We describe an algorithm that constructs formal (approximate) integrals of motion in time periodic Hamiltonian systems and apply it to a perturbed 1-d harmonic oscillator of frequency ${\omega }_1$ of the form $H=\frac{1}{2}\left({\omega }_1^2{x}^2+y^2\right)+\epsilon H_1$, where $H_1$ is an external time periodic field of frequency $\omega$ of the form: (1) $H_1=-x^{4}cos\left(\omega t\right)$, (2) $H_1=x^{4}exp\left(\frac{-x^2}{2}\right)cos\left(\omega t\right)$ and (3) truncated Taylor expansions of the exponential of case (2) in powers of $x$. The formal integrals are given as series in powers of the perturbation parameter $\epsilon$, which are constructed to high orders with the computer algebra system Maple. By use of stroboscopic Poincaré sections we find that in all cases ordered and chaotic orbits coexist. The latter are around unstable periodic orbits. Furthermore we find that the orbits close to the origin $(x,y)=(0,0)$ are ordered and can be represented accurately by invariant curves given by the approximate integral of motion. While in the case (2) the orbits starting far from the origin are bounded and ordered, for every truncated expansion of the exponential these orbits are chaotic for a finite time and then escape to infinity. The various types of orbits are described in detail.

我们描述了一种在时间周期哈密顿系统中构造运动的形式（近似）积分的算法，并将其应用于频率为1的扰动的一维谐波振荡器 ${\omega }_{\mathrm{1个}}$ 的形式 $H=\frac{\mathrm{1个}}{\mathrm{2个}}\left({\omega }_{\mathrm{1个}}^{\mathrm{2个}}{X}^{\mathrm{2个}}+{ÿ}^{\mathrm{2个}}\right)+\epsilon H_{\mathrm{1个}}$， 在哪里 $H_{\mathrm{1个}}$ 是频率的外部时间周期字段 $\omega$ 的形式：（1） $H_{\mathrm{1个}}=-X^{4}cos（\omega Ť）$，（2） $H_{\mathrm{1个}}=X^4经验值\left(\frac{-X^{\mathrm{2个}}}{\mathrm{2个}}\right)cos（\omega Ť）$ （3）将第（2）项的幂的幂的泰勒展开式截断 $X$。形式积分以摄动参数的幂次级数形式给出$\epsilon$，它们是通过计算机代数系统Maple高阶构建的。通过使用频闪庞加莱截面，我们发现在所有情况下有序轨道和混沌轨道共存。后者围绕不稳定的周期性轨道。此外，我们发现轨道接近原点$（X，ÿ）=（0，0）$是有序的，并且可以通过由运动的近似积分给出的不变曲线来准确表示。在情况（2）中，远离原点的轨道是有界的和有序的，对于指数的每次截断扩展，这些轨道在有限的时间内都是混沌的，然后逃逸到无穷大。详细描述了各种类型的轨道。