July  2011, 16(1): 151-171. doi: 10.3934/dcdsb.2011.16.151

Periodic and quasi--periodic orbits of the dissipative standard map

1. 

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma

2. 

Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 2, I-00185 Roma, Italy

Received  April 2010 Revised  July 2010 Published  April 2011

We present analytical and numerical investigations of the dynamics of the dissipative standard map. We first study the existence of periodic orbits by using a constructive version of the implicit function theorem; then, we introduce a parametric representation, which provides the interval of the drift parameter ensuring the existence of a periodic orbit with a given period. The determination of quasi--periodic attractors is efficiently obtained using the parametric representation combined with a Newton's procedure, aimed to reduce the error of the approximate solution provided by the parametric representation. These methods allow us to relate the drift parameter of the periodic orbits to that of the invariant attractors, as well as to constrain the drift of a periodic orbit within Arnold's tongues in the parameter space.
Citation: Alessandra Celletti, Sara Di Ruzza. Periodic and quasi--periodic orbits of the dissipative standard map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151
References:
[1]

V. I. Arnold (editor), "Encyclopaedia of Mathematical Sciences,", Dynamical Systems III, 3 (1988).   Google Scholar

[2]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I,, Physica D, 8 (1983), 381.  doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

[3]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-Periodic Motions in Families of Dynamical Systems. Order Amidst Chaos,", Lecture Notes in Mathematics {\bf 1645}, 1645 (1645).   Google Scholar

[4]

H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667.  doi: 10.1088/0951-7715/11/3/015.  Google Scholar

[5]

R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative standard map,, CHAOS, 20 (2010).  doi: 10.1063/1.3335408.  Google Scholar

[6]

A. Celletti, "Stability and Chaos in Celestial Mechanics,", Springer-Praxis, (2010).  doi: 10.1007/978-3-540-85146-2.  Google Scholar

[7]

A. Celletti and L. Chierchia, Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics,, Reg. Chaotic Dyn., 3 (1998), 107.  doi: 10.1070/rd1998v003n03ABEH000084.  Google Scholar

[8]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics,, Archive for Rational Mechanics and Analysis, 191 (2009), 311.  doi: 10.1007/s00205-008-0141-5.  Google Scholar

[9]

A. Celletti and C. Falcolini, Singularities of periodic orbits near invariant curves,, Physica D, 170 (2002), 87.  doi: 10.1016/S0167-2789(02)00543-2.  Google Scholar

[10]

A. Celletti and M. Guzzo, Cantori of the dissipative sawtooth map,, Chaos, 19 (2009).  doi: 10.1063/1.3094217.  Google Scholar

[11]

A. Celletti and R. S. MacKay, Regions of non-existence of invariant tori for a spin-orbit model,, Chaos, 17 (2007).  doi: 10.1063/1.2811880.  Google Scholar

[12]

B. V. Chirikov, A universal instability of many dimensional oscillator systems,, Physics Reports, 52 (1979), 264.  doi: 10.1016/0370-1573(79)90023-1.  Google Scholar

[13]

S. Y. Kim and D.S. Lee, Transition to chaos in a dissipative standardlike map,, Phys. Rev. A, 45 (1992), 5480.  doi: 10.1103/PhysRevA.45.5480.  Google Scholar

[14]

J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus,, Topology, 21 (1982), 457.  doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[15]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nach. Akad. Wiss. G\, 1 (1962), 1.   Google Scholar

[16]

I. C. Percival, Variational principles for invariant tori and cantori,, AIP Conf. Proc., 57 (1980), 302.  doi: 10.1063/1.32113.  Google Scholar

[17]

W. Wenzel, O. Biham and C. Jayaprakash, Periodic orbits in the dissipative standard map,, Phys. Rev. A, 43 (1991), 6550.  doi: 10.1103/PhysRevA.43.6550.  Google Scholar

show all references

References:
[1]

V. I. Arnold (editor), "Encyclopaedia of Mathematical Sciences,", Dynamical Systems III, 3 (1988).   Google Scholar

[2]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I,, Physica D, 8 (1983), 381.  doi: 10.1016/0167-2789(83)90233-6.  Google Scholar

[3]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-Periodic Motions in Families of Dynamical Systems. Order Amidst Chaos,", Lecture Notes in Mathematics {\bf 1645}, 1645 (1645).   Google Scholar

[4]

H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667.  doi: 10.1088/0951-7715/11/3/015.  Google Scholar

[5]

R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative standard map,, CHAOS, 20 (2010).  doi: 10.1063/1.3335408.  Google Scholar

[6]

A. Celletti, "Stability and Chaos in Celestial Mechanics,", Springer-Praxis, (2010).  doi: 10.1007/978-3-540-85146-2.  Google Scholar

[7]

A. Celletti and L. Chierchia, Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics,, Reg. Chaotic Dyn., 3 (1998), 107.  doi: 10.1070/rd1998v003n03ABEH000084.  Google Scholar

[8]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics,, Archive for Rational Mechanics and Analysis, 191 (2009), 311.  doi: 10.1007/s00205-008-0141-5.  Google Scholar

[9]

A. Celletti and C. Falcolini, Singularities of periodic orbits near invariant curves,, Physica D, 170 (2002), 87.  doi: 10.1016/S0167-2789(02)00543-2.  Google Scholar

[10]

A. Celletti and M. Guzzo, Cantori of the dissipative sawtooth map,, Chaos, 19 (2009).  doi: 10.1063/1.3094217.  Google Scholar

[11]

A. Celletti and R. S. MacKay, Regions of non-existence of invariant tori for a spin-orbit model,, Chaos, 17 (2007).  doi: 10.1063/1.2811880.  Google Scholar

[12]

B. V. Chirikov, A universal instability of many dimensional oscillator systems,, Physics Reports, 52 (1979), 264.  doi: 10.1016/0370-1573(79)90023-1.  Google Scholar

[13]

S. Y. Kim and D.S. Lee, Transition to chaos in a dissipative standardlike map,, Phys. Rev. A, 45 (1992), 5480.  doi: 10.1103/PhysRevA.45.5480.  Google Scholar

[14]

J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus,, Topology, 21 (1982), 457.  doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[15]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nach. Akad. Wiss. G\, 1 (1962), 1.   Google Scholar

[16]

I. C. Percival, Variational principles for invariant tori and cantori,, AIP Conf. Proc., 57 (1980), 302.  doi: 10.1063/1.32113.  Google Scholar

[17]

W. Wenzel, O. Biham and C. Jayaprakash, Periodic orbits in the dissipative standard map,, Phys. Rev. A, 43 (1991), 6550.  doi: 10.1103/PhysRevA.43.6550.  Google Scholar

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