Have a personal or library account? Click to login
Stability Analysis of Nonlinear Time–Delayed Systems with Application to Biological Models Cover

Stability Analysis of Nonlinear Time–Delayed Systems with Application to Biological Models

Open Access
|May 2017

References

  1. Aluru, S. (2005). Handbook of Computational Molecular Biology, CRC Press, Boca Raton, FL.10.1201/9781420036275
  2. Andrew, S.M., Baker, C.T. and Bocharov, G.A. (2007). Rival approaches to mathematical modelling in immunology, Journal of Computational and Applied Mathematics205(2): 669–686.10.1016/j.cam.2006.03.035
  3. Babbs, C.F. (2011). Predicting success or failure of immunotherapy for cancer: Insights from a clinically applicable mathematical model, American Journal of Cancer Research2(2): 204–213.
  4. Banerjee, S. (2008). Immunotherapy with interleukin-2: A study based on mathematical modeling, International Journal of Applied Mathematics and Computer Science18(3): 389–398, DOI: 10.2478/v10006-008-0035-6.10.2478/v10006-008-0035-6
  5. Bell, G.I. (1973). Predator–prey equations simulating an immune response, Mathematical Biosciences16(3): 291–314.10.1016/0025-5564(73)90036-9
  6. Bernot, G., Comet, J.-P., Richard, A., Chaves, M., Gouzé, J.-L. and Dayan, F. (2013). Modeling and analysis of gene regulatory networks, in F. Cazals and P. Kornprobst (Eds.), Modeling in Computational Biology and Biomedicine: A Multidisciplinary Endeavor, Springer, Berlin/Heidelberg, pp. 47–80.10.1007/978-3-642-31208-3_2
  7. Bo, W., Yang, L. and Jianquan, L. (2012). New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays, Mathematical and Computer Modelling55(3): 837–843.10.1016/j.mcm.2011.09.009
  8. Bodnar, M. (2015). General model of a cascade of reactions with time delays: Global stability analysis, Journal of Differential Equations259(2): 777–795.10.1016/j.jde.2015.02.024
  9. Chen, L. and Aihara, K. (2002). Stability of genetic regulatory networks with time delay, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications49(5): 602–608.10.1109/TCSI.2002.1001949
  10. De Jong, H. (2002). Modeling and simulation of genetic regulatory systems: A literature review, Journal of Computational Biology9(1): 67–103.10.1089/1066527025283320811911796
  11. d’Onofrio, A. (2005). A general framework for modeling tumor–immune-system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenomena208(3): 220–235.10.1016/j.physd.2005.06.032
  12. d’Onofrio, A. (2008). Metamodeling tumor–immune-system interaction, tumor evasion and immunotherapy, Mathematical and Computer Modelling47(5): 614–637.10.1016/j.mcm.2007.02.032
  13. d’Onofrio, A., Gatti, F., Cerrai, P. and Freschi, L. (2010). Delay-induced oscillatory dynamics of tumour–immune system interaction, Mathematical and Computer Modelling51(5): 572–591.10.1016/j.mcm.2009.11.005
  14. Eduardo, L. and Ruiz-Herrera, A. (2013). Attractivity, multistability, and bifurcation in delayed Hopfield’s model with non-monotonic feedback, Journal of Differential Equations255(11): 4244–4266.10.1016/j.jde.2013.08.007
  15. Goodwin, B.C. (1963). Temporal Organization in Cells: A Dynamic Theory of Cellular Control Processes, Academic Press, London/New York, NY.10.5962/bhl.title.6268
  16. Gu, K., Chen, J. and Kharitonov, V.L. (2003). Stability of Time-Delay Systems, Springer, New York, NY.10.1007/978-1-4612-0039-0
  17. Kao, C.-Y. and Pasumarthy, R. (2012). Stability analysis of interconnected Hamiltonian systems under time delays, IET Control Theory and Applications6(4): 570–577.10.1049/iet-cta.2011.0076
  18. Kao, C.-Y. and Rantzer, A. (2007). Stability analysis of systems with uncertain time-varying delays, Automatica43(6): 959–970.10.1016/j.automatica.2006.12.006
  19. Kauffman, S.A. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology22(3): 437–467.10.1016/0022-5193(69)90015-0
  20. Kolmanovskii, V. and Myshkis, A. (1999). Introduction to the Theory and Applications of Functional Differential Equations, Springer, Dordrecht.10.1007/978-94-017-1965-0
  21. Liu, Y., Xu, P., Lu, J. and Liang, J. (2016a). Global stability of Clifford-valued recurrent neural networks with time delays, Nonlinear Dynamics84(2): 767–777.10.1007/s11071-015-2526-y
  22. Liu, Y., Zhang, D., Lu, J. and Cao, J. (2016b). Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays, Information Sciences360: 273–288.10.1016/j.ins.2016.04.033
  23. Loiseau, J.J., Michiels, W., Niculescu, S.-I. and Sipahi, R. (2009). Topics in Time Delay Systems: Analysis, Algorithms and Control, Springer, Berlin/Heidelberg.10.1007/978-3-642-02897-7
  24. Mazenc, F. and Niculescu, S.-I. (2001). Lyapunov stability analysis for nonlinear delay systems, Systems & Control Letters42(4): 245–251.10.1016/S0167-6911(00)00093-1
  25. Melief, C.J. (2005). Cancer immunology: Cat and mouse games, Nature437(7055): 41–42.10.1038/437041a
  26. Papachristodoulou, A. (2004). Analysis of nonlinear time-delay systems using the sum of squares decomposition, American Control Conference, Boston, MA, USA, pp. 4153–4158.
  27. Papachristodoulou, A. and Prajna, S. (2005). A tutorial on sum of squares techniques for systems analysis, American Control Conference, Portland, OR, USA, pp. 2686–2700.
  28. Parrilo, P.A. (2000). Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD thesis, California Institute of Technology, Pasadena, CA.
  29. Pasumarthy, R. and Kao, C.-Y. (2009). On stability of time delay Hamiltonian systems, American Control Conference, St. Louis, MO, USA, pp. 4909–4914.
  30. Richard, J.-P. (2003). Time-delay systems: An overview of some recent advances and open problems, Automatica39(10): 1667–1694.10.1016/S0005-1098(03)00167-5
  31. Saleem, M. and Agrawal, T. (2012). Complex dynamics in a mathematical model of tumor growth with time delays in the cell proliferation, International Journal of Scientific and Research Publications2(6): 1–7.
  32. Sharma, A., Kohar, V., Shrimali, M. and Sinha, S. (2014). Realizing logic gates with time-delayed synthetic genetic networks, Nonlinear Dynamics76(1): 431–439.10.1007/s11071-013-1136-9
DOI: https://doi.org/10.1515/amcs-2017-0007 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Language: English
Page range: 91 - 103
Submitted on: Apr 5, 2016
Accepted on: Oct 12, 2016
Published on: May 4, 2017
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year

© 2017 H.A. Kruthika, Arun D. Mahindrakar, Ramkrishna Pasumarthy, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.