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July  2012, 17(5): 1507-1535. doi: 10.3934/dcdsb.2012.17.1507

## Period doubling and reducibility in the quasi-periodically forced logistic map

 1 Departament of Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Barcelona, Spain, Spain 2 Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, Netherlands

Received  July 2011 Revised  January 2012 Published  March 2012

We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, such as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and reducibility. This reveals that the parameter values for which the invariant curve doubles its period are contained in regions of the parameter space where the invariant curve is reducible. Then we present two additional studies to explain this fact. In first place we consider the images and the preimages of the critical set (the set where the derivative of the map w.r.t the non-periodic coordinate is equal to zero). Studying these sets we construct constrains in the parameter space for the reducibility of the invariant curve. In second place we consider the reducibility loss of the invariant curve as a codimension one bifurcation and we study its interaction with the period doubling bifurcation. This reveals that, if the reducibility loss and the period doubling bifurcation curves meet, they do it in a tangent way.
Citation: Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507
##### References:
 [1] R. A. Adomaitis, I. G. Kevrekidis and R. de la Llave, A computer-assisted study of global dynamic transitions for a noninvertible system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1305-1321. doi: 10.1142/S021812740701780X. [2] L. Alsedà and S. Costa, On the definition of strange nonchaotic attractor, Fund. Math., 206 (2009), 23-39. [3] A. Arneodo, P. H. Coullet and E. A. Spiegel, Cascade of period doublings of tori, Phys. Lett. A, 94 (1983), 1-6. doi: 10.1016/0375-9601(83)90272-4. [4] K. Bjerklöv, SNA's in the quasi-periodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161. doi: 10.1007/s00220-008-0626-y. [5] H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc., 83 (1990), viii+175 pp. [6] H. W. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity, 15 (2002), 1205-1267. doi: 10.1088/0951-7715/15/4/312. [7] H. W. Broer, C. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol'd resonance web, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 769-787. [8] E. Castellà and Á. Jorba, On the vertical families of two-dimensional tori near the triangular points of the bicircular problem, Celestial Mech. Dynam. Astronom., 76 (2000), 35-54. doi: 10.1023/A:1008321605028. [9] 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. [10] J. Figueras and A. Haro, A reliable computation of robust response tori on the verge of breakdown, to appear in SIAM J. Appl. Dyn. Syst., 2012. [11] V. Franceschini, Bifurcations of tori and phase locking in a dissipative system of differential equations, Physica D, 6 (1983), 285-304. doi: 10.1016/0167-2789(83)90013-1. [12] A. Haro and C. Simó, To be or not to be a SNA: That is the question, 2005. Available from: http://www.maia.ub.es/dsg/2005/index.html. [13] J. F. Heagy and S. M. Hammel, The birth of strange nonchaotic attractors, Phys. D, 70 (1994), 140-153. doi: 10.1016/0167-2789(94)90061-2. [14] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X. [15] T. H. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255. doi: 10.1088/0951-7715/16/4/303. [16] Á. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps, Nonlinearity, 14 (2001), 943-976. doi: 10.1088/0951-7715/14/5/303. [17] 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. [18] K. Kaneko, Doubling of torus, Progr. Theoret. Phys., 69 (1983), 1806-1810. doi: 10.1143/PTP.69.1806. [19] S. Kuznetsov, U. Feudel and A. Pikovsky, Renormalization group of scaling at the torus-doubling terminal point, Phys. Rev. E (3), 57 (1998), 1585-1590. doi: 10.1103/PhysRevE.57.1585. [20] A. S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors, Chaos, 5 (1995), 253-260. doi: 10.1063/1.166074. [21] A. Prasad, V. Mehra and R. Ramaskrishna, Intermittency route to strange nonchaotic attractors, Phys. Rev. Let., 79 (1995), 4127-4130. doi: 10.1103/PhysRevLett.79.4127. [22] A. Prasad, V. Mehra and R. Ramaskrishna, Strange nonchaotic attractors in the quasiperiodically forced logistic map, Phys. Rev. E, 57 (1998), 1576-1584. doi: 10.1103/PhysRevE.57.1576. [23] P. Rabassa, "Contribution to the Study of Perturbations of Low Dimensional Maps," Ph.D thesis, Universitat de Barcelona, 2010. [24] P. Rabassa, A. Jorba and J. C. Tatjer, Numerical evidences of universality and self-similarity in the forced logistic map, preprint, arXiv:1112.4143, 2011. [25] P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. I. Existence of reducibily loss bifurcations, preprint, arXiv:1112.4684, 2011. [26] P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. II. Asymptotic behavior of reducibility loss bifurcations, preprint, arXiv:1112.4686, 2011. [27] P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. III. Numerical support, preprint, arXiv:1112.4687, 2011. [28] C. Simó, On the analytical and numerical approximation of invariant manifolds, in "Les Méthodes Modernes de la Mecánique Céleste" (Course given at Goutelas, France, 1989) (eds.vD. Benest and C. Froeschlé), Editions Frontières, Paris, (1990), 285-329. Available from: http://www.maia.ub.es/dsg/2004/index.html. [29] L. van Veen, The quasi-periodic doubling cascade in the transition to weak turbulence, Phys. D, 210 (2005), 249-261. doi: 10.1016/j.physd.2005.07.020.

