Codimension-Three Flip-Hopf-Hopf Bifurcation in a General Discrete Time System

• Biliu Zhou ^[3] ; Huidong Xu ^[1] ; Wei Zhang ^[2]
  1. [1] Taiyuan University of Technology

     Taiyuan University of Technology

     China

     
  2. [2] Guangxi University

     Guangxi University

     China

     
  3. [3] Guangxi University, Suzhou University of Technology

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 2, 2026
• Idioma: inglés
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• Resumen

  • An explicit critical criterion for Flip-Hopf-Hopf bifurcation of discrete systems is proposed for detecting the existence of the codimension-three bifurcation. The explicit critical criteria including eigenvalue assignment, transversality condition and nonresonance condition are established based on the properties of the coefficients of characteristic polynomial equation, which is not involved with the computation of eigenvalues of the linearization matrix. The equivalence between the proposed criterion and the corresponding classical criterion is proved. The map is reduced to a five-dimensional map by the central manifold method. The expressions for the coefficients of the normal form are derived in detail to obtain the normal for corresponding to the reduced five-dimensional map. A three-degree-of-freedom vibro-impact system is taken as an example to show the effectiveness of the proposed explicit criterion and the local dynamical behaviors near bifurcation point.

• Referencias bibliográficas
  • 1. Hamzi, B.,Krener, A.J., Kang,W.: The controlled center dynamics of discrete time control bifurcations. Syst. Control Lett. 55, 585–596...
  • 2. di Bernardoa, M., Buddb, C.J., Champneys, A.R.: Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems....
  • 3. Gazor, M., Kazemi, M.: Z2-equivariant standard bases for submodules associated with Z2-equivariant singularities. ACM Commun. Comput. Algebra...
  • 4. Záthurecký, J.: A path inside Arnold tongue arising from Neimark-Sacker bifurcation. Commun. Nonlinear Sci. Numer. Simulat. 152, 109186...
  • 5. Kamiyama, K., Inaba, N., Sekikawa, M., Endo, T.: Bifurcation boundaries of three-frequency quasiperiodic oscillations in discrete-time...
  • 6. Luo, G.W., Yu, J.N., Xie, J.H.: Codimension-two bifurcation and chaos of a vibro-impact forming machine associated with 1:2 resonance case....
  • 7. Ding,W.C., Xie, J.H.: Torus T2 and its routes to chaos of a vibro-impact system. Phys. Lett. A 349(5), 324–330 (2006)
  • 8. Yue, Y., Xie, J.H.: Neimark-Sacker-pitchfork bifurcation of the symmetric period fixed point of the Poincaré map in a three-degree-of-freedom...
  • 9. He, X., Li, C.D., Shu, Y.L.: Fold-flip bifurcation analysis on a class of discrete-time neural network. Neural Comput. Appl. 22(2), 375–381...
  • 10. Wen, G.L.,Wang, Q.G., Chiu,M.S.: Delay feedback control for interaction ofHopf and period doubling bifurcations in discrete-time systems....
  • 11. Wang, D., Zheng, M., Peng, J.P.: Further reduction of normal forms of formal maps. C. R. Math. 343, 657–660 (2006)
  • 12. Wang, D., Zheng, M., Peng, J.P.: Further reduction of normal forms and unique normal forms of smooth maps. Int. J. Bifurcation Chaos 18,...
  • 13. Mokhtari, F., Roell, E., Sanders, J.A.: Normal form for maps with nilpotent linear part. R. Soc. Lond., A, Math. Phys. Eng. Sci Proc 478(2261),...
  • 14. Pryce, J.D., Ghaziani, R.K., De Witte, V., Govaerts, W.: Computation of normal form coefficients of cycle bifurcations of maps by algorithmic...
  • 15. Gazor, M., Kazemi, M.: Singularity: a Maple library for local zero bifurcation control of scalar smooth maps. Trans. Am. Soc.Mech. Eng...
  • 16. Govaerts,W., Ghaziani, R.K., Kuznetsov, Y.A., Meijer, H.G.E.: Numerical methods for two-parameter local bifurcation analysis of maps....
  • 17. Gimenoa, J., Jorba, À., Jorba-Cuscób, M., Zou, M.R.: Explicit numerical computation of normal forms for Poincaré maps. Commun. Nonlinear...
  • 18. Xie, J.H., Ding,W.C., Dowell, E.H., Virgin, L.N.: Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems....
  • 19. Wen, G.L., Yin, S., Xu, H.D., Zhang, S.J., Lv, Z.Y.: Parameterized center manifold for unfolding bifurcations with an eigenvalue +1...
  • 20. Lentjesa, B., Spek, L., Bosschaertc, M.M., Kuznetsov, Y.A.: Periodic normal forms for bifurcations of limit cycles in DDEs. J. Differ....
  • 21. Wen, G.L.: Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys. Rev. E 72, 026201 (2005)
  • 22. Wen, G.L., Chen, S.J., Jin, Q.T.: A new criterion of period-doubling bifurcation in maps and its application to an inertial impact shaker....
  • 23. Yao, S.J.: New bifurcation critical criterion of flip-neimark-sacker bifurcations for two parameterized family of n-dimensional discrete...
  • 24. Xu, H.D., Wen, G.L.: Alternative criterion for investigation of pitchfork bifurcations of limit cycle in relay feedback systems. J. Comput....
  • 25. Niu, W., Shi, J., Mou, C.Q.: Analysis of codimension 2 bifurcations for high-dimensional discrete systems using symbolic computation methods....
  • 26. Xu, H.D., Wen, G.L., Qin, Q.X., Zhou, H.A.: New explicit critical criterion of Hopf-Hopf bifurcation in a general discrete time system....
  • 27. Brown, B.M.: On the distribution of the zeros of a polynomial. Q. J. Math. 16, 241–256 (1965)
  • 28. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer-Verlag, New York (2004)
  • 29. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of the Hopf Bifurcation. Cambridge University Press, London (1981)
  • 30. Iooss, G.: Some codimension-two bifurcations formaps, leading to chaos. In:Kuramoto,Y. (ed.) Chaos and Statistical Methods, pp. 136–142....
  • 31. Iooss, G.: Persistance, d’un cercle invariant par une application voisine de"l’application temps "d’un champ de vecteurs intergrable....
  • 32. Los, J.E.: Non-normally hyperbolic invariant curves for maps in R3 and doubling bifurcation. Nonlinearity 2, 149–174 (1989)