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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 & 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.  doi: 10.1142/S021812740701780X.  Google Scholar

[2]

L. Alsedà and S. Costa, On the definition of strange nonchaotic attractor,, Fund. Math., 206 (2009), 23.   Google Scholar

[3]

A. Arneodo, P. H. Coullet and E. A. Spiegel, Cascade of period doublings of tori,, Phys. Lett. A, 94 (1983), 1.  doi: 10.1016/0375-9601(83)90272-4.  Google Scholar

[4]

K. Bjerklöv, SNA's in the quasi-periodic quadratic family,, Comm. Math. Phys., 286 (2009), 137.  doi: 10.1007/s00220-008-0626-y.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1088/0951-7715/15/4/312.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1023/A:1008321605028.  Google Scholar

[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., (2006).   Google Scholar

[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).   Google Scholar

[11]

V. Franceschini, Bifurcations of tori and phase locking in a dissipative system of differential equations,, Physica D, 6 (1983), 285.  doi: 10.1016/0167-2789(83)90013-1.  Google Scholar

[12]

A. Haro and C. Simó, To be or not to be a SNA: That is the question, 2005., Available from: , ().   Google Scholar

[13]

J. F. Heagy and S. M. Hammel, The birth of strange nonchaotic attractors,, Phys. D, 70 (1994), 140.  doi: 10.1016/0167-2789(94)90061-2.  Google Scholar

[14]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds,, Bull. Amer. Math. Soc., 76 (1970), 1015.  doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

[15]

T. H. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative,, Nonlinearity, 16 (2003), 1239.  doi: 10.1088/0951-7715/16/4/303.  Google Scholar

[16]

Á. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps,, Nonlinearity, 14 (2001), 943.  doi: 10.1088/0951-7715/14/5/303.  Google Scholar

[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.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[18]

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

[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.  doi: 10.1103/PhysRevE.57.1585.  Google Scholar

[20]

A. S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors,, Chaos, 5 (1995), 253.  doi: 10.1063/1.166074.  Google Scholar

[21]

A. Prasad, V. Mehra and R. Ramaskrishna, Intermittency route to strange nonchaotic attractors,, Phys. Rev. Let., 79 (1995), 4127.  doi: 10.1103/PhysRevLett.79.4127.  Google Scholar

[22]

A. Prasad, V. Mehra and R. Ramaskrishna, Strange nonchaotic attractors in the quasiperiodically forced logistic map,, Phys. Rev. E, 57 (1998), 1576.  doi: 10.1103/PhysRevE.57.1576.  Google Scholar

[23]

P. Rabassa, "Contribution to the Study of Perturbations of Low Dimensional Maps,", Ph.D thesis, (2010).   Google Scholar

[24]

P. Rabassa, A. Jorba and J. C. Tatjer, Numerical evidences of universality and self-similarity in the forced logistic map,, preprint, (2011).   Google Scholar

[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, (2011).   Google Scholar

[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, (2011).   Google Scholar

[27]

P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. III. Numerical support,, preprint, (2011).   Google Scholar

[28]

C. Simó, On the analytical and numerical approximation of invariant manifolds,, in, (1990), 285.   Google Scholar

[29]

L. van Veen, The quasi-periodic doubling cascade in the transition to weak turbulence,, Phys. D, 210 (2005), 249.  doi: 10.1016/j.physd.2005.07.020.  Google Scholar

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.  doi: 10.1142/S021812740701780X.  Google Scholar

[2]

L. Alsedà and S. Costa, On the definition of strange nonchaotic attractor,, Fund. Math., 206 (2009), 23.   Google Scholar

[3]

A. Arneodo, P. H. Coullet and E. A. Spiegel, Cascade of period doublings of tori,, Phys. Lett. A, 94 (1983), 1.  doi: 10.1016/0375-9601(83)90272-4.  Google Scholar

[4]

K. Bjerklöv, SNA's in the quasi-periodic quadratic family,, Comm. Math. Phys., 286 (2009), 137.  doi: 10.1007/s00220-008-0626-y.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1088/0951-7715/15/4/312.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1023/A:1008321605028.  Google Scholar

[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., (2006).   Google Scholar

[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).   Google Scholar

[11]

V. Franceschini, Bifurcations of tori and phase locking in a dissipative system of differential equations,, Physica D, 6 (1983), 285.  doi: 10.1016/0167-2789(83)90013-1.  Google Scholar

[12]

A. Haro and C. Simó, To be or not to be a SNA: That is the question, 2005., Available from: , ().   Google Scholar

[13]

J. F. Heagy and S. M. Hammel, The birth of strange nonchaotic attractors,, Phys. D, 70 (1994), 140.  doi: 10.1016/0167-2789(94)90061-2.  Google Scholar

[14]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds,, Bull. Amer. Math. Soc., 76 (1970), 1015.  doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

[15]

T. H. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative,, Nonlinearity, 16 (2003), 1239.  doi: 10.1088/0951-7715/16/4/303.  Google Scholar

[16]

Á. Jorba, Numerical computation of the normal behaviour of invariant curves of $n$-dimensional maps,, Nonlinearity, 14 (2001), 943.  doi: 10.1088/0951-7715/14/5/303.  Google Scholar

[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.  doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[18]

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

[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.  doi: 10.1103/PhysRevE.57.1585.  Google Scholar

[20]

A. S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors,, Chaos, 5 (1995), 253.  doi: 10.1063/1.166074.  Google Scholar

[21]

A. Prasad, V. Mehra and R. Ramaskrishna, Intermittency route to strange nonchaotic attractors,, Phys. Rev. Let., 79 (1995), 4127.  doi: 10.1103/PhysRevLett.79.4127.  Google Scholar

[22]

A. Prasad, V. Mehra and R. Ramaskrishna, Strange nonchaotic attractors in the quasiperiodically forced logistic map,, Phys. Rev. E, 57 (1998), 1576.  doi: 10.1103/PhysRevE.57.1576.  Google Scholar

[23]

P. Rabassa, "Contribution to the Study of Perturbations of Low Dimensional Maps,", Ph.D thesis, (2010).   Google Scholar

[24]

P. Rabassa, A. Jorba and J. C. Tatjer, Numerical evidences of universality and self-similarity in the forced logistic map,, preprint, (2011).   Google Scholar

[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, (2011).   Google Scholar

[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, (2011).   Google Scholar

[27]

P. Rabassa, A. Jorba and J. C. Tatjer, Towards a renormalization theory for quasi-periodically forced one dimensional maps. III. Numerical support,, preprint, (2011).   Google Scholar

[28]

C. Simó, On the analytical and numerical approximation of invariant manifolds,, in, (1990), 285.   Google Scholar

[29]

L. van Veen, The quasi-periodic doubling cascade in the transition to weak turbulence,, Phys. D, 210 (2005), 249.  doi: 10.1016/j.physd.2005.07.020.  Google Scholar

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