# American Institute of Mathematical Sciences

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 and 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, Springer-Verlag 3, 1988. [2] S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I, Physica D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6. [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 1645, Springer-Verlag, Berlin, 1996. [4] H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770. doi: 10.1088/0951-7715/11/3/015. [5] R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative standard map, CHAOS, 20 013121, (2010). doi: 10.1063/1.3335408. [6] A. Celletti, "Stability and Chaos in Celestial Mechanics," Springer-Praxis, 2010. doi: 10.1007/978-3-540-85146-2. [7] A. Celletti and L. Chierchia, Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics, Reg. Chaotic Dyn., 3 (1998), 107-121. doi: 10.1070/rd1998v003n03ABEH000084. [8] A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics, Archive for Rational Mechanics and Analysis, 191 (2009), 311-345. doi: 10.1007/s00205-008-0141-5. [9] A. Celletti and C. Falcolini, Singularities of periodic orbits near invariant curves, Physica D, 170 (2002), 87-102. doi: 10.1016/S0167-2789(02)00543-2. [10] A. Celletti and M. Guzzo, Cantori of the dissipative sawtooth map, Chaos, 19 (2009), pp.6. doi: 10.1063/1.3094217. [11] A. Celletti and R. S. MacKay, Regions of non-existence of invariant tori for a spin-orbit model, Chaos, 17 (2007), pp.12. doi: 10.1063/1.2811880. [12] B. V. Chirikov, A universal instability of many dimensional oscillator systems, Physics Reports, 52 (1979), 264-379. doi: 10.1016/0370-1573(79)90023-1. [13] S. Y. Kim and D.S. Lee, Transition to chaos in a dissipative standardlike map, Phys. Rev. A, 45 (1992), 5480-5487. doi: 10.1103/PhysRevA.45.5480. [14] J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4. [15] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II, 1 (1962), 1-20. [16] I. C. Percival, Variational principles for invariant tori and cantori, AIP Conf. Proc., 57 (1980), 302-310. doi: 10.1063/1.32113. [17] W. Wenzel, O. Biham and C. Jayaprakash, Periodic orbits in the dissipative standard map, Phys. Rev. A, 43 (1991), 6550-6557. doi: 10.1103/PhysRevA.43.6550.

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##### References:
 [1] V. I. Arnold (editor), "Encyclopaedia of Mathematical Sciences," Dynamical Systems III, Springer-Verlag 3, 1988. [2] S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I, Physica D, 8 (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6. [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 1645, Springer-Verlag, Berlin, 1996. [4] H. W. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770. doi: 10.1088/0951-7715/11/3/015. [5] R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative standard map, CHAOS, 20 013121, (2010). doi: 10.1063/1.3335408. [6] A. Celletti, "Stability and Chaos in Celestial Mechanics," Springer-Praxis, 2010. doi: 10.1007/978-3-540-85146-2. [7] A. Celletti and L. Chierchia, Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics, Reg. Chaotic Dyn., 3 (1998), 107-121. doi: 10.1070/rd1998v003n03ABEH000084. [8] A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics, Archive for Rational Mechanics and Analysis, 191 (2009), 311-345. doi: 10.1007/s00205-008-0141-5. [9] A. Celletti and C. Falcolini, Singularities of periodic orbits near invariant curves, Physica D, 170 (2002), 87-102. doi: 10.1016/S0167-2789(02)00543-2. [10] A. Celletti and M. Guzzo, Cantori of the dissipative sawtooth map, Chaos, 19 (2009), pp.6. doi: 10.1063/1.3094217. [11] A. Celletti and R. S. MacKay, Regions of non-existence of invariant tori for a spin-orbit model, Chaos, 17 (2007), pp.12. doi: 10.1063/1.2811880. [12] B. V. Chirikov, A universal instability of many dimensional oscillator systems, Physics Reports, 52 (1979), 264-379. doi: 10.1016/0370-1573(79)90023-1. [13] S. Y. Kim and D.S. Lee, Transition to chaos in a dissipative standardlike map, Phys. Rev. A, 45 (1992), 5480-5487. doi: 10.1103/PhysRevA.45.5480. [14] J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4. [15] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II, 1 (1962), 1-20. [16] I. C. Percival, Variational principles for invariant tori and cantori, AIP Conf. Proc., 57 (1980), 302-310. doi: 10.1063/1.32113. [17] W. Wenzel, O. Biham and C. Jayaprakash, Periodic orbits in the dissipative standard map, Phys. Rev. A, 43 (1991), 6550-6557. doi: 10.1103/PhysRevA.43.6550.
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