Algebraic Analysis of Bifurcations and Chaos for Discrete Dynamical Systems

  • Bo Huang
  • Wei NiuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11989)


This paper deals with the stability, bifurcations and chaotic behaviors of discrete dynamical systems by using methods of symbolic computation. We explain how to reduce the problems of analyzing the stability, bifurcations and chaos induced by snapback repellers to algebraic problems, and solve them by using an algorithmic approach based on methods for solving semi-algebraic systems. The feasibility of the symbolic approach is demonstrated by analyses of the dynamical behaviors for several discrete models.


Bifurcations Chaos Discrete systems Symbolic computation Snapback repeller 


  1. 1.
    Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer, Berlin (1996). Scholar
  2. 2.
    Aboites, V., Wilson, M., Bosque, L., del Campestre, L.: Tinkerbell chaos in a ring phase-conjugated resonator. Int. J. Pure Appl. Math. 54(3), 429–435 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Multidimensional Systems Theory, pp. 184–232. Reidel, Dordrecht (1985)Google Scholar
  4. 4.
    Bistritz, Y.: Zero location with respect to the unit circle of directe-time linear system polynomials. Proc. IEEE 72(9), 1131–1142 (1984)CrossRefGoogle Scholar
  5. 5.
    Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dickenstein, A., Millán, M., Shiu, A., Tang, X.: Multistationarity in structured reaction networks. Bull. Math. Biol. 81(5), 1527–1581 (2019)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davidchack, R.L., Lai, Y.C., Klebanoff, A., Bollt, E.M.: Towards complete detection of unstable periodic orbits in chaotic systems. Phys. Lett. A 287(1–2), 99–104 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Coelho, L.S., Mariani, V.C.: Firefly algorithm approach based on chaotic Tinkerbell map applied to multivariable PID controller tuning. Comput. Math. Appl. 64(8), 2371–2382 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases \((F4)\). J. Pure Appl. Algebra. 139(1–3), 61–88 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Galor, O.: Discrete Dynamical Systems. Springer, Berlin (2007). Scholar
  11. 11.
    Glendinning, P.: Bifurcations of snap-back repellers with application to border-collision bifurcations. Int. J. Bifurcat. Chaos 20(2), 479–489 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 69–76 (1976)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. J. Symb. Comput. 24(2), 161–187 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hong, H., Tang, X., Xia, B.: Special algorithm for stability analysis of multistable biological regulatory systems. J. Symb. Comput. 70(1), 112–135 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Huang, B., Niu, W.: Analysis of snapback repellers using methods of symbolic computation. Int. J. Bifurcat. Chaos 29(4), 1950054-1-13 (2019)Google Scholar
  16. 16.
    Kitajima, H., Kawakami, H., Mira, C.: A method to calculate basin bifurcation sets for a two-dimensional nonivertible map. Int. J. Bifurcat. Chaos 10(8), 2001–2014 (2000)CrossRefGoogle Scholar
  17. 17.
    Kaslik, E., Balint, S.: Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture. Neural Networks 22(10), 1411–1418 (2009)CrossRefGoogle Scholar
  18. 18.
    Li, C., Chen, G.: An improved version of the Marotto theorem. Chaos Solit. Fract. 18(1), 69–77 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lazard, D., Rouillier, F.: Solving parametric polynomial systems. J. Symb. Comput. 42(6), 636–667 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, X., Mou, C., Niu, W., Wang, D.: Stability analysis for discrete biological models using algebraic methods. Math. Comput. Sci. 5(3), 247–262 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mira, C., Barugola, A., Gardini, L.: Chaotic Dynamics in Two-Dimensional Nonvertible Map. World Scientific, Singapore (1996)CrossRefGoogle Scholar
  22. 22.
    Marotto, F.: Snap-back repellers imply chaos in \(\mathbb{R}^n\). J. Math. Anal. Appl. 63(1), 199–223 (1978)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Marotto, F.: On redefining a snap-back repeller. Chaos Solit. Fract. 25(1), 25–28 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Niu, W., Shi, J., Mou, C.: Analysis of codimension 2 bifurcations for high-dimensional discrete systems using symbolic computation methods. Appl. Math. Comput. 273, 934–947 (2016)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Stoyanov, B., Kordov, K.: Novel secure pseudo-random number generation scheme based on two Tinkerbell maps. Adv. Stud. Theor. Phys. 9(9), 411–421 (2015)CrossRefGoogle Scholar
  26. 26.
    Sang, B., Huang, B.: Bautin bifurcations of a financial system. Electron. J. Qual. Theory Differ. Equ. 2017(95), 1–22 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wu, W.-T.: Mathematics Mechanization. Science Press/Kluwer Academic, Beijing (2000)Google Scholar
  28. 28.
    Wang, D.: Elimination Methods. Springer, New York (2001). Scholar
  29. 29.
    Wen, G.: Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys. Rev. E. 72(2), 026201-1-4 (2005)Google Scholar
  30. 30.
    Yang, L., Xia, B.: Real solution classifications of parametric semi-algebraic systems. In: Algorithmic Algebra and Logic-Proceedings of the A3L, pp. 281–289. Herstel-lung und Verlag, Norderstedt (2005)Google Scholar

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Authors and Affiliations

  1. 1.LMIB-School of Mathematical SciencesBeihang UniversityBeijingChina
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Ecole Centrale de Pékin, Beihang UniversityBeijingChina
  4. 4.Beijing Advanced Innovation Center for Big Data and Brain ComputingBeihang UniversityBeijingChina

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