Periodically kicked hard oscillators

G.A. CECCHI; D.L. GONZÁLEZ; M. MAGNASCO; G.B. MINDLIN; O. PIRO; A. SANTILLAN

CHAOS AN INTERDISCIPLINARY JR OF NONLINEAR SCIENCE

Año: 1993 p. 1 - 10

1054-1500

A model of a hard oscillator with analytic solution is presented. Its behavior under periodickicking, for which a closed form stroboscopic map can be obtained, is studied. It isshown that the general structure of such an oscillator includes four distinct regions; the outertwo regions correspond to very small or very large amplitude of the external force andmatch the corresponding regions in soft oscillators (invertible degree one and degree zero circlemaps, respectively). There are two new regions for intermediate amplitude of theforcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; itis characterized by discontinuous circle maps with a flat segment. Region 2 (lowmoderate forcing) has a certain resemblance to a similar region in soft oscillators(noninvertible degree one circle maps); however, the limit set of the dynamics in this regionis not a circle, but a branched manifold, obtained as the tangent union of a circle andan interval; the topological structure of this object is generated by the finite size of the repellingset, and is therefore also intrinsic to hard oscillators.