Stephenson–Kapitza pendulum with an additional degree of freedom
In this case, the difference from the classical pendulum with a vibrating suspension axis of (the Stephenson–Kapitza pendulum) is that the pendulum bob may move along the rod.
The resulting domain of attraction of the stable upper position of the pendulum reaches 70° (by the angular deviation from the vertical), as compared to the few degrees for a fixed bob.
References
1. Blekhman I. I., Sperling L. Zum Einfluss eines inneren Freiheitsgrades auf das Verhalten eines Pendels mit vibrierender Aufhängung // Technische Mechanik. 2004. Band 24. Heft 3-4. S. 277–288.
2. Vasilkov V., Chubinsky A., Yakimova K. The Stephenson-Kapitsa pendulum: Area of the Attraction of the Upper Positions of the Balance // Technische Mechanik. 2007. Band 27. Heft 1. S. 61–66.