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Conjugation of cascades

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  • San Martín, Jesús
  • Rodríguez-Pérez, Daniel

Abstract

Presented in this work are some results relative to sequences found in the logistic equation bifurcation diagram, which is the unimodal quadratic map prototype. All of the different saddle-node bifurcation cascades, associated with every last appearance p-periodic orbit (p=3,4,5,…), can also be generated from the very Feigenbaum cascade. In this way it is evidenced the relationship between both cascades. The orbits of every saddle-node bifurcation cascade, mentioned above, are located in different chaotic bands, and this determines a sequence of orbits converging to every band-merging Misiurewicz point. In turn, these accumulation points form a sequence whose accumulation point is the Myrberg–Feigenbaum point. It is also proven that the first appearance orbits in the n-chaotic band converge to the same point as the last appearance orbits of the (n+1)-chaotic band. The symbolic sequences of band-merging Misiurewicz points are computed for any window.

Suggested Citation

  • San Martín, Jesús & Rodríguez-Pérez, Daniel, 2009. "Conjugation of cascades," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 666-681.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:2:p:666-681
    DOI: 10.1016/j.chaos.2007.01.073
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    References listed on IDEAS

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    1. Pastor, G. & Romera, M. & Montoya, F., 1996. "On the calculation of Misiurewicz patterns in one-dimensional quadratic maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 232(1), pages 536-553.
    2. Post, Thijs & Capel, Hans W., 1991. "Windows in one-dimensional maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 178(1), pages 62-100.
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    Cited by:

    1. Herrera, Ma Dolores Sotelo & Martín, Jesús San, 2009. "An analytical study in coupled map lattices of synchronized states and traveling waves, and of their period-doubling cascades," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 901-910.

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