The boundedness of almost-periodic oscillators with asymmetric potentials via normal form theorem

This paper investigates the boundedness of solutions for the semilinear asymmetric oscillator$\ddot{x}+a{x}^{+}-b{x}^{-}=p\left(t\right),$ where p is real analytic and almost-periodic function with infinitely many rationally independent frequencies. A key contribution is the development of a novel normal form theorem for planar almost-periodic mappings under a weighted Diophantine-type nonresonance condition (1.7). Unlike prior approaches relying on twist conditions or spatial averaging, our framework eliminates geometric constraints by leveraging the spatial structure of infinite-dimensional frequencies. As a direct consequence, we prove two main results: 1. The existence of infinitely many almost-periodic solutions; 2. The boundedness of all solutions for the asymmetric oscillator, even when traditional twist integrals (e.g., (1.5)) vanish. This work unifies periodic/quasi-periodic boundedness theories and extends them to the almost-periodic regime, resolving long-standing limitations in planar Hamiltonian systems with asymmetric potentials.