Analytical Approximation of the Dissipative Standard Map

Abstract

We investigate the dynamics of a dissipative standard mapping defined by the equationswhere y ∈ R, x ∈ T and ε is a real parameter, we refer to 0 < α < 1 as the “dissipative parameter” and to ψ as the “dissipative coefficient” (ε = α = 0 provides an integrable mapping). Notice that the dynamics is contractive, since the jacobian of the above mapping equals to 1 − α. In particular, we want to compare (see Celletti et al., 1997) the solutions associated to the conservative map (i.e., α = 0) with that related to (1) (α ≠ 0). For simplicity, we consider the case when α = ε2 and construct explicit approximate solutions to the conservative and dissipative systems, using a suitable parametrization like in (Celletti and Chierchia, 1988).