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February  2014, 34(2): 589-597. doi: 10.3934/dcds.2014.34.589

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

1. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585 , 08007 Barcelona

2. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona,, Spain

3. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08080 Barcelona, Spain

Received  February 2013 Revised  April 2013 Published  August 2013

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.
Citation: Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 589-597. doi: 10.3934/dcds.2014.34.589
References:
[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, 56 (2006). Google Scholar

[2]

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

[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. doi: 10.3934/dcdsb.2012.17.1507. Google Scholar

[4]

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. doi: 10.3934/dcdsb.2008.10.537. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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). doi: 10.1142/S0218127413500727. Google Scholar

show all references

References:
[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, 56 (2006). Google Scholar

[2]

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

[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. doi: 10.3934/dcdsb.2012.17.1507. Google Scholar

[4]

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. doi: 10.3934/dcdsb.2008.10.537. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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). doi: 10.1142/S0218127413500727. Google Scholar

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