A case of boundedness in Littlewood's problem on oscillatory differential equations

Abstract

It is shown that all solutions of x + 2x = pit) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p .The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almostperiodic solutions and, for each positive integer m , an infinity of periodic solutions of least period 2im (2TT being taken as the least period of p ).It is suggested that for a large class of equations the same attack would show all solutions of x + g(x) = pit) bounded.However, in order to show the method clearly, no generalisation is attempted here.

Resonance and nonlinearity: A survey

“…The first positive answer was given by Morris [40] in 1976 for the superlinear problemẍ + 2x 3 = e(t), and many generalizations have been given for this superlinear case. The idea consists in transforming the equation, outside of a large ball of the phase plane (x, x ), into a perturbation of an integrable Hamiltonian system, and to apply Arnold -KolmogorovMoser's twist theorem (see, e.g., [41]) to its Poincaré's map, which is shown to be closed to a twist map outside this ball.…”

Global attraction to the origin in a parametrically driven nonlinear oscillator

“…The answer is naturally of fundamental importance for the stability properties of dynamical systems and there currently exists a substantial body of work on this problem in the zerodamping case [10]- [15]. The essential idea used to show boundedness of solutions for these potentials is to transform the system, by a sequence of canonical transformations, into a near-integrable one, and then to apply KAM theory and the Moser twist theorem.…”

Boundedness in a Piecewise Linear Oscillator and a Variant of the Small Twist Theorem

“…In [15] Morris proved that all the solutions of (1.1) are bounded when gx 2x 3 and p is piecewise continuous. Later several authors have improved this result and the boundedness has been proved for a large class of functions g that are superlinear at in®nity in the sense that gx x 3 1 as jx j 3 1:…”