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Topology of the Liouville foliation in the generalized constrained three-vortex problem G. P. Palshin Phystech School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
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     § 1. IntroductionProblems of vortex dynamics have a rich and long history. Studies in this direction are based, for the most part, on the 1858 paper by von Helmholtz [1]. In this paper he established his famous principle of vortex conservation. In 1893 Poincaré noted that “the theory of vortex motions is based on Helmholtz’s theorem, which until the present, is the most significant contribution to hydrodynamics” (see the introduction in [2]). According to Helmholtz’s principle, the process of nucleation and disappearance of vortices is related to the presence of a passive friction force in fluids. In the ideal case the vortex intensities can be considered as constants, that is, as parameters of the dynamical system. In his lectures on mathematical physics Kirchhoff derived equations in Hamiltonian form for the motion of rectilinear parallel vortex filaments in an unbounded perfect fluid; he also proved the presence of four motion integrals (see [3], § 3, Lecture XX). Based on the Helmholtz–Kirchhoff equations, Gröbli, in his doctoral thesis, solved the motion problems for three vortex filaments (see [4], §§ 2–12), four vortex filaments with the presence of a symmetry plane (see [4], §§ 13–20) and also $2n$ filaments with $n$ symmetry planes (see [4], § 21). Joukowsky, a prominent figure in the Russian science, also contributed to the development of vortex dynamics. For example, in 1894, at the 9th Congress of Russian Naturalists and Physicians he gave a talk on the motion of a vortex near a knife-edge submerged in a fluid; later he published the paper [5] on this topic in Matematicheskii Sbornik. In 1899 Goryachev, a student of Joukowsky, formalized a master’s thesis in which he studied some cases of the motion of four (see Ch. II in [6]), five (see Ch. III in [6]) and $2n+1$ vortices (see Ch. IV in [6]). In the second half of the 20th century the concept of vortex motion found a number of applications in new areas of physics. As a good example here, one can mention the vortex structures discovered in a Bose–Einstein condensate (see, for example, [7] and [8]) and also magnetic vortices (see, for example, [9]–[11]), with which we are concerned in this paper. In particular, [11] presents Gröbli’s solution for the problem of three magnetic vortices. Vortex models attract attention because they have a large number of motion integrals and can frequently be written in the Hamiltonian form. Owing to their properties, integrable Hamiltonian systems yield to investigation by a variety of effective analytical tools. For example, Smale, Kharlamov, Fomenko, Zieschang, Oshemkov, Bolsinov and others (see [12]–[20]) developed a rich theory of topological classification of completely Liouville integrable Hamiltonian systems. One of the main approaches to their phase topology is an analysis of the momentum map (or integral map), which establishes a correspondence between points in the phase space and the set of values of the first integrals. By the Arnold–Liouville theorem the inverse images of regular values of this map (regular common levels of the first integrals) are smooth invariant manifolds. The connected components of the inverse images of all values of the integral map form the Liouville foliation in the phase space. Critical points of the momentum map lie on singular leaves through which regular integral manifolds bifurcate. For integrable systems with $n$ degrees of freedom and with compact leaves, the inverse image of a regular value of the momentum map is a disjoint union of $n$-dimensional Liouville tori. In the case of integrable systems with two degrees of freedom bifurcations of two-dimensional tori can conveniently be described in terms of 3-atoms, which are neighbourhoods of singular leaves considered up to leafwise homeomorphisms. In actual integrable systems of classical mechanics and mathematical physics 3-atoms of the form $A$, $B$, $A^*$, $C_2$ or $D_1$ are the most frequent ones. The topology of various vortex models was studied, for example, by Sokolov and Ryabov [21]–[24]. The topology of the classical problem of motion of three point vortices in a perfect fluid was examined by Selivanova [25]. In the present paper we continue the bifurcation analysis of one generalized system of vortex dynamics — this system covers the model of motion of three vortex filaments in an unbounded perfect fluid and the model of three magnetic vortices in a ferromagnetic medium (see [26]). This system is subject to the constraint that one of the vortices is fixed at the origin; the resulting system has two degrees of freedom and preserves its Liouville integrability. In this paper we prove some general results, which simplify substantially the analysis of the phase topology of a system for any set of parameters. In particular, we prove the presence of bifurcations of Liouville tori of the form $\mathbb T^2 \to 3\mathbb T^2$ and $2\mathbb T^2 \to 2\mathbb T^2$, which pass through the singular leaf $\mathbb S^1 \times (\mathbb S^1 \,\dot{\cup}\, \mathbb S^1 \,\dot{\cup}\, \mathbb S^1)$ and can be described by the 3-atoms $D_1$ and $D_2$, respectively. As already noted, the atom $D_1$ is slightly more common and was found, for example, in the Goryachev–Chaplygin–Sretenskii integrable case of rigid body dynamics (see [16]) and also in another generalized problem of vortex dynamics (see [22]). The bifurcation of tori described by the atom $D_2$ is quite rare and was previously found numerically by Moskvin [27] in his study of the topology of the Dullin–Matveev integrable case on the two-dimensional sphere (see [28]). Vedyushkina and Kharcheva realized the 3-atom $D_2$ by a billiard book (see [29], § 8, Example 1); this was an illustration of the prospects for modelling any nondegenerate three-dimensional bifurcations of Liouville tori by a new class of billiards. For the above generalized system of three vortices with a constraint we discovered earlier [26] cases of noncompact common levels of the first integrals and found noncritical bifurcations of two-dimensional integral surfaces. In the present paper we analyze the topology of three-dimensional isoenergy manifolds (surfaces of constant level of the Hamiltonian of the system) for some subsets of values of the parameters. Such manifolds are found to be noncompact, despite the fact that all leaves of the corresponding Liouville foliation are compact. In particular, this results in noncritical bifurcations of Liouville tori of the same form as in [26]. Noncompact Liouville foliations were extensively studied