Long‐time stability estimates for the non‐periodic Littlewood boundedness problem

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.