Boundedness in a Piecewise Linear Oscillator and a Variant of the Small Twist Theorem
Rafael Ortega · Proceedings of the London Mathematical Society · 1999
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 hL 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 / π where p ^ n : = 1 2 π ∫ 0 2 π p ( t ) e − i n t d t . In this paper I prove that if p is of class C5 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.