Singularities of periodic orbits near invariant curves

Computation of domains of analyticity for the dissipative standard map in the limit of small dissipation

“…We note that the shapes of the domains that we present here are remarkably different from what one sees in the symplectic case, see [CF02,BG04,dlLT95,CdlL10b]. This is partly due to the fact that in the symplectic case or in the dissipative case, the small divisors do not depend on the conformal factor b(ε) which in our case is a function of ε.…”

Perturbative Series Expansions: Theoretical Aspects and Numerical Investigations

“…Notice that inserting (9) in (7), one recovers the solution ( 5). In view of this example,w ea re ledt ot he following Definition: Ac hangeo fc oordinates Ψ verifying (6)i ss aid to be canonical.…”

“…The tools adopted to investigate the shapeo ft he analyticity domains are basedondifferentmethods: Greene's technique [13], the computation of Padé approximantsa nd ar ecentm ethodd eveloped in [5].…”

“…The zeros of the denominators provide the poles, whichc ontribute to the determination of theanalyticity domain. In order to evaluate theradiusofconvergence onec an apply an alternative technique developed in [5], to whichw er efer as Lyapunov's method, based on the computation of aq uantityr elated to the numerical definition of Lyapunov's exponents.…”

“…We consider also variantso fs uchm apping (adding suitable Fourierc omponents)a nd we analyze theb ehavior of invariant curvesw ith three different frequencies. Notice thatt he results( Tablesa nd Figures)p rovided in sections 5a nd 6a re takenf rom [ 5].…”