Boundedness in Asymmetric Quasi-Periodic Oscillations (Communications in Mathematical Research, 2019)
In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation $x′′+ax^+−bx^−+ ϕ(x) = p(t)$, where $a ≠ b$ are two positive constants and $ϕ(s)$, $p(t)$ are real analytic functions. Moreover, the $p(t)$ is quasi-periodic coefficient, whose frequency vectors are Diophantine. The results we obtained also imply that, under some conditions, the quasi-periodic oscillator has the Lagrange stability.
Keywords: boundedness; quasi-periodic; KAM theorem