N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

Panels (a–f) of Figure 5 show early-time trajectories of the equivalent heton for $\tilde{M}/N=-0.28$, $N=3$, $\Phi \left(0\right)={\Phi }_0=\pi /3$ and ${\varrho }^{+}=0.864,0.878,0.8892,0.9,0.9144$ and $0.9145$. The parameter values $({\varrho }^{+},\tilde{M}/N)$ are indicated by white markers in Figure 4e. When (${\varrho }^{+},\tilde{M}/N$) = (0.864, −0.28), as shown on Figure 5a, the initial conditions belong to the red region of Figure 4e. This region, symbolised by $\left[-/+\right]$, corresponds to cases where the average angular velocity of the orbital motion of the upper layer vortex is negative, and the lower layer vortex is positive. In the general case, in addition to circular motion, the vortices also perform nutational oscillations relative to their mean position. Thus, their trajectories over long times fill concentric annular regions.

Panels (b–e) show examples of motions in the $\left[-0+/+\right]$ region (black area of Figure 4). Symbols ‘−’, ‘0’ and ‘+’, respectively, correspond to an anticyclonic displacement, the absence of any average displacement, and a cyclonic displacement of the upper layer vortex. The symbol ‘+’ after the forward slash corresponds to a cyclonic rotation in lower layer. In each panel of Figure 5, the corresponding symbol is highlighted in red. A characteristic feature of this class of motions is that the sign of the total angular velocity (not averaged) of the upper-layer vortices alternates (nutational trajectories are loop-shaped). In panel (c), the outer vortex only performs periodic nutational motions, while the inner vortex periodically rotates along a fixed quasi-triangular curve (the corresponding marker in Figure 4a lies on the white line). In reality, we have a six-vortex structure, in which three peripheral vortices in the upper layer rotate along individual closed trajectories shifted relative to each other by an angle $\pi /3$, and three vortices in the lower layer move along a common central quasi-triangular trajectory one after the other with the same angular displacement. For this reason, the trajectories in panel (c) are called ‘absolute complex choreographies’.

It should be noted that the values of the values of the parameter ${\varrho }^{+}$ in panels (b–d) differ only slightly and, therefore, the markers in Figure 4e are almost undistinguishable. The last pair of trajectories of this type in Figure 4e corresponds to initial conditions close to the border of the black region. There, the loops almost disappear, and the trajectory of the outer vortex approaches a circular form. An increase in ${\varrho }^{+}$ by just 0.0001 shifts the initial conditions into the region of unbounded motions, when three two-layer pairs (hetons) scatter in different directions—see Figure 5f. It should be noted that the example of absolute choreographies, shown in Figure 5c, is not unique. Trajectories of this type correspond to points belonging to the white lines in Figure 4a–c,e. The case $N=2$ was studied in detail in SKDR.

Figure 6 shows a gallery of absolute choreographies for $2\le N\le 6$, corresponding to the maximum values of ${\varrho }^{+}$ (the closed curves or orbits were obtained numerically). We call these periodic trajectories the ‘limiting’ ones. The parameters ${\varrho }^{+}$ and $\tilde{M}/N$ for the cases $N=2,3$ and 6 correspond to the white markers in the upper right corners of Figure 4a–c.

Figure 6 shows that the size of internal N-shaped (N-gons with smoothed corners) choreographies increase with increasing of N.