in recent years. For a detailed account of key problems in this direction, see the survey by Fedoseev and Fomenko [30], which was motivated by the discovery of a wide range of systems whose invariant phase surfaces are not always compact. In this regard we mention the natural Hamiltonian systems on Bertrand manifolds, which were examined by Fedoseev in [31]. Fedoseev and Fomenko [30] also introduced the concept of an augmented bifurcation diagram, which generalizes a bifurcation diagram of the momentum map and includes the images of both critical and (possibly empty) noncritical singular leaves. Some other recent studies on this topic are also worth mentioning. For example, it has been discovered that a Liouville foliation with noncompact singularities appears in pseudo-Euclidean analogues of mechanical systems introduced by Borisov and Mamaev. In particular, for the analogue of the Kovalevskaya system, a criterion for the presence of a noncompact leaf in the case of a nonzero area constant was put forward [32], and noncompact integral manifolds and their bifurcations on the zero level of the Kovalevskaya integral were studied [33]. The topology of a pseudo-Euclidean Euler top was considered by Altuev and Kibkalo [34], who constructed an augmented bifurcation diagram and studied bifurcations of compact and noncompact leaves (including, in particular, noncritical ones). It should also be noted that the billiard systems mentioned above are capable of modelling some noncompact bifurcations if billiards on unbounded tables are considered. Such systems were studied in detail in [35]. For the problem of the description of the topology of Hamiltonian systems with noncompact leaves, we also mention Nikolaenko [36] and [37], who obtained a classification of a wide class of nondegenerate noncompact singularities of completely Liouville integrable systems with one or two degrees of freedom, respectively. § 2. The model and definitions2.1. Magnetic vorticesIn the planar problem of motion of $N$ magnetic vortices in a ferromagnetic medium, a vortex has coordinates $r_\alpha=(x_\alpha,y_\alpha)$, where $\alpha=1,\dots, N$, nonzero intensity $\Gamma_\alpha$ and polarity $\lambda_\alpha$, which assumes the value $+1$ or $-1$ depending on the magnetization direction of the vortex. The motion equations in the Hamiltonian form are as follows:
$$
\begin{equation}
H=\sum_{\alpha<\beta}^{N} \frac{\Gamma_\alpha \Gamma_\beta}{\lambda_\alpha \lambda_\beta}\log |r_\alpha-r_\beta|, \qquad \Gamma_\alpha \dot{x}_\alpha=\frac{\partial H}{\partial y_\alpha}, \quad \Gamma_\alpha \dot{y}_\alpha=-\frac{\partial H}{\partial x_\alpha},
\end{equation}
\tag{2.1}
$$
where $|r_\alpha|=\sqrt{x_\alpha^2+y_\alpha^2}$. Equations (2.1) are generalizations of the equations written by Kirchhoff in [3], because they take the vortex polarities into account. In the case when $\lambda_\alpha=\lambda_\beta$ for all $(\alpha, \beta)$, system (2.1) reduces to its hydrodynamic analogue. The additional motion integrals are independent of the vortex polarities and coincide with the integrals obtained by Kirchhoff:
$$
\begin{equation*}
P_x=\sum_{\alpha=1}^{N}\Gamma_\alpha x_\alpha, \qquad P_y=\sum_{\alpha=1}^{N}\Gamma_\alpha y_\alpha\quad\text{and} \quad F=\sum_{\alpha=1}^{N}\Gamma_\alpha (x_\alpha^2+y_\alpha^2).
\end{equation*}
\notag
$$
The Poisson bracket
$$
\begin{equation}
\{f,g\}=\sum_{\alpha=1}^{N} \frac{1}{\Gamma_\alpha}\biggl( \frac{\partial f}{\partial x_\alpha}\,\frac{\partial g}{\partial y_\alpha} - \frac{\partial f}{\partial y_\alpha}\,\frac{\partial g}{\partial x_\alpha} \biggr)
\end{equation}
\tag{2.2}
$$
is defined on the system phase space
$$
\begin{equation*}
\mathcal{P}_N=\bigl\{ (r_1,\dots,r_N) \mid r_\alpha \neq r_\beta \bigr\}.
\end{equation*}
\notag
$$
For $N=3$, (2.1) is a completely Liouville integrable system with three degrees of freedom because one can find the triple of integrals $(H, F, P_x^2+P_y^2)$ which are pairwise in involution with respect to the Poisson bracket (2.2). The first motion integrals for the Hamiltonian system describing the dynamics of magnetic vortices were obtained by Papanicolaou and Tomaras, who identified in [9] the conservation laws corresponding to analogues of the angular and linear momenta. The generalized system (2.1), which we consider here, has been obtained on the basis of the later paper [11] by Komineas and Papanicolaou, in which a system of magnetic vortices was derived using the method of collective variables. This approach, which even dates back to Thiele [38], is widely useful in the dynamics of magnetic structures. In [11] Komineas and Papanicolaou also showed, by adapting Gröbli’s solution, that the generalized Helmholtz–Kirchhoff equations are integrable in the case of three magnetic vortices. We let
$$
\begin{equation*}
M=\sum_{\alpha=1}^{N} \Gamma_\alpha(x_\alpha \dot{y}_\alpha-y_\alpha \dot{x}_\alpha)
\end{equation*}
\notag
$$
denote the total vortical moment. Let us show that it is constant in the case of magnetic vortices. Theorem 1. The total vortex moment $M$ is a constant of motion of the Hamiltonian system (2.1). Proof. Our argument generalizes that of Poincaré (see [2], Ch. 4, § 76). Consider the function
$$
\begin{equation*}
\ell_{\alpha,\beta}=\sqrt{(x_\alpha-x_\beta)^2+(y_\alpha-y_\beta)^2}.
\end{equation*}
\notag
$$
It is a positive homogeneous function of degree 1 with respect to the variables $(x_\alpha, y_\alpha, x_\beta, y_\beta)$. By Euler’s theorem,
$$
\begin{equation*}
\ell_{\alpha,\beta}= x_\alpha \frac{\partial\ell_{\alpha,\beta}}{\partial x_\alpha}+ y_\alpha \frac{\partial\ell_{\alpha,\beta}}{\partial y_\alpha}+ x_\beta \frac{\partial\ell_{\alpha,\beta}}{\partial x_\beta}+ y_\beta \frac{\partial\ell_{\alpha,\beta}}{\partial y_\beta}.
\end{equation*}
\notag
$$
Dividing by $\ell_{\alpha,\beta}$ we have
$$
\begin{equation*}
x_\alpha \frac{\partial \log\ell_{\alpha,\beta}}{\partial x_\alpha}+ y_\alpha \frac{\partial \log\ell_{\alpha,\beta}}{\partial y_\alpha}+ x_\beta \frac{\partial \log\ell_{\alpha,\beta}}{\partial x_\beta}+ y_\beta \frac{\partial \log\ell_{\alpha,\beta}}{\partial y_\beta}=1.
\end{equation*}
\notag
$$
Next, multiplying by $(\Gamma_\alpha \Gamma_\beta)/(\lambda_\alpha \lambda_\beta)$ and summing we find that
$$
\begin{equation*}
\sum_{\alpha<\beta}^{N} \frac{\Gamma_\alpha \Gamma_\beta}{\lambda_\alpha \lambda_\beta}= \sum_{\alpha=1}^{N} \biggl( x_\alpha \frac{\partial H}{\partial x_\alpha}+ y_\alpha \frac{\partial H}{\partial y_\alpha} \biggr).
\end{equation*}
\notag
$$
Now the required result
$$
\begin{equation*}
M= \sum_{\alpha=1}^{N} \Gamma_\alpha(x_\alpha \dot{y}_\alpha-y_\alpha \dot{x}_\alpha)= -\sum_{\alpha<\beta}^{N} \frac{\Gamma_\alpha \Gamma_\beta}{\lambda_\alpha \lambda_\beta}=\mathrm{const}
\end{equation*}
\notag
$$
follows by the substitution of (2.1).
