Boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequency

Bounded Non-response Solutions with Liouvillean Forced Frequencies for Nonlinear Wave Equations

“…Forced frequencies and internal frequencies of this model work at the same time. Similar to Wang-You [27], who proved that all solutions of…”

“…Avila-Fayad-Krikorian in [1] obtained a KAM scheme for SL(2, R) cocycles with Liouvillean frequencies and Hou-You in [19] gave the rotation result of Liouvillean quasi-periodic systems. Wang-You [27] studied nonlinear quasi-periodic differential equations with Liouvillean frequency and Wang-You-Zhou [28] got the response solutions for quasi-periodically forced harmonic oscillators. In [24], Lou-Geng proved the existence of the response solutions for forced reversible ODE with Liouvillean frequencies.…”

“…If the solutions depend on not only the forced frequencies ω 1 but also internal frequencies ω 2 , thus the solutions we construct are non-response ones. The goal of this paper is to further develop the methods of [23,24,27,28,30] and prove the existence of bounded solutions in time t (which is not necessarily quasi-periodic ones) with Liouvillean forced frequency for NLW. More precisely, consider the following Hamiltonian NLW: (1.2) satisfying Dirichlet boundary conditions u(t, 0) = u t, π ξ 1 = 0, ξ 1 > 0.…”

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

“…Remark that mechanical systems of the formq 1 + q 2l−1 1 = β(t) or similar have been widely studied in the literature, and conditions on β(t) are known to guarantee either the boundedness of all solutions, or the existence of unbounded ones, see e.g. [28,27,1,24,33,14,25,26,38,36] and reference therein. For example if t → β(t) is periodic or quasi-periodic in time with a Diophantine frequency vector and d = 1, then each orbit of (1.9) in bounded [38].…”

Persistence of Invariant Tori in Integrable Hamiltonian Systems Under Almost Periodic Perturbations

“…A natural question is whether the boundedness for all solutions, called Lagrangian stability, still holds if ω = (ω 1 , ω 2 , · · · , ω m ) is not Diophantine but Liouvillean? Wang and You [31] proved the boundedness of all solutions of (1.3) with m = 2 and ω = (1, α), α ∈ R \ Q, without assuming α to be Diophantine.…”