Conjugation of cascades

Jesús San Martín and Daniel Rodríguez-Pérez

Chaos, Solitons & Fractals, 2009, vol. 39, issue 2, 666-681

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.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:39:y:2009:i:2:p:666-681

DOI: 10.1016/j.chaos.2007.01.073

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