This proves the theorem. It is worth pointing out that, despite it is constant, the total vortex moment $M$ is not a new additional integral of motion because it is obtained from the Hamiltonian. However, $M$ describes an interesting property of the motion of vortices. As we will see below, the value of the total vortical moment has an effect on the phase topology of the system. 2.2. Constrained three-vortex systemConsider a special case of the three-vortex problem. Namely, we study the motion of two unconstrained vortices with intensities $\widetilde{\Gamma}_1$ and $\widetilde{\Gamma}_2$, with a vortex of intensity $\widetilde{\Gamma}_0$ fixed at the origin. Changing time and passing to the relative intensities $\Gamma_\alpha=\widetilde{\Gamma}_\alpha/\widetilde{\Gamma}_0$ and relative polarities $\lambda_\alpha=\widetilde{\lambda}_\alpha/\widetilde{\lambda}_0$ the motion equations can be written in the Hamiltonian form
$$
\begin{equation}
\Gamma_\alpha\dot{x}_\alpha=\frac{\partial H}{\partial y_\alpha}, \quad \Gamma_\alpha\dot{y}_\alpha=-\frac{\partial H}{\partial x_\alpha}, \qquad \alpha=1,2,
\end{equation}
\tag{2.3}
$$
with Hamiltonian
$$
\begin{equation}
H=\frac{\Gamma_1}{\lambda_1}\log |r_1|+ \frac{\Gamma_2}{\lambda_2}\log |r_2|+ \frac{\Gamma_1\Gamma_2}{\lambda_1\lambda_2}\log |r_1-r_2|.
\end{equation}
\tag{2.4}
$$
Now we omit the word ‘relative’ when speaking about parameters of system (2.3). In addition to the energy integral (2.4), the system features the additional first integral of motion which is an analogue of the angular momentum. Together with $H$, these integrals form a complete set of first integrals on the phase space
$$
\begin{equation*}
\mathcal{P}_{\mathrm{c}}=\bigl\{(r_1, r_2) \mid r_1^2 \neq 0, \,r_2^2 \neq 0, \,r_1 \neq r_2 \bigr\},
\end{equation*}
\notag
$$
which commute with respect to the Poisson bracket (2.2). So (2.3) is a completely Liouville integrable system with two degrees of freedom. The total vortex moment of the generalized system of three vortices with constraint has the form
$$
\begin{equation}
M_{\mathrm{c}}=\Gamma_1 (x_1 \dot{y}_1-y_1 \dot{x}_1) +\Gamma_2 (x_2 \dot{y}_2-y_2 \dot{x}_2) =- \frac{\Gamma_1}{\lambda_1} -\frac{\Gamma_2}{\lambda_2} -\frac{\Gamma_1\Gamma_2}{\lambda_1\lambda_2}.
\end{equation}
\tag{2.6}
$$
A problem statement of this type, which is widely useful in hydrodynamics, was inspired by the phenomenon of topographical vortices appearing over mountains in the ocean and in the atmosphere (see [39]). The presence of topographical irregularities has a marked effect on the dynamics of the whole system, which leads to interest in problems with a fixed vortex. For some properties of the dynamics of a vortex pair in a perfect fluid with a fixed vortex, see, for example, [40] and [41]. We are interested in the more general problem of the phase topology of the above generalized vortex system. Some machinery is required for dealing with this problem. 2.3. Necessary definitions and theoremsIn this section we present the main definitions and theorems used below. Definition 1. Consider a differential 2-form $\omega$ on a smooth manifold $\mathcal{P}$. Let $\Omega(\boldsymbol{x})=(\omega_{ij}(\boldsymbol{x}))$ be the matrix of this form. Assume that $\omega$ Then this form (and the underlying manifold) is called symplectic. By definition a symplectic manifold is even-dimensional. In what follows we denote such objects by $\mathcal{P}^{2n}$. On $(\mathcal{P}^{2n}, \omega)$ we consider an arbitrary function $f \in C^\infty(\mathcal{P}^{2n})$ and define its skew-symmetric gradient vector by
$$
\begin{equation*}
(\operatorname{sgrad} f)^i=\omega^{ij} \frac{\partial f}{\partial x^j},
\end{equation*}
\notag
$$
where the $\omega^{ij}$ are elements of the matrix $\Omega^{-1}$ and the Einstein summation convention is adopted. For arbitrary smooth functions $f$ and $g$ on $\mathcal{P}^{2n}$ the Poisson bracket defined by a nondegenerate symplectic structure $\omega$ is
$$
\begin{equation*}
\{ f, g \}=\omega^{ij} \frac{\partial f}{\partial x^i}\,\frac{\partial g}{\partial x^j}.
\end{equation*}
\notag
$$
Definition 2. Functions $F_i$ and $F_j$ on a symplectic manifold $(\mathcal{P}^{2n}, \omega)$ with a prescribed Poisson bracket $\{ \,\cdot\, {,} \,\cdot\, \}$ are said to be in involution if $\{ F_i, F_j \} \equiv 0$. A nonconstant function $F_i \colon \mathcal{P}^{2n} \to \mathbb R$ is a first integral of a dynamical system $\dot{\boldsymbol{x}}=v(\boldsymbol{x})$, $\boldsymbol{x} \in \mathcal{P}^{2n}$, if $F_i(\boldsymbol{x}(t)) \equiv \mathrm{const}$ for all motions $\boldsymbol{x}(t)$. A vector field of the form $v=\operatorname{sgrad} H$ is called a Hamiltonian vector field, and the function $H$ itself is the Hamiltonian of the system. It is known that a function $F_i$ is a first integral of a Hamiltonian system $v$ with Hamiltonian $H$ if and only if $H$ and $F_i$ are in involution. Definition 3. A Hamiltonian system $v$ on a symplectic manifold $(\mathcal{P}^{2n}, \omega)$ is a completely Liouville integrable system with $n$ degrees of freedom if there exists a set of $n$ functionally independent pairwise involutive first integrals $F_1, \dots, F_n$ and the vector fields $\operatorname{sgrad} F_i$ are complete. An important property of a completely Liouville integrable system $v$ with Hamiltonian $H=F_1$ is the integrability of its canonical equations by quadratures. The partition of a manifold $\mathcal{P}^{2n}$ into the connected components of common level surfaces of the first integrals $F_1, \dots, F_n$ is called a Liouville foliation. The first integrals are preserved by the Hamiltonian flow, and so each leaf of this foliation is an invariant surface of the system. In the general case a common level of the first integrals
$$
\begin{equation*}
\mathcal{T}_f=\bigl\{\boldsymbol{x} \in \mathcal{P}^{2n} \mid F_i(\boldsymbol{x})=f_i, \, i=1,\dots,n\bigr\}
\end{equation*}
\notag
$$
is a disjoint union of some leaves. A leaf (connected component of $\mathcal{T}_f$) is called regular if the $dF_i$ are linearly independent at each point on the leaf and the Liouville foliation is trivial in a small invariant neighbourhood of this leaf. Otherwise, the leaf is called singular. Theorem 2 (Liouville’s theorem). Consider a completely Liouville integrable system with $n$ degrees of freedom on a symplectic manifold $\mathcal{P}^{2n}$. Let $F_1, \dots, F_n$ be first integrals of this system and $\mathcal{T}_f$ be a regular common level set. Then: Liouville’s theorem, which is well known in analytic mechanics, underlies the study of the phase topology of Liouville integrable Hamiltonian systems. Properties of the momentum map (or integral map)
$$
\begin{equation*}
\mathcal{F}(\boldsymbol{x})=(F_1(\boldsymbol{x}), \dots, F_n(\boldsymbol{x}))
\end{equation*}
\notag
$$
are important in the bifurcation analysis of invariant integral manifolds. In the case of integrable systems with two degrees of freedom this map has the form
$$
\begin{equation*}
\mathcal{F} \colon \mathcal{P}^4 \to \mathbb R^2, \qquad \mathcal{F}(\boldsymbol{x})=(F(\boldsymbol{x}), H(\boldsymbol{x})).
