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Bogdanov–Takens and Hopf Bifurcations Analysis of a Genetic Regulatory Network

  • Ming Liu [1] ; Fanwei Meng [1] ; Dongpo Hu [1]
    1. [1] Qufu Normal University

      Qufu Normal University

      China

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 2, 2022
  • Idioma: inglés
  • Enlaces
  • Resumen

    • The stability and two kinds of bifurcations of a genetic regulatory network are considered in this paper. We have given a complete stability analysis involved in mentioned model. The Hopf bifurcation of codimension 1 and Bogdanov–Takens bifurcation of codimension 2 for the nonhyperbolic equilibria of the model is characterized analytically. In order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Three bifurcation curves related to Bogdanov–Takens bifurcation, namely saddle-node, Hopf and homoclinic bifurcation curves, are given explicitly by calculating a universal unfolding near the cusp. Moreover, the numerical continuation results show that the model has other bifurcation types, including saddle-node and cusp bifurcations. The bifurcation diagram and phase portraits are also given to verify the validity of the theoretical results. These results show that there exists rich bifurcation behavior in the genetic regulatory network.

  • Referencias bibliográficas
    • 1. Xue, Y., Zhang, L.N., Zhang, X.: Reachable set estimation for genetic regulatory networks with timevarying delays and bounded disturbances....
    • 2. Liu, C.Y., Wang, X., Xue, Y.: Global exponential stability analysis of discrete-time genetic regulatory networks with time-varying discrete...
    • 3. Gnanakkumaar, P., Murugesan, R., Ahmed, S.: Gene regulatory networks in peripheral mononuclear cells reveals critical regulatory modules...
    • 4. Edwards, D.R.: Cell signaling and the control of gene transcription. Trends Pharmacol. Sci. 15, 239–244 (1994)
    • 5. Pandiselvi, S., Raja, R., Cao, J.D., Rajchakit, G.: Stabilization of switched stochastic genetic regulatory networks with leakage and impulsive...
    • 6. Ghosh, A., Greenberg, M.: Calcium signaling in neurons: molecular mechanisms and cellular consequences. Science 268, 239–247 (1995)
    • 7. Shen, H., Huo, S.C., Yan, H.C., Park, J.H., Sreeram, V.: Distributed dissipative state estimation for Markov jump genetic regulatory networks...
    • 8. Zhang, X., Wu, L.G., Cui, S.C.: An improved integral inequality to stability analysis of genetic regulatory networks with interval time-varying...
    • 9. Sun, Q.S., Xiao, M., Tao, B.B.: Local bifurcation analysis of a fractional-order dynamic model of genetic regulatory networks with delays....
    • 10. Liu, H.H., Yan, F., Liu, Z.R.: Oscillatory dynamics in a gene regulatory network mediated by small RNA with time delay. Nonlinear Dyn....
    • 11. Mussel, C., Hopfensitz, M., Kestler, H.A.: BoolNet-an R package for generation, reconstruction and analysis of Boolean networks. Bioinformatics...
    • 12. Liu, F., Zhang, S.W., Guo, W.F., Wei, Z.G., Chen, L.N.: Inference of gene regulatory network based on local Bayesian networks. PLOS Comput....
    • 13. Krämer, N., Schäfer, J., Boulesteix, A.L.: Regularized estimation of large-scale gene association networks using graphical Gaussian models....
    • 14. Pan, W., Wang, Z.D., Gao, H.J., Liu, X.H.: Monostability and multistability of genetic regulatory networks with different types of regulation...
    • 15. Braniff, N., Richards, A., Ingalls, B.: Optimal experimental design for a bistable gene regulatory network. IFAC-PapersOnLine 52, 255–261...
    • 16. Duan, L., Di, F.J., Wang, Z.Y.: Existence and global exponential stability of almost periodic solutions of genetic regulatory networks...
    • 17. Zang, H., Zhang, T.H., Zhang, Y.D.: Bifurcation analysis of a mathematical model for genetic regulatory network with time delays. Appl....
    • 18. Poignard, C.: Inducing chaos in a gene regulatory network by coupling an oscillating dynamics with a hysteresis-type one. J. Math. Biol....
    • 19. Khlebodarova, T.M., Kogai, V.V., Fadeev, S.I., Likhoshvai, V.A.: Chaos and hyperchaos in simple gene network with negative feedback and...