We next consider the topological properties of the choreographies for the case $N=6$ ($\Phi \left(0\right)={\Phi }_0=\pi /6$) in more detail in Figure 7. We show a family of choreographies over an interval for ${\varrho }^{+}\in [0.60;0.97]$ with an increment of 0.01. This interval is the green portion of the white line in Figure 4c. Numerical experiments show that as ${\varrho }^{+}$ decreases, the quasi-hexagonal choreographies decrease in size in the lower layer and the trajectories become nearly circular. In the upper layer, the situation is more complicated. For ${\varrho }^{+}\ge 0.97$, the vortices move along thin drop-shaped trajectories, whose innermost tips point towards the origin. For ${\varrho }^{+}$ = 0.96, figure-of-eight orbits begin to form due to the emergence of a second closed loop to the left of the tip. Such doubly connected patterns occur in the interval ${\varrho }^{+}\in [0.93,0.96]$. In this interval, as ${\varrho }^{+}$ decreases, the left part of the ‘eight’ grows, while the right part shrinks. Finally, for ${\varrho }^{+}$ = 0.93, the right part completely disappears, and now the drop-shaped trajectory has a tip pointing away from the origin. As we decrease ${\varrho }^{+}$ further, the size of the drop first increases to reach a local maximum at ${\varrho }^{+}\approx 0.90$, and then decreases, eventually shrinking to a point.

We next consider the evolution of the system close to the figure-of-eight stationary states for ${\varrho }^{+}=0.95$ (purple curves in Figure 7) and for ${\varrho }^{+}=0.97$ (black curves). The next three figures are devoted to ${\varrho }^{+}=0.95$, followed by another devoted to ${\varrho }^{+}=0.97$.

Figure 8 shows a series of vortex trajectories in the vicinity of an equivalent heton with $N=6$ and ${\varrho }^{+}=0.95$. Panel (b) shows the absolute choreography of the [-0+/+] type, for which the vortex trajectory of the upper layer has a figure-of-eight shape, and panels (a) and (c) show the absolute vortex trajectories of types [-0+/+] and [-0+/+],, respectively, when the upper layer vortex moves along a loop-like trajectory, unwinding in the anticyclonic and cyclonic directions, respectively. This behaviour is found in the black region of Figure 4c.

We construct the relative choreographies for these cases in Figure 8(a₁,c₁). The relative choreographies are the closed trajectories obtained in a reference frame rotating with the average angular velocity $\omega$ of the vortex in the upper layer. One may see that the relative choreographies have loop-like absolute upper-layer trajectories (found inside the black region of Figure 4c) also belong to a type of figure-of-eight trajectory. The figure-of-eight relative choreographies also occur in some parts of the red and yellow areas of the diagram, where the upper-layer vortices move along zigzag trajectories—see Figure 9.

Further away from the stationary state, it is possible to obtain drop-shaped relative choreographies, both in the red region (for $\tilde{M}/N=-0.2$) and in the yellow region (for $\tilde{M}/N=-0.5$) in Figure 4c, as shown in Figure 10.

We next consider the second class of motions in the vicinity of the stationary state, when the absolute choreography for the vortex of the upper layer has a drop-like shape, i.e., for ${\varrho }^{+}=0.97$. Results are presented in Figure 11, where all absolute trajectories of the upper-layer vortices are loop-shaped. The initial conditions are in the black region of Figure 4c, outside the yellow [+/+]-region. All relative choreographies are drop-shaped.

Let us point out another new type of absolute choreography in the N-symmetric dynamics of N hetons. In this choreography, trajectories are circular synchronous rotations of the lower-layer and upper-layer vortices around a common centre. Clearly, these movements are of the [+/+] type (the yellow areas in Figure 4). For a circular motion, the right-hand side of Equation (5) must be zero, i.e., $\varrho \equiv 0$, and we must have ${\omega }_1={\dot{\phi }}_1={\omega }_2={\dot{\phi }}_2$ (see Equations (7) and (8)). These conditions are satisfied for a discrete set of parameters in the plane (${\varrho }^{+},\tilde{M}/N$) in a narrow neighbourhood of the yellow region near the boundary of the region [∞]. Two examples of such circular choreographies for $N=5$ and 6 are shown in Figure 12. Markers in the figure only show the initial location of the two vortices of the equivalent heton. In reality, five (respectively, six) anticyclones/cyclones in the upper/lower layer move counter-clockwise, with the same angular velocity, thereby preserving the regular pentagonal or hexagonal shape.