\end{equation*}
\notag
$$
The set of its critical points of rank $m<n$ is denoted by
$$
\begin{equation*}
\mathcal{C}^m=\bigl\{ \boldsymbol{x} \mid \operatorname{rank} d\mathcal{F}(\boldsymbol{x})=m \bigr\}.
\end{equation*}
\notag
$$
The set of all critical points of the momentum map is the union of its strata; that is, $\mathcal{C}=\bigcup_{m}\mathcal{C}^m$. It is clear that in integrable systems with two degrees of freedom (that is, for $n=2$) only critical points of rank 0 and 1 are possible. Definition 4. The bifurcation diagram of the momentum map is the set of its critical values $\Sigma=\mathcal{F}(\mathcal{C} \cap \mathcal{P}^4)$. It was shown earlier (see [26]) that, in addition to critical bifurcation curves, this problem involves also noncritical ones, over which a bifurcation of invariant phase manifolds takes place without decreasing the rank of the momentum map. Let $\Sigma_0$ denote the noncritical bifurcation set. Following [30], the union ${\Sigma'=\Sigma \cup \Sigma_0}$ is called the augmented bifurcation diagram. A chamber of the augmented bifurcation diagram is a path-connected component of the set $\mathbb R^2(f,h)/\Sigma'$, that is, a two-dimensional open domain on the plane. A singular point of the diagram $\Sigma'$ is a point of intersection or tangency, a cusp point on a bifurcation curve, or an isolated point. Definition 5. The set of points defined by the equation $H(\boldsymbol{x})=\mathrm{const}$ is known as an isoenergy surface. For an integrable system with two degrees of freedom we use the notation
$$
\begin{equation*}
Q^3_h=\{\boldsymbol{x} \in \mathcal{P}^4 \mid H(\boldsymbol{x})=h\}.
\end{equation*}
\notag
$$
Note that $Q^3_h$ is a three-dimensional manifold invariant under the Hamiltonian field $v=\operatorname{sgrad} H$. An isoenergy surface is said to be nonsingular if $dH \neq 0$ everywhere on $Q^3_h$. An integrable Hamiltonian system is called a topologically stable system on $Q^3_{h_0}$ if sufficiently small variations of the energy level $h_0$ do not change the structure of the Liouville foliation of the system, that is, the systems $(v,Q^3_{h_0})$ and $(v,Q^3_{h_0+\varepsilon})$ are Liouville equivalent for sufficiently small $\varepsilon$. Two systems are called Liouville equivalent if there exists a leafwise diffeomorphism $Q_1^3 \to Q_2^3$ which preserves the orientation of the manifolds $Q_1^3$ and $Q_2^3$, and also the orientation of all critical circles of the additional integral. Definition 6. We say that $F$ is a Bott function on the manifold $Q_h^3$ if all of its critical points are organized into nondegenerate critical submanifolds. By nondegeneracy we mean here that of the form $d^2F$ on a subspace transversal to the submanifold (in other words, the restriction of $F$ to this transversal is a Morse function). It is also known (see [19], Ch. 1, Proposition 1.15) that in the case of a compact nonsingular isoenergy surface $Q_h^3$ such submanifolds are diffeomorphic either to a circle, a torus or to a Klein bottle. In our problem (as in the majority of real physical and mechanical problems) all critical submanifolds are diffeomorphic to $\mathbb S^1$, as we demonstrate below. A consideration of critical points of the momentum map on the whole of $\mathcal{P}^4$ suggests the following practical definition of the nondegeneracy of a point of rank 1. Definition 7. Let $\boldsymbol{x} \in \mathcal{C}^1$. Then there exist $\lambda$ and $\mu$ such that Consider the symmetric 2-form on the three-dimensional subspace orthogonal (in the sense of the symplectic form) to the subspace generated by the linearly dependent vectors $\operatorname{sgrad} F$ and $\operatorname{sgrad} H$. A point $\boldsymbol{x}$ is a nondegenerate point of the momentum map $\mathcal{F}$ if the rank of this 2-form is 2. The index of a nondegenerate critical circle is the number of negative eigenvalues of this form, that is, the number of independent directions in which the function $F$ is decreasing. So the index of a maximal (respectively, minimal) critical circle is $2$ (respectively, $0$). The index $1$ corresponds to a saddle-type singularity. Definition 8. A 3-atom is the class of Liouville equivalence of a small neighbourhood of a singular leaf of the Liouville foliation on a nonsingular isoenergy surface $Q^3_h$. The complexity of a compact 3-atom is the number of critical circles on its singular leaf. A compact 3-atom can be elliptic (the case of a minimal or a maximal critical circle) or saddle (containing only critical circles of saddle type). There exists only one compact elliptic 3-atom $A$; this atom has complexity 1 and is a solid torus foliated by tori and a single singular circle. The remaining compact 3-atoms, which are of saddle (hyperbolic) type, are subdivided into two types. In the first type a saddle 3-atom is obtained as the Cartesian product of some compact orientable saddle 2-atom and the circle $\mathbb S^1$ (in this case the letter notation for the atom is usually preserved). By a 2-atom one means here a small neighbourhood of a critical leaf of a Morse function considered up to a leafwise homeomorphism, that is, the topological type of a singularity of a Morse function. The second type of compact saddle 3-atoms consists of the so-called atoms with stars, which are nondegenerate singularities with nonorientable separatrix diagram. For a detailed account of these objects and also for an algorithm of the construction of a complete list of 2- and 3-atoms, see [19]. It should be noted that the problem under consideration involves only atoms without stars. The atoms describing the bifurcations of Liouville tori play an important role in the classification of integrable Hamiltonian systems. Two integrable Hamiltonian systems are called roughly Liouville equivalent if there exists a homeomorphism between the bases of the corresponding Liouville foliations which can locally (in a neighbourhood of each point of the base) be lifted to a leafwise homeomorphism of the Liouville foliations. So rough Liouville equivalence is a weakening of Liouville equivalence. A rough molecule (a Fomenko invariant) is the Kronrod–Reeb graph of the function $F$ on $Q^3_h$, whose vertices carry 3-atoms describing bifurcations of two-dimensional Liouville tori. According to Fomenko’s theorem (see [19], Ch. 3, Theorem 3.5), this graph is a complete invariant of rough Liouville equivalence. There also exists a complete classifying invariant of Liouville equivalence, namely, a marked molecule (the Fomenko–Zieschang invariant). Such a molecule is constructed by endowing the Fomenko invariant with a set of numerical marks. In the present paper we are only concerned with the description of bifurcations of integral manifolds and the construction of rough molecules. § 3. Critical circles and bifurcation diagram3.1. A parametrization of the critical circlesIn [26] the set of critical points $\mathcal{C}$ of the momentum map $\mathcal{F}$ was found; this set consists only of critical points of rank 1:
