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

PHYSICA D-NONLINEAR PHENOMENA

卷:395

Article

Bustamante, Adrian P.¹  Calleja, Renato C.² 

[1] Georgia Inst Technol, Sch Math, 686 Cherry St, Atlanta, GA 30332 USA

[2] Natl Autonomous Univ Mexico UNAM, IIMAS, Dept Math & Mech, Apdo Postal 20-126, Mexico City 01000, DF, Mexico

关键词: Dissipative systems;    Quasi-periodic solutions;    Invariant manifolds;    Bifurcation;   

DOI  :  10.1016/j.physd.2019.02.006

来源: Elsevier

【 摘 要 】

Conformally symplectic systems include mechanical systems with a friction proportional to the velocity. Geometrically, these systems transform a symplectic form into a multiple of itself making the systems dissipative or expanding. In the present work we consider the limit of small dissipation. The example we study is a family of conformally symplectic standard maps of the cylinder for which the conformal factor, b(epsilon), is a function of a small complex parameter, epsilon. We assume that for epsilon = 0 the map preserves the symplectic form and the dependence on epsilon is cubic, i.e., b(epsilon) = 1 - epsilon(3). We compute perturbative expansions formally in epsilon and use them to estimate the shape of the domains of analyticity of invariant circles as functions of epsilon. We also give evidence that the functions might belong to a Gevrey class at epsilon = 0. We also perform numerical continuation of the solutions as they pass through the boundary of the domain to illustrate that the monodromy of the solutions is trivial. The numerical computations we perform support conjectures on the shape of the domains of analyticity. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free