Since we assume throughout that $r_1>r_2$ (as in Figure 1), the external N-gon is in the upper layer. Thus, ‘external/internal’ and ‘upper-layer/lower-layer’ are synonymous in this context. The internal vortices always maintain their direction of rotation, while the external ones can change it.

The direction of rotation of the external vortices depends on the region of the parameter space. The external vortices are expected to rotate in the opposite (anticyclonic) direction if the interaction between the two layers is sufficiently weak. In the parameter space $({\varrho }^{+};\tilde{M})$ in Figure 4, this anticyclonic motion occurs in the red [$-/+$] region. To determine the boundary of this region, we determine the value of the angular impulse $\tilde{M}\left({\varrho }^{+}\right)$ at which the angular velocity of the external vortices vanishes. This leads to the implicit equation:

Analytical progress can be made assuming $\varrho \ll 1$. Then, Equation (22) reduces to

We can estimate the solution by manipulating the sum in the equation. On the one hand, if we neglect the sum, we obtain the approximate value of $R=1.257$, which is smaller than the actual solution to Equation (23). On the other hand, we can find an upper bound for the sum in question: from which we obtain the upper bound $R=2$ for the solution to (23). Thus, we obtain the following bounds for the solution:

Numerical solutions to Equation (23) for $50\le N\le 100$ suggest that $R\left({\varrho }^{+}=0\right)$ tends to $R_0^{\infty }\approx 1.72$.

From this result, we conclude that the boundary of the [$-/+$] region corresponds to when $\varrho \to 0$, and decreases with N. The limiting lower boundary $\tilde{M}/N\approx -1.4792$ of the red regions is shown in Figure 4a–c by the dashed horizontal lines.

We next consider the direction of rotation of the internal (lower-layer) vortices. Their angular velocity at the minimum of $r_2=R\varrho$ is found from

We can see that ${{\dot{\phi }}_2}^{-}$ is always positive. In Equation (27), only the second sum can be negative, but at sufficiently large R this sum is exponentially small. For small R one can use (A6) and approximate ${\mathrm{K}}_0$ as a natural logarithm and use (A2) to obtain the estimate $\frac{N{\varrho }^N}{\left(1-{\varrho }^N\right)}>0$ for this sum.

At the maximum distance of $r_2$, the value of ${\dot{\Phi }}^{+}$ vanishes only at the edge of the blue [∞] region. It follows that ${{\dot{\phi }}^{+}}_2$ must be opposite in sign to ${{\dot{\phi }}^{+}}_1$ at this distance, i.e., it is also positive at maximum distance. This confirms that the internal vortices always rotate in the same direction.

We next analyse the boundary of the yellow [$+/+$] region shown in Figure 4, where the vortices in the two layers rotate in the same direction. Along this boundary, given by the curve $\tilde{M}\left({\varrho }^{+}\right)$, the angular velocity of the external vortices vanishes at the minimum of $r_1=R$. It thus cannot change sign for any $r_1$. The equation for $\tilde{M}\left({\varrho }^{+}\right)$ is

This curve coincides with the boundary of the red $[-/+]$ region shown in Figure 4, in the limit $\varrho \ll 1$, and reaches the same limiting point given by Equation (26). There are no solutions for $\tilde{M}\to -\infty$ or $\tilde{M}\to -0$. It follows that the boundaries of the yellow $[+/+]$ and the blue $[\infty ]$ regions intersect at a finite value of $\tilde{M}$ (as has been confirmed numerically).

There is a small region between the $[-/+]$ and $[+/+]$ regions, shown in black in Figure 4, where angular velocity of external vortices changes sign over the period $T_2$ of oscillation. The white curve lying inside this region indicates where the time-averaged angular velocity, ${\langle {\dot{\phi }}_1\rangle }_{T_2}$, vanishes. Along this curve, the external vortices only nutate, with no net rotation over a period.