$$
\begin{equation*}
\mathcal{C}=\mathcal{C}^0 \cup \mathcal{C}^1,\qquad \mathcal{C}^0=\varnothing\quad\text{and} \quad \mathcal{C}^1=\bigl\{(x_1, y_1, x_2, y_2)\mid E_1=0,\, E_2=0 \bigr\},
\end{equation*}
\notag
$$
where and
$$
\begin{equation*}
\begin{aligned} \, E_2 &=x_1x_2\bigl[(x_1-x_2)^2+(y_1-y_2)^2\bigr] \bigl[\lambda_2(x_2^2+y_2^2)-\lambda_1(x_1^2+y_1^2)\bigr] \\ &\qquad +(x_1^2+y_1^2)(x_2^2+y_2^2)(x_1-x_2)(\Gamma_1x_1+\Gamma_2x_2). \end{aligned}
\end{equation*}
\notag
$$
In the polar coordinates
$$
\begin{equation*}
x_1=\rho_1\cos\theta_1, \quad y_1=\rho_1\sin\theta_1, \quad x_2=\rho_2\cos\theta_2, \quad y_2=\rho_2\sin\theta_2,
\end{equation*}
\notag
$$
the set $\mathcal{C}$ is determined by the following two systems
$$
\begin{equation}
\begin{cases} W(\rho_1, \rho_2)=0, \\ \theta_1=\pi+\theta_2, \end{cases}\quad\text{and} \quad \begin{cases} W(\rho_1, -\rho_2)=0, \\ \theta_1=\theta_2, \end{cases}
\end{equation}
\tag{3.1}
$$
where
$$
\begin{equation*}
W(\rho_1, \rho_2)=(\rho_{1}+\rho_{2}) (\lambda_{1}\rho_{1}^{2}-\lambda_{2}\rho_{2}^{2}) +\rho_{1}\rho_{2}(\Gamma_{1}\rho_{1}-\Gamma_{2}\rho_{2})
\end{equation*}
\notag
$$
is a homogeneous polynomial of degree $3$ with respect to $(\rho_1, \rho_2)$. Lemma 1. Any critical submanifold of system (2.3) with Hamiltonian (2.4) is diffeomorphic to the circle $\mathbb S^1$. Proof. Consider the set of critical points (3.1). Let $k=\rho_2 / \rho_1$. Then the equation $W(\rho_1, k \rho_1)=0$ can be written as
$$
\begin{equation}
\lambda_2 k^3 +(\Gamma_2+\lambda_2) k^2 -(\Gamma_1+\lambda_1) k -\lambda_1=0.
\end{equation}
\tag{3.2}
$$
To each set of parameters of system (2.3) there correspond three roots of equation (3.2). The positive real roots $k_i=k_i(\Gamma_1, \Gamma_2, \lambda_1, \lambda_2)$ are included in the case $\theta_1=\pi+\theta_2$ and the negative real roots in the case $\theta_1=\theta_2$, with the exception of $k_i=-1$ because $(\rho_1,\theta_1)\neq(\rho_2,\theta_2)$. In this case the critical submanifolds can be parametrized as follows
$$
\begin{equation}
\Pi_i(t_i) \colon \begin{cases} x_1=- \operatorname{sgn} k_i \cos\theta_2t_i, \quad x_2=|k_i|\cos\theta_2t_i, \\ y_1=- \operatorname{sgn} k_i \sin\theta_2t_i, \quad y_2=|k_i|\sin\theta_2t_i, \end{cases} \qquad \theta_2 \in [0,2\pi),
\end{equation}
\tag{3.3}
$$
where the parameter $t_i > 0$ controls the choice of the common level set of the first integrals. For fixed $t_i^0$ the parametrization (3.3) gives a circle.
This proves the lemma. An arrangement of vortices at a critical point $\boldsymbol{x}_0 \in \mathcal{C}$ for $k_i>0$ is shown in Figure 1. From Lemma 1 it follows that the critical points of the integral $F$ on $Q_h^3$ cannot be organized into critical tori or Klein bottles. If the critical circles are nondegenerate, then the bifurcations of two-dimensional Liouville tori can be described by 3-atoms. Using parametrization (3.3) one can ascertain analytically some important properties of the critical submanifolds. This simplifies significantly the analysis of the topology of the Liouville foliation. 3.2. The stability of critical motionsWith the help of the parametrization (3.3) one can find the stability types of the critical trajectories. For convenience we write equation (3.2) as where
$$
\begin{equation}
A(k)=\Gamma_1 k+\lambda_1(k+1)\quad\text{and} \quad B(k)=\Gamma_2+\lambda_2(k+1).
\end{equation}
\tag{3.4}
$$
We use the following notation: $A_i=A(k_i)$ and $B_i=B(k_i)$. Note that that is, this equivalence holds at the critical points corresponding to a root of the form $k_i^0=-\Gamma_2\lambda_2-1$ of (3.2). In addition, the root $k_i^0$ is possible only for $M_{\mathrm{c}}=0$. So in the case of a nonzero total vortex moment (2.6) each surface $Q_h^3$ is nonsingular. A similar conclusion of nonsingular isoenergy surfaces was made in [25] for the system of three vortices in a perfect fluid.Consider the following numerical characteristic of the critical circle: The following result holds.Theorem 3. The singular periodic motions $\Pi_i$ have the elliptic (hyperbolic) stability type if $C_i>0$ (respectively, $C_i<0$). The singularity is degenerate if $C_i=0$. Proof. Consider the function containing the unknown factor $\lambda$; the value of this function is constant along each trajectory. Using the parametrization (3.3) we can find $\lambda$ such that the critical points $\boldsymbol{x}_0 \in \mathcal{C}^1$ are fixed under the flow $\operatorname{sgrad} L$, that is, Since and since $dF$ and $dH$ are linearly dependent at each critical point of rank 1, the factor is well defined. The linearization of the Hamiltonian field $\operatorname{sgrad} L$ at the point $\boldsymbol{x}_0$ is given by the symplectic operator $A_L$, whose characteristic polynomial has the form
We have $\operatorname{sgn} c_i=\operatorname{sgn} C_i$, and so the sign of the constant $C_i$ controls the stability type of the critical trajectories. In the case $C_i>0$ the critical circle is a stable manifold, which is the limit of a concentric family of regular two-dimensional tori. In the case $C_i<0$ the one-dimensional orbit is of unstable saddle type, that is, there exist motions which are asymptotic to this solution and lie on two-dimensional separatrix surfaces. If $C_i=0$, then the singularity is degenerate. This proves the theorem. For integrable systems with two degrees of freedom singularities of the integral map were analyzed in [19] by using characteristic polynomials. The relationship between the type of the singularity and the orbital stability of the solution was considered in [42]. 3.3. Indices of critical circlesWith the help of the parametrization (3.3) one can also determine analytically the indices of critical points of the function $F$ on a level surface of the integral $H$. If $dH \neq 0$ at a critical point $\boldsymbol{x}_0$ of rank 1, then
$$
\begin{equation*}
dF(\boldsymbol{x}_0)+\lambda^{-1} dH(\boldsymbol{x}_0)=0,
\end{equation*}
\notag
$$
where $\lambda$ is defined by (3.6). In this case we can use the approach from [19], Ch. 14, Lemma 14.1. Let $d^2H$ and $d^2F$ be the Hessians of the first integrals. Then one can consider the quadratic form
$$
\begin{equation}
\mathbf{G}=d^2F(\boldsymbol{x}_0)+\lambda^{-1}d^2H(\boldsymbol{x}_0)
\end{equation}
\tag{3.7}
$$
on the tangent vectors at the critical point. The quadratic form defined by the Hessian of the function $F|_{Q^3_h}(\boldsymbol{x}_0)$ is the restriction of the form (3.7) to the tangent space $T_{\boldsymbol{x}_0}Q_h^3$. To find the restriction of the form $\mathbf{G}$, consider the orthogonal complement to $dH(\boldsymbol{x}_0)$ and choose basis vectors in this subspace. For example,
$$
\begin{equation*}
\begin{gathered} \, e_1 =(\sin\theta_2,\,-\cos\theta_2,\,0,\, 0), \\ e_2 =(0,\,0,\,\sin\theta_2,\,-\cos\theta_2) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
e_3 =(\Gamma_2 k_i\cos\theta_2,\, \Gamma_2 k_i\sin\theta_2,\,\Gamma_1 \cos\theta_2,\,\Gamma_1 \sin\theta_2).
