A mechanical counterexample to KAM theory with low regularity

PHYSICA D-NONLINEAR PHENOMENA

卷:283

Article

Maro, Stefano

关键词: KAM theory;    Bouncing ball;    Unbounded orbits;   

DOI  :  10.1016/j.physd.2014.05.010

来源: Elsevier

【 摘 要 】

We give a mechanical example concerning the fact that some regularity is necessary in KAM theory. We consider the model given by the vertical bouncing motion of a ball on a periodically moving plate. Denoting with f the motion of the plate, some variants of Moser invariant curve theorem apply if f(over dot) is small in norm C-5 and every motion has bounded velocity. This is not possible if the function f is only C-1. Indeed we construct a function f is an element of C-1 with arbitrary small derivative in norm C-0 for which a motion with unbounded velocity exists. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

Free