The Global Phase Space of the Three-Vortex Interaction System

TL;DR

We develop a two-step geometric reduction of the planar three-vortex dynamics by applying Jacobi coordinates to remove translations, followed by a Lie-Poisson momentum-map reduction to remove rotations, yielding a unified reduced system that captures the full range of nonzero-total-circulation dynamics. This framework reveals the global phase-space structure as a quadric surface in $(X,Y,Z)$ with conserved angular impulse $\Theta$, enabling complete enumeration of relative equilibria (equilateral and collinear) and their bifurcations as $(\Gamma_1,\Gamma_2,\Gamma_3)$ vary, including the deltoid boundary and dipole lines, and clarifies the topology of phase portraits on spheroidal and hyperboloidal phase surfaces. The paper also treats the singular case of vanishing total circulation with an alternative two-vortex reduction, deriving a reduced Hamiltonian in two dimensions and describing triple-collision dynamics and dipole scattering. Altogether, the work reconciles and extends the Conte–Aref bifurcation diagrams within a cohesive geometric framework and points to broader applicability in vortex and celestial mechanics problems.

Abstract

We derive a symplectic reduction of the evolution equations for a system of three point vortices and use the reduced system to succinctly explain a kind of bifurcation diagram that has appeared in the literature in a form that was difficult to understand and interpret. Using this diagram, we enumerate and plot all the global phase-space diagrams that occur as the circulations of the three vortices are varied. The reduction proceeds in two steps: a reduction to Jacobi coordinates and a Lie-Poisson reduction. In a recent paper, we used a different method in the second step. This took two forms depending on a sign that arose in the calculation. The Lie-Poisson equations unify these into a single form. The Jacobi coordinate reduction fails when the total circulation vanishes. We adapt the reduction method to this case and show how it relates to the non-vanishing case.

Figures (15)

• Figure 2.1: Left: Jacobi coordinates for three particles with nonzero total circulation. The coordinate $\mathbf{Z}_1$ is the vector from $\mathbf{z}_2$ to $\mathbf{z}_1$, $\mathbf{Z}_2$ is the vector from $\mathbf{z}_3$ to $\tilde{\mathbf{z}}_2$ (the center of vorticity of the first two particles), and the transformed variable $\mathbf{Z}_3$ is the conserved center of vorticity. Note carefully that the variables $z_1$, $z_2$, $z_3$, $\tilde{z}_2$, and $Z_3$ are locations in space, represented by dots, whereas $Z_1$ and $Z_2$ are displacement vectors, represented by arrows. Right: The alternative to Jacobi coordinates for two point-vortices with $-\Gamma_2=\Gamma_1>0$.
• Figure 3.1: The trilinear diagram identifying regimes of phase space behavior. The three arrows point to the direction in which the indicated scaled circulation is positive so that each vanishes on the line the arrow emerges from. In the shaded region, the quadric surface defining the phase space (hereafter, the phase surface) in Eq. \ref{['rankoneXYZ']} is a spheroid, and in the unshaded region, it is a hyperboloid.
• Figure 3.2: Bifurcation diagram displaying the equilibria and singularities defined by Eqs. \ref{['XZ_singularities']}, \ref{['equilateral_configs']} and \ref{['symmetric_equilibria']} for $\Theta=-1$ and $\Theta=1$ as a function of $\eta_3$ along the vertical center line of Fig. \ref{['fig:trilinear']}. The $X$-component of the collinear equilibria and singularities is shown using the $y$-axis scale on the left. The $Y$ component of the triangular equilibria is shown using the $y$-axis scale on the right and the same shade of red as the $Y$-scale along the right edge of the figure. Solid lines show stable equilibria, dashed lines show unstable equilibria, and dotted lines show singularities. Dashed vertical lines show the $\eta_3$ values indicated by the letter labels in Fig. \ref{['fig:trilinear']}. Cases a and j lie outside the plotted region.
• Figure 4.1: The global phase space for the case h with $\Gamma_3 =\frac{1}{3}$, showing "front" and "back" views of the sphere. The singularities are represented by black dots, the collinear equibria by blue, and the triangular equilibria by gray. The separatrix orbits are denoted by thicker curves than the periodic orbits. The equilibrium $\mathcal{E}_3$ lies on the positive $Z$-axis here and in all subsequent plots. The figures depict periodic orbits that are roughly equally spaced, rather than equally spacing the level sets of the Hamiltonian, which would lead to an accumulation of curves near each singularity. The markings are consistent across the remaining plots.
• Figure 4.2: The global phase space for case g with $\Gamma_3=\frac{1}{15}$. The invariant manifolds connected to $\mathcal{E}_3$ are homoclinic orbits oriented along the equator. The collinear equilibria $\mathcal{E}_1$ and $\mathcal{E}_2$ have migrated to the front.