\end{equation*}
\notag
$$
The restriction of the quadratic form $\mathbf{G}$ to the tangent space $T_{\boldsymbol{x}_0}Q_h^3$ is given by the matrix
$$
\begin{equation}
\mathbf{G}_{Q_h^3}= \frac{2\Gamma_1\Gamma_2}{B_i(k_i+1)} \begin{pmatrix} k_i & 1 & 0 \\ 1 & k_i^{-1} & 0 \\ 0 & 0 & \dfrac{(k_i+1)^2}{k_i^3} C_i \end{pmatrix},
\end{equation}
\tag{3.8}
$$
where $C_i$ is defined by (3.5). The eigenvalues of $\mathbf{G}_{Q_h^3}$ are
$$
\begin{equation}
\mu_1=0, \qquad \mu_2=\frac{2\Gamma_1\Gamma_2}{B_i}\,\frac{k_i^2+1}{k_i(k_i+1)}\quad\text{and} \quad \mu_3=\frac{2\Gamma_1\Gamma_2}{B_i}\,\frac{k_i+1}{k_i^3} C_i.
\end{equation}
\tag{3.9}
$$
So the form (3.8) always has an eigenvalue $\mu_1=0$ and a nonzero eigenvalue $\mu_2 \neq 0$. We have $\operatorname{sgn} C_i=\operatorname{sgn}(\mu_3/\mu_2)$, and so in the case $C_i < 0$ the index of the critical point is 1, and in the case $C_i > 0$ the index is 0 or 2 (a minimal or a maximal critical circle, respectively), and in the case $C_i=0$ the critical point is degenerate (there are two zero eigenvalues), which also follows from Theorem 3. 3.4. The bifurcation diagram of the momentum mapConsider a system with nonzero total vortex moment (2.6). Then the following result holds. Theorem 4. If $M_{\mathrm{c}} \neq 0$, then the bifurcation diagram $\Sigma$ of the momentum map $\mathcal{F}$ on the plane of the first integrals $\mathbb R^2(f,h)$ is the union of the disjoint bifurcation curves Proof. For each family of critical circles $\Pi_i$ its image on the plane of first integrals $\Sigma_i=\mathcal{F}(\Pi_i) \subset \mathbb R^2(f,h)$ is a connected subset of critical values $\Sigma_i \subseteq \Sigma$. The parameter $t_i$ controls the position of the image $\Pi_i(t_i)$ on $\Sigma_i$. If $M_{\mathrm{c}} \neq 0$, then the $\Sigma_i$ are strictly monotone smooth curves defined by (3.10).
The total vortex moment $M_{\mathrm{c}}$ depends only on the parameters of the system and assumes the same value for all $\Sigma_i$. For $M_{\mathrm{c}} \neq 0$ all bifurcation curves have the same asymptote $f=0$, and $h_i(f) \to \operatorname{sgn}(M_{\mathrm{c}})\cdot\infty$ as $f \to 0$. The difference between the images of critical circles is reflected by $\eta_i$ (the ‘height’ of the curve) and $\operatorname{sgn} m_i$ (the half-plane where $\operatorname{sgn} f=\operatorname{sgn} m_i$), which depend both on the parameters of the system and the choice of the root $k_i$. Thus,
$$
\begin{equation}
\Sigma_i=\Sigma_j \quad\Longleftrightarrow\quad \eta_i=\eta_j \land \operatorname{sgn}(m_im_j)=1.
\end{equation}
\tag{3.11}
$$
In addition, we either have $\Sigma_i=\Sigma_j$, that is, the images of the critical circles coincide (we denote such a bifurcation curve by $\Sigma_{i,j}$), or $\Sigma_i \cap \Sigma_j=\varnothing$, that is, the bifurcation curves $\Sigma_i$ and $\Sigma_j$ have no common points.
This proves the theorem. If $\bigcup_{j \colon j \neq i} \Sigma_j \cap \Sigma_i=\varnothing$, then the singular leaf in the inverse image of a point $\Sigma_i^0=\Sigma_i(\Pi_i(t_i^0))$ contains one critical circle $\Pi_i(t_i^0)$. If the leaves in a neighbourhood of the singular leaf $\mathcal{F}^{-1}(\Sigma_i^0)$ are compact, then the corresponding bifurcation of Liouville tori can be described either by the elliptic 3-atom $A$ or by the saddle 3-atoms $B$ or $A^*$. If $\mathcal{F}(\Pi_i)=\mathcal{F}(\Pi_j)$ and $\Pi_i \neq \Pi_j$, then in the inverse images of points on the curve $\Sigma_{i,j}$ the critical circles $\Pi_i$ and $\Pi_j$ lie on the same common level. If both circles are nondegenerate and lie on the same connected component of the level, then a neighbourhood of such a leaf is described by a complicated 3-atom. If $M_{\mathrm{c}} \neq 0$, then the bifurcation diagram does not contain singular points, and the bifurcation curves are strictly monotone functions of $f$ with range $h \in (-\infty,+\infty)$. In the inverse image of any point on $\Sigma_i$ a neighbourhood of the singular leaf has the same type and does not depend on the choice of $t_i^0$. Leaves have the same topology throughout a chamber of the bifurcation diagram, and therefore the structure of the Liouville foliation is preserved on all isoenergy surfaces, that is, it does not depend on the choice of the energy level. In other words, the integrable Hamiltonian system is topologically stable on $Q^3_h$ for all $h$. Let us find out how the bifurcation diagram looks like for $M_{\mathrm{c}}=0$. As already pointed out, in this case the root $k_1^0=-\Gamma_2\lambda_2-1$ always exists, and $dH$ vanishes on the corresponding set of critical points. If the discriminant of equation (3.2) is negative, then the bifurcation diagram $\Sigma$ consists of the single horizontal interval
$$
\begin{equation*}
\Sigma=\Sigma_1=\biggl\{ (f,h) \biggm| f\operatorname{sgn} m_1 > 0, \, h=\frac{\eta_1}{2} \biggr\}.
