In this paper, we explore the stability of an inverted pendulum under ageneralized parametric excitation described by a superposition of $N$ cosineswith different amplitudes and frequencies, based on a simple stabilitycondition that does not require any use of Lyapunov exponent, for example. Ouranalysis is separated in 3 different cases: $N=1$, $N=2$, and $N$ very large.Our results were obtained via numerical simulations by fourth-order Runge Kuttaintegration of the non-linear equations. We also calculate the effectivepotential also for $N>2$. We show then that numerical integrations recover awider region of stability that are not captured by the (approximated)analytical method. We also analyze stochastic stabilization here: firstly, welook the effects of external noise in the stability diagram by enlarging thevariance, and secondly, when $N$ is large, we rescale the amplitude by showingthat the diagrams for time survival of the inverted pendulum resembles theexact case for $N=1$. Finally, we find numerically the optimal number ofcosines corresponding to the maximal survival probability of the pendulum.
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机译：在本文中，我们基于一个简单的不需要使用Lyapunov指数的稳定条件，探索了广义参量激励下倒立摆的稳定性，该参量由具有不同幅度和频率的$ N $余弦的叠加描述。我们的分析分为3种情况：$ N = 1 $，$ N = 2 $和$ N $非常大。我们的结果是通过非线性方程的四阶Runge Kuttaintegration的数值模拟获得的。我们还计算了$ N> 2 $的有效电位。然后，我们证明了数值积分恢复了未被（近似）分析方法捕获的更宽广的稳定性区域。我们还在这里分析随机稳定度：首先，我们通过扩大方差来查看稳定性图中的外部噪声影响；其次，当$ N $很大时，我们通过显示倒立摆的时间生存图与精确图相似来重新调整幅度。 $ N = 1 $的情况。最后，我们在数值上找到与摆的最大生存概率相对应的最佳余弦数。
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