Bifurcations and Invariant Sets in a Class of Two-Dimensional Endomorphisms

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Djellit Ilham
Fakroune Yamina
Selmani Wissame


Several endomorphisms of the plane have been constructed by simple maps. We study the dynamics occuring in one of them, which is rich in global bifurcations. The invariants sets are stable manifolds of saddle type points or cycles, as well as closed curves issued from Hopf bifurcations. The present paper focuses some bifurcations related with attractors or basins which produce other attractors which coexist with invariant sets.

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  1. G. S. Agarwal, Existence of multistability in systems with complex order parameters, Phys. Rev. A, 26 (1982), 888-891.
  2. F. T. Arecchi, R. Meucci, G. Puccioni and and J. Tredicce, Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a q-switched gas laser, Phys. Rev. Lett., 49 (1982), 1217-1220.
  3. R. L. Clerc and C. Hartmann, Invariant manifolds of separable discrete dynamic systems, Dynamics Days, La Jolla, California, (1982).
  4. I. Djellit and Y. Soula, On riddled sets and bifurcations of chaotic attractors, Appl. Math. Sci., 1 (13) (2007), 603-614.
  5. M. R. Ferchichi, I. Djellit and J. C. Sprott, Broken symmetry in modified Lorenz model, Int. J. Dyn. Syst. Differential Equations, 5 (2) (2015), 136-148.
  6. U. Feudel and C. Grebogi, Multistability and the control of complexity, Chaos, 7 (1997), 597-604.
  7. C. Grebogi, E. Kostelich, E. Ott and J. A. Yorke, Multi-dimensioned intertwined basin boundaries: Basin structure of the kicked double rotor, Physica D, 25 (1987), 347-360.
  8. I. Gumowski, C. Mira, Dynamique chaotique, Ed. Cepadues, Toulouse, (1980).
  9. M. Henon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50, (1976), 69-77.
  10. K. Ikeda, Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. Commun., 30 (1976), 257-261.
  11. H. W. Lorenz, Multiple attractors, complex basin boundaries, and transient motion in deterministic economic systems, in Dynamic Economic Models and Optimal Control, ed. Feichtinger, G. (Elsevier), (1992).
  12. C. Mira, Chaotic dynamics, World Scientific, (1987).
  13. C. Mira, L. Gardini, A. Barugola and J. C. Cathala, Chaotic dynamics in two-dimensional non invertible maps, World Scientific, (1996).
  14. B. Razafimandimby, Domaine dinfluence de certaines singularit ´es stables dun endomophisme de R2, These de 3eme cycle, Univ. Paul Sabatier, Toulouse, (1981).