# 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 & Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151
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