\end{equation*}
\notag
$$
If the discriminant is positive, then $\Sigma=\Sigma_1 \cup \Sigma_2 \cup \Sigma_3$, where
$$
\begin{equation*}
\Sigma_2=\biggl( 0,\frac{\eta_2}{2} \biggr)\quad\text{and} \quad \Sigma_3=\biggl( 0,\frac{\eta_3}{2} \biggr).
\end{equation*}
\notag
$$
If the discriminant (3.2) is zero, then $\Gamma_1=-\Gamma_2=2\lambda_2$ and $\lambda_1\neq\lambda_2$, and the bifurcation diagram consists of the single isolated point
§ 4. Reduction to a system with one degree of freedom4.1. Equations of motionIn [26] system (2.3) was reduced to a Hamiltonian system with one degree of freedom by the change of coordinates
$$
\begin{equation}
\begin{gathered} \, x_1 =\frac{\operatorname{sgn}(\Gamma_1\Gamma_2)}{\sqrt{|\Gamma_1|}} [u\cos(\alpha)-v\sin(\alpha)], \qquad x_2=\frac{\cos(\alpha)}{\sqrt{|\Gamma_2|}}\sqrt{R(u,v)}, \\ y_1 =\frac{\operatorname{sgn}(\Gamma_1\Gamma_2)}{\sqrt{|\Gamma_1|}} [u\sin(\alpha)+v\cos(\alpha)], \qquad y_2=\frac{\sin(\alpha)}{\sqrt{|\Gamma_2|}}\sqrt{R(u,v)}, \end{gathered}
\end{equation}
\tag{4.1}
$$
where
$$
\begin{equation*}
R(u,v)=\operatorname{sgn}(\Gamma_1\Gamma_2)(f\operatorname{sgn}\Gamma_1-u^2-v^2).
\end{equation*}
\notag
$$
The choice of the variables $(u,v,\alpha)$ was suggested by the presence of the first integral $F$ which is invariant under the group of rotations $\operatorname{SO}(2)$. The existence of a one-parameter symmetry group makes it possible to reduce the original system to a system with one degree of freedom, as in mechanical systems with symmetry (see [16]). A similar substitution was applied with success to reduce other integrable Hamiltonian systems of vortex dynamics which admit the additional integral $F$ (see [22]–[24], and [43]). The inverse substitution
$$
\begin{equation}
U=\frac{\sqrt{|\Gamma_1|}\,(x_1x_2+ y_1y_2)}{\operatorname{sgn}(\Gamma_1\Gamma_2)\sqrt{x^2_2+y^2_2}}, \qquad V=\frac{\sqrt{|\Gamma_1|}\,(x_2y_1- x_1y_2)}{\operatorname{sgn}(\Gamma_1\Gamma_2)\sqrt{x^2_2+y^2_2}}
\end{equation}
\tag{4.2}
$$
provides the canonical variables with respect to the Poisson bracket (2.2),
$$
\begin{equation*}
\{U,V\}=-\{V,U\}=\operatorname{sgn}\Gamma_1, \qquad \{U,U\}=\{V,V\}=0.
\end{equation*}
\notag
$$
The reduced system remains a Hamiltonian system with respect to $(u,v)$:
$$
\begin{equation}
\dot{u}=\frac{\partial H_{\mathrm{r}}}{\partial v}\operatorname{sgn}\Gamma_1, \qquad \dot{v}=-\frac{\partial H_{\mathrm{r}}}{\partial u}\operatorname{sgn}\Gamma_1,
\end{equation}
\tag{4.3}
$$
with Hamiltonian
$$
\begin{equation}
\begin{aligned} \, \notag H_{\mathrm{r}} &=\frac{1}{2}\biggl\{\frac{\Gamma_1}{\lambda_1} \log\biggl[\frac{u^2+v^2}{|\Gamma_1|}\biggr] +\frac{\Gamma_2}{\lambda_2}\log \biggl[\frac{R(u,v)}{|\Gamma_2|}\biggr] \\ &\qquad +\frac{\Gamma_1\Gamma_2}{\lambda_1\lambda_2}\log\biggl[ \frac{|\Gamma_2|v^2+\bigl(\sqrt{|\Gamma_1| R(u,v)}- u\operatorname{sgn}(\Gamma_1\Gamma_2)\sqrt{|\Gamma_2|}\, \bigr)^2}{|\Gamma_1 \Gamma_2|} \biggr] \biggr\}. \end{aligned}
\end{equation}
\tag{4.4}
$$
The value of the parameter $f$ of the reduced system (4.3) corresponds to the level of the additional first integral $F$ (see (2.5)). The fixed points of the reduced system (4.3) are determined by critical points of the Hamiltonian (4.4). The phase space of the reduced system has the form
$$
\begin{equation*}
\mathcal{P}_{\mathrm{r}}=\{(u,v) \mid R(u,v) > 0\}\setminus\{w_0,w_1 \},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
w_0=(0, 0), \qquad w_1=(u_1, 0)\quad\text{and} \quad u_1=\operatorname{sgn}(\Gamma_1\Gamma_2) \sqrt{\frac{f|\Gamma_1|}{\Gamma_1+\Gamma_2}}.
\end{equation*}
\notag
$$
The points $w_0$ and $w_1$ are naturally excluded from the phase space $\mathcal{P}_{\mathrm{r}}$ because the point $w_0$ of the original system (2.3) corresponds to $r_1=(0,0)$, and the point $w_1$ corresponds to the prohibited position $r_1=r_2$ of the vortices. Lemma 2. The change from the phase variables $(x_1, y_1, x_2, y_2)$ to the variables $(u,v,\alpha)$ takes the critical circles (3.3) to fixed points of system (4.3) which lie on the straight line $v=0$. The fixed points of the reduced system determine the set of critical points $\mathcal{C}$ of the integral map $\mathcal{F}$ of the original system (2.3). Proof. A direct substitution of the parametrization (3.3) into the transition formulae (4.2) shows that the critical circles of the original system with two degrees of freedom are sent to the points with coordinates
$$
\begin{equation}
(u_i^*, v_i^*)=\bigl( -\operatorname{sgn}(\Gamma_1 \Gamma_2 k_i) \sqrt{|\Gamma_1|}t_i, 0 \bigr).
\end{equation}
\tag{4.5}
$$
Now let us show that (4.5) are fixed points of the reduced system (4.3). Indeed, using equation (3.2) and the fact that $f_i=(\Gamma_2 k_i^2+\Gamma_1)t_i^2$ at the critical point, we find that
$$
\begin{equation*}
\operatorname{sgrad} H_{\mathrm{r}}(u_i^*, v_i^*) \equiv (0,0).