show all references

##### References:
 [1] R. A. Adomaitis, I. G. Kevrekidis and R. de la Llave, A computer-assisted study of global dynamic transitions for a noninvertible system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1305-1321. doi: 10.1142/S021812740701780X. [2] L. Alsedà and S. Costa, On the definition of strange nonchaotic attractor, Fund. Math., 206 (2009), 23-39. [3] A. Arneodo, P. H. Coullet and E. A. Spiegel, Cascade of period doublings of tori, Phys. Lett. A, 94 (1983), 1-6. doi: 10.1016/0375-9601(83)90272-4. [4] K. Bjerklöv, SNA's in the quasi-periodic quadratic family, Comm. Math. Phys., 286 (2009), 137-161. doi: 10.1007/s00220-008-0626-y. [5] H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc., 83 (1990), viii+175 pp. [6] H. W. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity, 15 (2002), 1205-1267. doi: 10.1088/0951-7715/15/4/312. [7] H. W. Broer, C. Simó and R. Vitolo, The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol'd resonance web, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 769-787. [8] E. Castellà and Á. Jorba, On the vertical families of two-dimensional tori near the triangular points of the bicircular problem, Celestial Mech. Dynam. Astronom., 76 (2000), 35-54. doi: 10.1023/A:1008321605028. [9] 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. [10] J. Figueras and A. Haro, A reliable computation of robust response tori on the verge of breakdown, to appear in SIAM J. Appl. Dyn. Syst., 2012. [11] V. Franceschini, Bifurcations of tori and phase locking in a dissipative system of differential equations, Physica D, 6 (1983), 285-304. doi: 10.1016/0167-2789(83)90013-1. [12] A. Haro and C. Simó, To be or not to be a SNA: That is the question, 2005. Available from: http://www.maia.ub.es/dsg/2005/index.html. [13] J. F. Heagy and S. M. Hammel, The birth of strange nonchaotic attractors, Phys. D, 70 (1994), 140-153. doi: 10.1016/0167-2789(94)90061-2. [14] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X. [15] T. H. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255. doi: 10.1088/0951-7715/16/4/303. [16] Á. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps, Nonlinearity, 14 (2001), 943-976. doi: 10.1088/0951-7715/14/5/303. [17] 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. [18] K. Kaneko, Doubling of torus, Progr. Theoret. Phys., 69 (1983), 1806-1810. doi: 10.1143/PTP.69.1806. [19] S. Kuznetsov, U. Feudel and A. Pikovsky, Renormalization group of scaling at the torus-doubling terminal point, Phys. Rev. E (3), 57 (1998), 1585-1590. doi: 10.1103/PhysRevE.57.1585. [20] A. S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors, Chaos, 5 (1995), 253-260. doi: 10.1063/1.166074. [21] A. Prasad, V. Mehra and R. Ramaskrishna, Intermittency route to strange nonchaotic attractors, Phys. Rev. Let., 79 (1995), 4127-4130. doi: 10.1103/PhysRevLett.79.4127. [22] A. Prasad, V. Mehra and R. Ramaskrishna, Strange nonchaotic attractors in the quasiperiodically forced logistic map, Phys. Rev. E, 57 (1998), 1576-1584. doi: 10.1103/PhysRevE.57.1576. [23] P. Rabassa, "Contribution to the Study of Perturbations of Low Dimensional Maps," Ph.D thesis, Universitat de Barcelona, 2010. [24] P. Rabassa, A. Jorba and J. C. Tatjer, Numerical evidences of universality and self-similarity in the forced logistic map, preprint, arXiv:1112.4143, 2011. [25] P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. I. Existence of reducibily loss bifurcations, preprint, arXiv:1112.4684, 2011. [26] P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. II. Asymptotic behavior of reducibility loss bifurcations, preprint, arXiv:1112.4686, 2011. [27] P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. III. Numerical support, preprint, arXiv:1112.4687, 2011. [28] C. Simó, On the analytical and numerical approximation of invariant manifolds, in "Les Méthodes Modernes de la Mecánique Céleste" (Course given at Goutelas, France, 1989) (eds.vD. Benest and C. Froeschlé), Editions Frontières, Paris, (1990), 285-329. Available from: http://www.maia.ub.es/dsg/2004/index.html. [29] L. van Veen, The quasi-periodic doubling cascade in the transition to weak turbulence, Phys. D, 210 (2005), 249-261. doi: 10.1016/j.physd.2005.07.020.
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