Dynamics of three noncorotating vortices in Bose-Einstein condensates

In this work we use standard Hamiltonian-system techniques in order to study the dynamics of three vortices with alternating charges in a confined Bose-Einstein condensate. In addition to being motivated by recent experiments, this system offers a natural vehicle for the exploration of the transition of the vortex dynamics from ordered to progressively chaotic behavior. In particular, it possesses two integrals of motion, the energy (which is expressed through the Hamiltonian $H$) and the angular momentum $L$ of the system. By using the integral of the angular momentum, we reduce the system to a 2-degrees-of-freedom one with $L$ as a parameter and reveal the topology of the phase space through the method of Poincaré surfaces of section. We categorize the various motions that appear in the different regions of the sections and we study the major bifurcations that occur to the families of periodic motions of the system. Finally, we correspond the orbits on the surfaces of section to the real space motion of the vortices in the plane.

Figure 1

The Poincaré section for $L=-0.25$. We can distinguish two regions of predominantly ordered dynamics, as discussed in detail in the text. The red (light colored) lines depict the boundaries of motion.

Figure 2

An orbit in the $x$-$y$ space for $L=-0.25$ is shown. It corresponds to the “central” periodic orbit of Fig. 1, which in turn represents a quasiperiodic orbit in the $x$-$y$ space. Here it is shown for three values of the time, $t=0.2$, $t=1.9$, and $t=20$, from left to right. The dots are showing the initial configuration. The yellow, orange, and blue (light, intermediate and dark gray) dots correspond to the $S_1$, $S_2$, and $S_3$ vortices, respectively. The same colors are used for the solid lines which denote the corresponding trajectories. The black dot represents the center of the condensate, while the gray circle represents the Thomas-Fermi radius. This color code will be used in the rest of this work. In addition, in the first two panels there are arrows indicating the main rotation direction of the vortices, which is dictated by the gyroscopic precession due to the trapping potential.

Figure 3

An orbit for $L=-0.25$ is shown which is a representation of the “satellite” regime orbits. It corresponds to $({\varphi }_1,J_1)\simeq (0,0.03)$ and it is depicted for $t=3$ (left panel) and $t=10$ (right panel).

Figure 4

At $L=-0.218$ a supercritical pitchfork bifurcation occurs, replacing the central stable periodic orbit with an unstable and two stable ones. In the diagram the value of $J_1$ of these orbits on the section is shown as a function of $L$. The value of ${\varphi }_1$ is constant and equal to ${\varphi }_1=\pi$, i.e., all of them are symmetric (in the meaning of the term indicated in the text). The Poincaré section of the system after the first pitchfork bifurcation which occurs for $L\simeq -0.218$ is shown in the bottom panels of the figure. The left panel corresponds to a magnification of the section for $L=-0.215$ around ${\varphi }_1=\pi$. In the right panel we show the full section for $L=-0.2$.

Figure 5

Top panels: The two stable periodic orbit configurations for $L=-0.215$ for $t=2.1$. The two orbits are similar. In the one in the left panel which corresponds to $({\varphi }_1,J_1)\simeq (\pi ,0.06)$ the $S_1$ vortex rotates closer to the center than $S_3$, while in the configuration of the right panel, which corresponds to $({\varphi }_1,J_1)\simeq (\pi ,0.0918)$, the situation is reversed. Bottom panels: The motion of the vortices which corresponds to the central unstable periodic orbit for $L=-0.215$. In the left panel the evolution for $t=2.1$ is shown while in the right-hand panel the system has evolved for $t=20$. As we can see the orbits of the vortices lie in an almost one-dimensional subspace of the $x$-$y$ plane, indicating that this orbit is very close to the “exact” periodic orbit.

Figure 6

The graphs show the Poincaré section of the tripole system in the same format as, e.g., in Fig. 1 but for the case of progressively increasing $L$ in the range $-0.09\le L\le 0.2$. As can be observed, the small chaotic region around the unstable central periodic orbit increases and finally spreads to almost all the phase space as the value of $L$ is increased, while ordered dynamics becomes confined into two regions in the top and bottom of the section.

Figure 7

The form of two regular orbits in the $x$-$y$ plane for $L=-0.05$ and $t=15$. In the left panel, the orbit which corresponds to $({\varphi }_1,J_1)\simeq (\pi ,0.03)$ is depicted where the orbit of $S_3$ approaches the one of $S_2$. In the right panel, the $({\varphi }_1,J_1)\simeq (\pi ,0.23)$ is shown, where $S_1$ approaches $S_2$.

Figure 8

A chaotic mixing orbit for $L=0.1$ for $({\varphi }_1,J_1)=(\pi ,0.15)$. In the left panel, the orbit has evolved for $t=15$, while in the right one it has evolved for $t=50$.

Figure 9

The Poincaré section for $L=0.25$. In this area of values of $L$, the two persisting regular regions of the section have occupied their maximal area before shrinking again in size for larger $L$.

Figure 10

A magnification of the Poincaré sections is shown for the period-doubling scenario described in the text and occurring for $0.2<L<0.247$.

Figure 11

Top two rows: The dependence of the relevant Floquet multipliers of the destabilizing and restabilizing periodic orbit associated with the period-doubling scenario described in the text. Bottom row: The real and imaginary parts of the associated multipliers.

Figure 12

The Floquet multipliers for the inverse period-doubling scenario of the asymmetric periodic orbit which leads to its stabilization.

Figure 13

Left panel: The Poincaré section for $L=0.35$. We can observe the two additional regions of regular orbits around the asymmetric double period periodic orbit. Right panel: The form of the asymmetric periodic orbits in the $x$-$y$ plane is shown.

Figure 14

The Poincaré sections are shown in a similar way as in the earlier figures but now for $0.365\le L\le 0.43$.

Figure 15

The Poincaré sections are shown for $0.44\le L\le 0.55$.

Figure 16

Comparison of vortex trajectories in the case of highly regular (top), less regular but still nonchaotic (middle), and, finally, in the case of a weakly chaotic trajectory. The solid lines indicate the ODE results, while the corresponding symbols the PDE ones. The gray circle denotes the Thomas-Fermi radius.