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Long‐time stability estimates for the non‐periodic Littlewood boundedness problem
Author(s) -
Kunze Markus,
Ortega Rafael
Publication year - 2013
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pds089
Subject(s) - mathematics , bounded function , countable set , plane (geometry) , interval (graph theory) , phase plane , mathematical analysis , nonlinear system , stability (learning theory) , forcing (mathematics) , pure mathematics , combinatorics , geometry , physics , quantum mechanics , machine learning , computer science
We consider the nonlinear oscillator equationx ¨ +| x |α − 1 x = p ( t ) for α⩾3 and non‐periodic forcing p . For any solution x which is unbounded in the( x , x ˙ ) ‐phase plane, it is shown that, for ε >0 small enough, there is a solution x ε which is bounded in the phase plane and such that the actions of x ε and x remain close on a time interval of length ε −2 ; further precise information is available on the location of the zeros of x ε . In addition, it is possible to take the bounded solution x ε from a fixed countable family of such solutions.