Birkhoff normal form and twist coefficients of asymmetric oscillations

In this paper, we study the Birkhoff normal form around the elliptic fixed point for the asymmetric oscillation$x^{″}+a{x}^{+}-b{x}^{-}=f\left(t\right),$ where $x^{+}=\max \{x,0\}$, $x^{-}=\max \{-x,0\}$, $f\left(t\right)\in C^6\left(R\right)$ is 2π periodic with zero mean value with respect to t, a and b are two different positive constants satisfying ${\omega }_0:=\frac{1}{2}(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}})\in R﹨Q$. We shall establish an explicit construction of the Birkhoff transformation and thus obtain the explicit formula for the third twist coefficient where the first two twist coefficients are both zero. As an application, we propose a sufficient condition for f such that for a given parameter $a>0$ and any given bounded closed interval $I\subset (\frac{1}{2\sqrt{a}},+\infty )$, there exists a finite set $C_I\subset I$ such that for any ${\omega }_0\in I﹨(Q\cup C_I)$, the solutions of the asymmetric oscillation are all bounded.