Analytical Approximation of the Solution of the Dissipative Standard Map

Abstract

We consider a dissipative mapping derived from a modification of the Chirikov standard mapping. For simplicity, we assume that the dissipative strength is of the order of the square of the perturbing parameter of the conservative model. Under this assumption, we derive an analytical approximation of the solution associated to the dissipative mapping. The equations are explicitly solved up to the order 7 in the perturbing parameter. Having fixed a frequency ω, a comparison of the associated to the dissipative solutions shows that the two curves coincide for low values of the perturbing parameter, while in most cases they diverge as the breakdown threshold of the invariant curve with rotation number ω is approached.