Driven toroidal helix as a generalization of Kapitzas pendulum

TL;DR

This paper generalizes the Kapitza pendulum by studying a particle on a driven toroidal helix, revealing new stability behaviors, phase space structures, and chaotic dynamics influenced by helix radius and external forces.

Contribution

It introduces a driven toroidal helix model as a generalization of the Kapitza pendulum and analyzes its stability and chaotic behavior both analytically and numerically.

Findings

Presence of static fixed points similar to Kapitza pendulum

Stability depends on helix radius, driving amplitude, and static potential

Unusual transition to chaos and directed transport observed

Abstract

We explore a model system consisting of a particle confined to move along a toroidal helix while being exposed to a static potential as well as a driving force due to a harmonically oscillating electric field. It is shown that in the limit of a vanishing helix radius the governing equations of motion coincide with those of the well-known Kapitza pendulum - a classical pendulum with oscillating pivot - implying that the driven toroidal helix represents a corresponding generalization. It is shown that the two dominant static fixed points present in the Kapitza pendulum are also present for a finite helix radius. The dependence of the stability of these two fixed points on the helix radius, the driving amplitude, and the static potential are analyzed both analytically and numerically. Additionally, the most prominent deviations of the driven helix from Kapitzas pendulum with respect to the resulting phase space are investigated and analyzed in some detail. These effects include an unusual transition to chaos and an effective directed transport due to the simultaneous presence of multiple chaotic phase space regions.