• Previous Article
    Dynamics of random selfmaps of surfaces with boundary
  • DCDS Home
  • This Issue
  • Next Article
    Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces
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 and Continuous Dynamical Systems, 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, Series A: Monographs and Treatises, 56, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.

[2]

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.

[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.

[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), 537-567. doi: 10.3934/dcdsb.2008.10.537.

[5]

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

[6]

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

[7]

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.

[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), 1350072, 11 pp. doi: 10.1142/S0218127413500727.

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, Series A: Monographs and Treatises, 56, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.

[2]

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.

[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.

[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), 537-567. doi: 10.3934/dcdsb.2008.10.537.

[5]

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

[6]

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

[7]

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.

[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), 1350072, 11 pp. doi: 10.1142/S0218127413500727.

[1]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139

[2]

Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249

[3]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133

[4]

Jordi-Lluís Figueras, Àlex Haro. A note on the fractalization of saddle invariant curves in quasiperiodic systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1095-1107. doi: 10.3934/dcdss.2016043

[5]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006

[6]

Vladimir Sobolev. Canard cascades. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 513-521. doi: 10.3934/dcdsb.2013.18.513

[7]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1

[8]

Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755

[9]

Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497

[10]

Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102

[11]

Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063

[12]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098

[13]

Àngel Jorba, Joan Carles Tatjer. A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 537-567. doi: 10.3934/dcdsb.2008.10.537

[14]

Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure and Applied Analysis, 2021, 20 (2) : 559-582. doi: 10.3934/cpaa.2020281

[15]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839

[16]

Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589

[17]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061

[18]

Yu-Hao Liang, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1075-1105. doi: 10.3934/dcds.2020071

[19]

Ananta Acharya, R. Shivaji, Nalin Fonseka. $ \Sigma $-shaped bifurcation curves for classes of elliptic systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022067

[20]

Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure and Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]