We next consider the characteristics of the localised, bounded vortex motions. First, we examine angular velocity profiles in a few examples. Figure 13 shows the initial upper-layer and lower-layer profiles ${\omega }_{1,2}^{+}={\dot{\phi }}_{1,2}^{+}$ (the right-hand sides of Equations (7) and (8) at $t=0$), for $\Phi =\pi /N$ and $N=3$. In Figure 13a the profiles vary with ${\varrho }^{+}=r_2/r_1$ for two values of $R=r_1$, while in (b) the profiles vary with R for two values of ${\varrho }^{+}$. In panel (a), the angular velocity profile does not change sign over the entire interval of the parameter ${\varrho }^{+}$, but the signs are opposite for the upper-layer and lower-layer vortices when $R=1$ (they move in the direction corresponding to their strength). However, when $R=4$, both angular velocities are positive, due to the dominant role of the cyclonic lower layer. In Figure 13b in both cases (${\varrho }^{+}=0.6$ and ${\varrho }^{+}=0.8$), ${\omega }_1^{+}$ changes sign as R increases, when the upper-layer vortices fall under the prevailing influence the lower-layer vortices.

Following SKDR, we introduce a characteristic of the direction of the upper-layer vortex rotation: where $T_0$ is the time it takes a vortex to move from its maximum radius to its minimum radius and back, $T_1$ is the upper-layer vortex rotation period around the origin, and ${\langle \dots \rangle }_T$ denotes the time average over the interval $[0,T]$. For further insight, the parameter $\delta =T_1/T_2$, which characterises the ratio of the rotational periods of the vortices in the upper and lower layers, is also useful.

Figure 14 shows examples of the distributions ${\delta }_{max}\left({\varrho }^{+}\right)$, for several values of the angular impulse, $\tilde{M}$, and for $N=6$. Here, the red line curve shows ${\delta }_{max}\left({\varrho }^{+}\right)$ when $\tilde{M}/N=-0.1326$, corresponding to the white marker in the upper right corner in Figure 4c, as well as the red vortex trajectories (absolute choreography) in Figure 6, in which ${\varrho }^{+}=0.9854$. The nonmonotonic character of ${\delta }_{max}\left({\varrho }^{+}\right)$ is consistent with the [$-/+$]-type of motion, but the monotonically decreasing branch of this distribution corresponds to the [$+/+$]-type of motion.

According to Figure 4c, when $\tilde{M}/N=-1$ only the [$+/+$] type of bounded motion occurs (the green line on Figure 14). The two cases $\tilde{M}/N=-0.3$ and $-0.4$ by contrast contain both bounded and unbounded types of motion. The black markers in Figure 14 indicate the initial conditions for the periodic motions shown in Figure 15 and Figure 16. We see that ${\delta }_{max}$ varies continuously in certain intervals of the parameters ${\varrho }^{+}$ and $\tilde{M}/N$, with the exception of the resonance values of ${\varrho }^{+}$ when ${\omega }_1=0$. Of particular interest are the cases with integer values of ${\delta }_{max}$; they correspond to closed periodic trajectories of the vortex of the upper layer with an integer number of maxima. In view of Remark 1, in this case, the vortex in the lower layer must also move along a closed trajectory, which must also have an integer number of maxima. Thus, the value of $\delta$ for such cases is always rational or an integer. Figure 15 shows four examples of stationary periodic trajectories for an equivalent heton. In all cases, the closed shape of the vortex trajectory of the upper layer is such that, as noted above, the number of maxima coincides with value of ${\delta }_{max}$: at ${\delta }_{max}$ = 2 it resembles an ellipse symmetrically flattened above and below. In the other cases, the trajectories are quasi ${\delta }_{max}$-gonal in structure. In the lower layer, the situation as a whole is more complicated. For ${\delta }_{max}$ = 2, the trajectory is closed with three maxima of the radial coordinate $r_2$ and three self-intersection points. For ${\delta }_{max}$ = 3, the trajectory has an oval shape. The number of maximum increases as ${\delta }_{max}$ increases.

