We study the ratio $\epsilon_c(\omega)/\exp(-\eta B(\omega))\,$, where $\epsilon_c(\omega)$ is the breakdown threshold function for invariant tori, $\eta$ is a parameter and $B(\omega)$ is the Bruno function, which is purely arithmetic (i.e. it only depends by the continued fractions expansion of number $\omega$). We consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency. Our numerical experiments, together with some heuristic consideration, show that $\epsilon_c(\omega)/\exp(-\eta B(\omega))$ is not a continuous function on diophantine numbers $\omega\,$, for all values of $\eta\,$.
Locatelli, U., Froeschlé, C., Lega, E., Morbidelli, A. (2000). On the Relationship between the Bruno Function and the Breakdown of Invariant Tori. PHYSICA D-NONLINEAR PHENOMENA, 139, 48-71 [10.1016/S0167-2789(99)00221-3].
On the Relationship between the Bruno Function and the Breakdown of Invariant Tori
LOCATELLI, UGO;
2000-01-01
Abstract
We study the ratio $\epsilon_c(\omega)/\exp(-\eta B(\omega))\,$, where $\epsilon_c(\omega)$ is the breakdown threshold function for invariant tori, $\eta$ is a parameter and $B(\omega)$ is the Bruno function, which is purely arithmetic (i.e. it only depends by the continued fractions expansion of number $\omega$). We consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency. Our numerical experiments, together with some heuristic consideration, show that $\epsilon_c(\omega)/\exp(-\eta B(\omega))$ is not a continuous function on diophantine numbers $\omega\,$, for all values of $\eta\,$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