    • 20. Lai, Q., Chen, Q.: Stability and bifurcation of delayed gene regulatory network with self-feedback, positive feedback and negative feedback....
    • 21. Takens, F.: Singularities of vector fields. Publ. Math. IHES. 43, 47–100 (1974)
    • 22. Bogdanov, R.I.: Versal deformations of a singular point on the plane in the case of zero eigenvalues. Funct. Anal. Appl. 9, 144–145 (1975)
    • 23. Steindl, A.: Numerical investigation of the Hopf-Bogdanov-Takens mode interaction for a fluidconveying tube. Procedia Eng. 199, 857–862...
    • 24. Algaba, A., Domínguez-Moreno, M.C., Merino, M., Rodríguez-Luis, A.J.: Takens-Bogdanov bifurcations of equilibria and periodic orbits in...
    • 25. de Blank, H.J., Kuznetsov, Y.A., Pekkér, M.J., Veldman, D.W.M.: Degenerate Bogdanov-Takens bifurcations in a one-dimensional transport...
    • 26. Yu, P., Zhang, W.J.: Complex dynamics in a unified SIR and HIV disease model: a bifurcation theory approach. J. Nonlinear Sci. 29, 2447–2500...
    • 27. Lu, M., Huang, J.C., Ruan, S.G., Yu, P.: Bifurcation analysis of an SIRS epidemic model with ageneralized nonmonotone and saturated incidence...
    • 28. Hu, D.P., Cao, H.J.: Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting. Nonlinear...
    • 29. Titz, S., Kuhlbrodt, T., Feudel, U.: Homoclinic bifurcation in an ocean circulation box model. Int. J. Bifurcat. Chaos 12, 869–875 (2002)
    • 30. Lenz, E., Pagano, D.J., Tahim, A.P.N.: Codimension-two bifurcation analysis in DC microgrids under droop control. Int. J. Bifurcat. Chaos...
    • 31. Marwan, M., Ahmad, S., Aqeel, M., Sabir, M.: Control analysis of rucklidge chaotic system. J. Dyn. Sys. Meas. Control 141, 041010 (2019)
    • 32. Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)
    • 33. Novák, B., Tyson, J.J.: Design principles of biochemical oscillators. Nat. Rev. Mol. Cell Biol. 9, 981–991 (2008)
    • 34. Liu, M., Meng, F.W., Hu, D.P.: Impacts of multiple time delays on a gene regulatory network mediated by small noncoding RNA. Int. J. Bifurcat....
    • 35. Sun, Q.S., Xiao, M., Tao, B.B.: Local bifurcation analysis of a fractional-order dynamic model of genetic regulatory networks with delays....
    • 36. Yue, D.D., Guan, Z.H., Li, J., Liu, F., Xiao, J.W., Ling, G.: Stability and bifurcation of delay-coupled genetic regulatory networks with...
    • 37. Wang, G.Y., Yang, Z.Q., Turcotte, M.: Dynamic analysis of the time-delayed genetic regulatory network between two auto-regulated and mutually...
    • 38. Smolen, P., Baxter, D.A., Byrne, J.H.: Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory...
    • 39. Wan, A.Y., Zou, X.F.: Hopf bifurcation analysis for a model of genetic regulatory system with delay. J. Math. Anal. Appl. 356, 464–476...
    • 40. Yu, T.T., Zhang, X., Zhang, G.D., Niu, B.: Hopf bifurcation analysis for genetic regulatory networks with two delays. Neurocomputing 164,...
    • 41. Cheng, X.J., Wang, H., Wang, X., Duan, J.Q., Li, X.F.: Most probable transition pathways and maximal likely trajectories in a genetic...
    • 42. Wang, H., Cheng, X.J., Duan, J.Q., Kurths, J., Li, X.F.: Likelihood for transcriptions in a genetic regulatory system under asymmetric...
    • 43. Liu, F., Ren, J., Ling, G., Wei, L.S., Zheng, S.Q., Wang, H.: Stability analysis and bifurcation control of a delayed fractional order...
    • 44. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in applied mathematics, vol. 2. Springer, New York (2003)
    • 45. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd edn. CRC Press (2018)
    • 46. Kuznetsov, Y.A.: Elements of applied bifurcation theory. In: Texts in Applied Mathematical Sciences, vol. 112, 2nd edn. Springer, New...
    • 47. Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MatCont: a MATLAB package for numerical bifurcation analysis of ODEs. ACM TOMS 29, 141–164...
    • 48. Zhu, Q.H., Shen, J.W., Han, F., Lu, W.L.: Bifurcation analysis and probabilistic energy landscapes of two-component genetic network. IEEE...
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