Figure 16 shows the periodic trajectories for ${\delta }_{max}$ = 20, $\tilde{M}/N=-0.1326,{\varrho }^{+}=0.96550$ (left panel) and $\tilde{M}/N=-0.3,{\varrho }^{+}=0.96447$ (right panel). In the first case, the corresponding marker in Figure 14 lies on the red non-monotonic curve, and in the second it lies on the blue monotonic one. Since the values ${\varrho }^{+}$ are very close for both cases, the markers in Figure 14 are visually indistinguishable. It is important to note that in the first case the solution belongs to the [$-/+$] type of motion; that is, when the upper-layer vortices rotate, on average, anticyclonically and the lower-layer vortices rotate cyclonically. In the second case, all vortices rotate cyclonically (region [$+/+$]).

Figure 17 shows the trajectories of all the vortices for the cases shown in Figure 15 with $N=6$. The combined trajectories are 6-fold symmetric, unlike that of the equivalent heton, but are also highly tangled. We discuss the cases with different integer values of ${\delta }_{max}$ in turn.

For ${\delta }_{max}=2$, there are three distinct trajectories in the upper layer (plotted in black, red and blue) along which vortices (1,4), (2,5) and (4,6) move, respectively. In the lower layer, we observe two distinct trajectories. Vortices 1, 3 and 5 move along the black trajectory, while vortices 2, 4 and 6 move along the red one.

For ${\delta }_{max}$ = 3, now there are only two distinct trajectories in the upper layer, with vortices 1, 3 and 5 moving along the trajectory plotted in black, and vortices 2, 4 and 6 along the trajectory plotted in red. In the lower layer, there are three distinct trajectories along which vortices (1,4), (2,5) and (4,6) move, respectively. This behaviour is exactly the reverse of what was found for ${\delta }_{max}=2$. The fact that the vortices move along independent closed curves is characteristic of the vortex motion for low-mode periodic solutions (low integer values of ${\delta }_{max}$) for $N>2$.

For ${\delta }_{max}$ = 4, as in the first case, three distinct trajectories are observed in the upper layer, while now all vortices of the lower layer move along a single trajectory. During the full period of revolution around the centre, the radial coordinate of the vortices reaches a maximum twelve times.

The case ${\delta }_{max}=5$ stands out from the previous ones, because now, in the upper layer, each of the six vortices moves along its own individual trajectory. This is because $N=6$ and ${\delta }_{max}=5$ are co-prime. Each trajectory is plotted in a different colour in Figure 17. In the lower layer, as before, all the vortices follow one another along one common trajectory, and now the the radial coordinate reaches a maximum thirty times per period.

Thus, in all cases with integer values of ${\delta }_{max}$, all trajectories are absolute choreographies. In the first two cases, the choreographies are convoluted in both layers, and in last two cases, the choreographies are convoluted in the upper layer, but simple in the lower layer.

Figure 18 shows three more examples of the trajectories of all six two-layer pairs of vortices for $\tilde{M}/N=-0.1396$. The trajectories of the vortices of the upper layer are plotted in red, while those of the lower layer are plotted in blue. The central part of the figure shows the trajectories of all the vortices for ${\varrho }^{+}=0.96550$ corresponding to $R=1.97761$ and $\delta =7/4$, as previously shown in Figure 16. In this case, vortices (1,6), (2,5) and (3,6) of the upper layer move along three distinct trajectories, while the vortices of the lower layer move along a single trajectory. In the second case, for ${\varrho }^{+}=0.8954$ corresponding to $R=3.024729$ and $\delta \to \infty$, the vortices of the lower layer follow a single trajectory resembling an hexagon with smoothed edges. The vortices of the upper layer move along distinct, closed, peripheral trajectories, as previously shown in Figure 6. In the third case, for ${\varrho }^{+}=0.8960$ corresponding to $R=3.088396$ (i.e., a small increase in ${\varrho }^{+}$ from the second case), the vortices scatter as six heton pairs. This is justified by the fact that the second case lies on the boundary between the regions of bounded and unbounded motion shown in the regime diagram in Figure 4.