1999.2282

[12]   Beretta, E. and Kuang, Y. (2002) Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters. SIAM Journal on Mathematical Analysis, 33, 1144-1165.
https://doi.org/10.1137/S0036141000376086

[13]   Cooke, K.L. and van den Driesche, P. (1986) On Zeros of Some Transcendental Equaions. Funkcialaj Ekvacioj, 29, 77-90.

[14]   Ma, S.Q., Lu, Q.S. and Mei, S. L. (2005) Dynamics of a Logistic Population Model with Maturation Delay and Nonlinear Birth Rate. Discrete and Continuous Dynamical Systems-B, 5, 735-752. http://dx.doi.org/10.3934/dcdsb.2005.5.735

[15]   Aiello, W.G., Freedman, H.I. and Wu, J. (1992) A Model of Stage Structural Population Growth with Density Dependent Time Delay. SIAM Journal on Applied Mathematics, 52, 855-869. https://doi.org/10.1137/0152048

[16]   Beretta, E. and Kuang, Y. (2001) Modeling and Analysis of a Marine Bacteriophage Infection with Latency Period. Nonlinear Analysis: Real World Applications, 2, 35-74. https://doi.org/10.1016/S0362-546X(99)00285-0

[17]   Ma, S.Q., Feng, Z.S. and Lu, Q.S. (2008) A Two Parameter Criteria for Delay Differential Equations. Discrete & Continuous Dynamical Systems-B, 9, 397-413.
http://dx.doi.org/10.3934/dcdsb.2008.9.397

[18]   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

[19]   Xu, J. and Chung, K.W. (2003) Effects of Time Delayed Position Feedback on a Van Der Pol-Duffing Oscillator. Physica D: Nonlinear Phenomena, 180, 17-39.
https://doi.org/10.1016/S0167-2789(03)00049-6

[20]   Wang, Z.H., Hu, H.Y., Xu, Q. and Stepan, G. (2016) Effect of Delay Combinations on Stability and Hopf Bifurcation of an Oscillator with Acceleration-Derivative Feedback. International Journal of Nonlinear Mechanics, 94, 392-399.
https://doi.org/10.1016/j.ijnonlinmec.2016.10.008

[21]   Hale, J.K. and Lunel, S.M.V. (1993) Introduction of Functional Differential Equations. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4612-4342-7