Driven toroidal helix as a generalization of the Kapitza pendulum

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 analytically. These analytical results are subsequently compared to results corresponding of numerical simulations. Additionally, the most prominent deviations of the driven helix from the Kapitza 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.

Figure 1

(a) A 3D sketch of the torus and the toroidal helix with the parametric function $\mathbf{r}\left(u\right)$, for $M=10$, $r=0.8$, and $R=2.5$. The inset in the top right visualizes the direction of the driving electric field. (b) The potential energy created by the driving electric field $E\left(t\right)$ (TIP, orange and WIP, green) and the static potential $V\left(u\right)$ (blue) shown for a toroidal helix with $M=10$, $R=2.5$, $V_0=5$, and a helix radius of $r=0.1$. (c) Visualization of the Kapitza limit $r\to 0$. The toroidal helix becomes a circle in the xy plane. The potential energy induced by the static potential is indicated by the color. For comparison, a schematic of the Kapitza pendulum is shown in the inset on the left. (d) Visualization of the Ince-Strutt diagram highlighting the regions where the two major fixed points of the Kapitza pendulum are stable (white) or unstable (red). (e) Poincare surface of section (PSOS) in the Kapitza limit $r\to 0$ for $V_0=5$ and $E_0=3$. The most prominent types of trajectories are shown: (I) rotators that are not significantly affected by the driving, (II) trajectories circling in phase around the ring with the driving, (III) bounded trajectories centered at the minimum of the static potential, and (IV) chaotic trajectories. Trajectories marked $\overline{\text{II}}$ circle around the ring in the opposite direction as those trajectories marked II.

Figure 2

Comparison of numerical calculations to the analytically predicted stability of the major fixed points in the generalized Kapitza pendulum. (a)–(c) Maximal distance in phase space between the fixed point (at $u=M\pi$ for $\alpha >0$ and at $u=0$ for $\alpha <0$) and a trajectory starting at a distance of ${10}^{-8}$ from this point after a simulation time of 1000 driving periods. White color indicates that the particle moves at least once around the torus. The parameter regions where our analytical calculations based on Eqs. (14, 15, 16) predict the fixed points to be stable are marked by the dotted red lines. (d)–(f) The time needed for the particle to move once around the torus. The used trajectories are the same as in (a)–(c). Again, the corresponding Ince-Strutt diagram is indicated by the dotted red lines. (g), (h) Example PSOS for trajectories from the purple and white regions.

Figure 3

(a) Poincare surface of sections (PSOS) for $r=0.1$, $E_0=5$, and $V_0=10$. Trajectories close to the fixed point at $(u,p)=(M\pi ,0)$ (indicated by the arrow) become chaotic while (quasi)periodic trajectories with greater phase space distances from the fixed point prevail. (b) Potential energy due to the WIP, as well as the range of kinetic energy values taken by one of the (quasi)periodic trajectories that separates the two chaotic regions shown in (a). (c), (d) PSOS for $r=0.5$, $E_0=2$ and $V_0=0$ (c), $V_0=2$ (d). The two separated chaotic seas marked V and $\overline{\text{V}}$ in (c) become connected when $V_0$ is of similar order as $E_0$. (e), (f) PSOS for $r=0.1$, $E_0=7$, and $V_0=0.9$ showing the presence of three distinct chaotic regions. The upper chaotic region is highlighted in (f).