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Boundedness in a Piecewise Linear Oscillator and a Variant of the Small Twist Theorem
Author(s) -
Ortega Rafael
Publication year - 1999
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/s0024611599012034
Subject(s) - mathematics , twist , piecewise linear function , invariant (physics) , piecewise , differential equation , combinatorics , integer (computer science) , function (biology) , mathematical analysis , pure mathematics , mathematical physics , geometry , evolutionary biology , computer science , biology , programming language
Consider the differential equation x ¨ + n 2 x + h L ( x ) = p ( t ) ,where n =1,2,… is an integer, p is a 2π‐periodic function and h L is the piecewise linear function h L ( x ) = {L if x ⩾ 1 ,L x if | x | ⩽ 1 ,− L if x ⩽ − 1 . A classical result of Lazer and Leach implies that this equation has a 2π‐periodic solution if and only if|p ∈ n | < 2 L / πwherep ^ n : = 1 2 π∫ 0 2 π p ( t ) e − i n td t .In this paper I prove that if p is of class C 5 then the condition ref(ll) is also necessary and sufficient for the boundedness of all the solutions of the equation. The proof of this theorem motivates a new variant of Moser's Small Twist Theorem. This variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle. 1991 Mathematics Subject Classification : 34C11, 58F35.