Hamiltonian bifurcation perspective on two interacting vortex pairs: From symmetric to asymmetric leapfrogging, period doubling, and chaos

In this work we study the dynamical behavior of two interacting vortex pairs, each one of them consisting of two point vortices with opposite circulation in the two-dimensional plane. The vortices are considered as effective particles and their interaction can be described in classical mechanics terms. We first construct a Poincaré section, for a typical value of the energy, in order to acquire a picture of the structure of the phase space of the system. We divide the phase space in different regions which correspond to qualitatively distinct motions and we demonstrate its different temporal evolution in the “real” vortex space. Our main emphasis is on the leapfrogging periodic orbit, around which we identify a region that we term the “leapfrogging envelope” which involves mostly regular motions, such as higher order periodic and quasiperiodic solutions. We also identify the chaotic region of the phase plane surrounding the leapfrogging envelope as well as the so-called walkabout and braiding motions. Varying the energy as our control parameter, we construct a bifurcation tree of the main leapfrogging solution and its instabilities, as well as the instabilities of its daughter branches. We identify the symmetry-breaking instability of the leapfrogging solution (in line with earlier works), and also obtain the corresponding asymmetric branches of periodic solutions. We then characterize their own instabilities (including period doubling ones) and bifurcations in an effort to provide a more systematic perspective towards the types of motions available to this dynamical system.

Figure 1

Contours showing the boundary of the permissible area of the Poincaré section in the $x_2-y_1$ plane for values of $h$ ranging from 0 to 1.2. We can see how this area shrinks for increasing values of $h$ which are depicted as labels of the corresponding contours.

Figure 2

The Poincaré section for $h=0.7458$. (a) The permissible area of motion of the system lies inside the blue curve. Inside this area we can distinguish the leapfrogging envelope around $(0,0)$ [see also (b) for a detailed zoom into that area]. The leapfrogging envelope is surrounded by a chaotic region beyond which one finds the area of walkabout motion which extends up to approximately $|x_2|<1.5$. In the outer parts of the section lies the area of the braiding motion. (b) Zoom into the leapfrogging envelope. A close-up view of the region around the $(0,0)$ point which corresponds to the main leapfrogging motion. We can distinguish many higher order resonances around the central periodic orbit. These higher-period periodic orbits, correspond to motions which have the same characteristics as the leapfrogging one.

Figure 3

The temporal evolution of the main leapfrogging motion for $h=0.7458$. Crosses are used for vortices with positive circulation; circles are used for vortices with negative circulation. (a) The motion in the vortex $X-Y$ plane. (b) The leapfrogging motion in the $t-X$ plane. We can observe that the orbit is completely symmetric in the sense that the $X$ coordinates of the vortices of the same pair coincide. Every other point in (b) corresponds to one point on the Poincaré section.

Figure 4

(a) Temporal evolution of a walkabout orbit for $h=0.7458$ in the $X-Y$ plane. (b) The walkabout motion in the $t-X$ plane.

Figure 5

A single walkabout orbit in the PSS for $h=0.7458$. We can observe its attribute of crossing the $x_2=1$ asymptote.

Figure 6

A magnification of Fig. 2 around the islands corresponding to the mixed period leapfrogging motion.

Figure 7

(a) Temporal evolution of the mixed period leapfrogging motion for $h=0.7458$ in the $X-Y$ plane. (b) Evolution of mixed period leapfrogging in the $t-X$ plane. This kind of motion is similar to leapfrogging, but with vortex pairs of like circulation orbiting each other at different speeds.

Figure 8

(a) Temporal evolution of a braiding orbit for $h=0.7458$ in the $X-Y$ plane. (b) Evolution of the braiding orbit in the $t-X$ plane.

Figure 9

PSS for various values of $h$ showing the formation of the leapfrogging envelope and its growth as $h$ increases.

Figure 10

In the upper panel of this figure the bifurcation tree diagram for the main leapfrogging solution is shown. The main branch of this tree is the main leapfrogging solution with the bifurcated solutions spawning from it (distinguished by independent markers and colors). Only half of the tree is shown; the other half is the same but mirrored across the $y_1=0$ plane. Solid lines represent linearly stable motions while the dotted ones represent unstable motions. In the bottom panel of the figure, a two-dimensional projection of the diagram in the $(Y_1,h)$ plane is also given with an identification of the relevant bifurcations (PF stands for pitchfork and PD for period doubling).

Figure 11

Leapfrogging envelope for $h=0.69$. This value lies below the stability threshold ($h\approx 0.6931$). We can see here that the central stable periodic orbit has been replaced by an unstable one (a saddle point) while two new stable orbits have been created above and below the central one. We can still observe the higher resonances.

Figure 12

Temporal evolution of the asymmetric leapfrogging orbit for $h=0.69, x_2=0, y_1=0.052$. (a) The motion in the $X-Y$ plane. (b) The motion in the $t-X$ plane. We can observe that one member of each pair is always lagging with respect to the other. Here and henceforth, by the same line color we denote the vortices of the same pair, while by the same symbol we denote vortices of the same circulation.

Figure 13

Close-up view of a period-doubled region on the Poincarè section for $h=0.64$. The parent orbits remains stable after the period doubling, which generates two new period-two orbits which introduce four stability islands on the section.

Figure 14

A plot of the square of the normalized Floquet exponent of the asymmetric leapfrogging mode. It acquires the value of ${\mu }^2=-{\pi }^2$ around $h=0.641$ (where the period doubling occurs), while it remains negative for the range $0.5988<h<0.6931$ indicating the linear stability of the orbit. The value of ${\mu }^2$ becomes positive below $h=0.5988$ where a pitchfork bifurcation occurs and the branch becomes unstable.

Figure 15

Temporal evolution of the period-doubled leapfrogging with $h=0.64$. (a) The motion in the $X-Y$ plane. (b) The motion in the $t-X$ plane. We can see here that the lag between the two vortex pairs takes two periods of leapfrogging to return to its initial state.

Figure 16

Section in the region of one of the branches after undergoing a second pitchfork bifurcation, $h=0.5988$. We can see that the central stable periodic orbit has been replaced by an unstable one (a saddle point), while two new stable more asymmetric orbits has been generated above and below the original one.

Figure 17

Temporal evolution of the orbit resulting from the second pitchfork bifurcation. (a) The motion in the $X-Y$ plane. (b) The motion in the $t-X$ plane. Now, the characteristic lag between the two vortex pairs is again constant at all of the section hits. The magnitude of the lag, however, has gotten much bigger than the case of Fig. 12, and we see much more apparent asymmetries than in the parent branch.

Figure 18

(a) A similar example to the instability observed in [2]. (b) Another orbit that undergoes a variant of this instability (with a different outcome). Both orbits shown are for a value of $h=0.1116$. (a) has $(x_2,y_1)=(0.001,0)$ and (b) has $(x_2,y_1)=(-0.01,0.001)$.