Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. 

For any $x \in E$, there is a decomposition $TE_x=TE_x^h \oplus TE_x^v$, $TE^v=\ker d\pi$ and $TE_x^h=\{v \in TE_x: \Omega(v,w)=0,\forall w \in TE_x^v\}$. Let $\partial_t$ be the coordinate vector field on $S^1=\partial D$,  $R$ be the horizontal lift of  $\partial_t$  and $\phi_{\Omega}$ be the flow generated by $R$. Taking base point $1 \in S^1$, then we get a map $\phi_{\Omega}: \pi^{-1}(1)\to\pi^{-1}(1)$, so the boundary $\partial E=\pi^{-1}(\partial D)$ is a mapping torus $Y_{\phi_{\Omega}}$.

For the mapping torus, we can define a periodic orbit to be an integral curve of vector field $R$. I don't know whether the periodic orbit are non degenerate. 

> Can I do a perturbation $\Omega'$ of $\Omega$ such that the periodic orbits with period $\le d$ on the mapping torus $Y_{\phi_{\Omega'}}=\pi^{-1}(\partial D)$ are non-degenerate?