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Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss

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  • Let $g_{\alpha}$ be a one-parameter family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, a superstable periodic orbit is known to exist. An example of such a family is the well-known logistic map. In this paper we deal with the effect of a quasi-periodic perturbation (with only one frequency) on this cascade. Let us call $\varepsilon$ the perturbing parameter. It is known that, if $\varepsilon$ is small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on $\alpha$ and $\varepsilon$) of the perturbed system. In this article we focus on the reducibility of these invariant curves.
        The paper shows that, under generic conditions, there are both reducible and non-reducible invariant curves depending on the values of $\alpha$ and $\varepsilon$. The curves in the space $(\alpha,\varepsilon)$ separating the reducible (or the non-reducible) regions are called reducibility loss bifurcation curves. If the map satifies an extra condition (condition satisfied by the quasi-periodically forced logistic map) then we show that, from each superattracting point of the unperturbed map, two reducibility loss bifurcation curves are born. This means that these curves are present for all the cascade.
    Mathematics Subject Classification: Primary: 37C55; Secondary: 37E99, 37G35.


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  • [1]

    U. Feudel, S. Kuznetsov and A. Pikovsky, "Strange Nonchaotic Attractors. Dynamics Between Order and Chaos in Quasiperiodically Forced Systems," World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 56, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.


    C. Grebogi, E. Ott, S. Pelikan and J. A. Yorke, Strange attractors that are not chaotic, Phys. D, 13 (1984), 261-268.doi: 10.1016/0167-2789(84)90282-3.


    À. Jorba, P. Rabassa and J. C. Tatjer, Period doubling and reducibility in the quasi-periodically forced logistic map, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1507-1535.doi: 10.3934/dcdsb.2012.17.1507.


    À. Jorba and J. C. Tatjer, A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567.doi: 10.3934/dcdsb.2008.10.537.


    K. Kaneko, Doubling of torus, Progr. Theoret. Phys., 69 (1983), 1806-1810.doi: 10.1143/PTP.69.1806.


    K. Kaneko, Oscillation and doubling of torus, Progr. Theoret. Phys., 72 (1984), 202-215.doi: 10.1143/PTP.72.202.


    A. Prasad, S. S. Negi and R. Ramaswamy, Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 291-309.doi: 10.1142/S0218127401002195.


    P. Rabassa, À. Jorba and J. C. Tatjer, A numerical study of universality and self-similarity in some families of forced logistic maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350072, 11 pp.doi: 10.1142/S0218127413500727.

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