\end{equation*}
\notag
$$
The converse result also holds: considering the set of fixed points of the reduced system $\mathcal{C}_{\mathrm{r}}=\{(u,v) \mid \operatorname{sgrad} H_{\mathrm{r}}=0\}$, using the inverse substitution (4.2) and simplifying we obtain the set of critical points $\mathcal{C}$ of the original integrable system with two degrees of freedom.
This proves the lemma. 4.2. Energy levels of the reduced systemLet $Q_h(f)$ be the energy level $h$ of the reduced Hamiltonian (4.4) for some value of the parameter $f$ of the additional integral, that is, Now we prove a statement reducing the analysis of the topology of the original system with two degrees of freedom to that of the reduced Hamiltonian system (4.3) with Hamiltonian (4.4).Theorem 5. The common level set $\mathcal{T}_{h,f}$ of the first integrals (2.4) and (2.5) of system (2.3) is diffeomorphic to the Cartesian product of the circle $\mathbb S^1$ and the energy level $Q_h(f)$ of the reduced energy (4.4). Proof. Consider the manifold
$$
\begin{equation*}
\mathcal{P}_\alpha=\bigl\{(u,v,\alpha,f) \mid (u,v) \in \mathcal{P}_{\mathrm{r}},\, \alpha \in \mathbb S^1, \, f \in \mathbb R\bigr\},
\end{equation*}
\notag
$$
the map $g \colon \mathcal{P}_{\mathrm{c}} \to \mathcal{P}_\alpha$ and the inverse map $g^{-1}$. Let $g$ be given by the functions (4.2) and (2.5), where $\alpha$ is the polar angle of the vortex with coordinates $(x_2,y_2)$. Hence (4.1) defines the inverse map $g^{-1}$. The Jacobian is distinct from zero everywhere, and $g$ and $g^{-1}$ are continuous and smooth on $\mathcal{P}_{\mathrm{c}}$ and $\mathcal{P}_\alpha$, respectively. So the map $g$ defines a diffeomorphism.
Now consider the common level set of the first integrals $\mathcal{T}_{h_0,f_0} \subset \mathcal{P}_{\mathrm{c}}$ of system (2.3) with two degrees of freedom. Its image
$$
\begin{equation*}
g(\mathcal{T}_{h_0,f_0})=\bigl\{ (u,v,\alpha, f_0) \mid (u,v) \in Q_{h_0}(f_0), \, \alpha \in \mathbb S^1 \bigr\}
\end{equation*}
\notag
$$
is the Cartesian product of the level set of the reduced energy (4.4) and the one-dimensional circle. So $\mathcal{T}_{h_0,f_0} \cong Q_{h_0}(f_0) \times \mathbb S^1$, which is the result required.
The theorem is proved. Corollary 1. It follows from Theorem 5, in particular, that the regular common level sets of the first integrals $H$ and $F$ of system (2.3) are represented by a disjoint union of several two-dimensional Liouville tori $\mathbb T^2=\mathbb S^1 \times \mathbb S^1$ and cylinders ${\mathrm{Cyl}=\mathbb R \times \mathbb S^1}$. Corollary 2. It also follows from Theorem 5 that a small neighbourhood of a singular leaf of system (2.3) with Hamiltonian (2.4) can be described by a 3-atom without stars, that is, as the Cartesian product of some 2-atom and a circle. The following lemma is useful in the evaluation of the number of Liouville tori in the inverse image of a regular value of the momentum map. Lemma 3. Any connected component of a compact regular energy level $Q_h(f)$ has two different points on the straight line $v=0$. Proof. By Liouville’s theorem (see Theorem 2) any compact energy level $Q_h(f)$ of system (4.3) is a disjoint union of several smooth manifolds diffeomorphic to $\mathbb S^1$. We have $H_{\mathrm{r}}(u,v)=H_{\mathrm{r}}(u,-v)$, and therefore the levels $Q_h(f)$ are symmetric about $v=0$. So, if a regular connected component intersects the level $v=0$, then the set of intersection consists of two different points.
Consider $Q_h(f)$ as $|h| \to \infty$. In the first case $h$ tends to infinity with sign $-\operatorname{sgn}(\Gamma_1\lambda_1)$ as $(u,v) \to w_0$. In the second case, as $R(u,v) \to 0$, the connected component shrinks towards the circle $u^2+v^2=f\operatorname{sgn}\Gamma_1$, and the value of the energy integral goes off to infinity with sign $-\operatorname{sgn}(\Gamma_2\lambda_2)$. In the third case the connected component shrinks to the point $w_1$ as $h$ goes off to infinity with sign $-\operatorname{sgn}(\Gamma_1\Gamma_2\lambda_1\lambda_2)$. In all these cases the regular connected components have a nonempty intersection with the straight line $v=0$. Now let us assume that there exists an energy level $h_0$ on which there exists a connected component $X \subset Q_{h_0}(f)$ such that $X \cap \{(u,v)\mid v=0\}=\varnothing$. Then there exist a finite energy level $h_1$ and a fixed point $(u_1^*,v_1^*) \in Q_{h_1}(f)$ such that $(u_1^*,v_1^*)$ lies inside the closed curve $X$. Hence $v_1^*\neq0$, and so the assumption is wrong by Lemma 2. This proves the lemma. The number of tori in the case of a compact $\mathcal{T}_{h,f}$ can conveniently be evaluated using the following approach. By Theorem 4, if $M_{\mathrm{c}}\neq0$, then we can consider an arbitrary energy level; for example, $h_0=0$. The choice of the chamber of the augmented bifurcation diagram depends on the choice of an arbitrary regular value $f_0$ of the additional integral in this chamber. So by Theorem 5 and Lemma 3, it suffices to evaluate the number of roots of the equation Each regular connected component has two points at the level $v=0$, and so the number of such roots is $2n$, where $n$ is the number of Liouville tori in the inverse image of $(f_0,h_0)$.It is worth pointing out that this approach can only be used if all leaves on the level under consideration are compact. The picture is more involved in the case of noncompact common levels of the first integrals because regular connected components can be diffeomorphic to $\mathbb S^1$ and $\mathbb R$, alike, as demonstrated in [26]. § 5. Bifurcation of one Liouville torus into three tori5.1. The bifurcation diagramConsider system (2.3) with parameters $\Gamma_1=\Gamma_2=\nu$ and $\lambda_1=\lambda_2=\delta$. Let $\kappa=\delta\nu$, and $\kappa \in (-\infty,-4)\cup(0,+\infty)$. In this case equation (3.2) has the three different admissible roots
$$
\begin{equation*}
k_1=1\quad\text{and} \quad k_{2,3}=-\frac12\bigl[\kappa+2 \pm \sqrt{\kappa(\kappa+4)}\bigr].
\end{equation*}
\notag
$$
A direct substitution shows that condition (3.11) is satisfied for $k_2$ and $k_3$, that is, the images of the corresponding critical circles $\Pi_2$ and $\Pi_3$ coincide on the plane of first integrals $\mathbb R^2(f,h)$. The total vortex moment is Hence by (3.10), for all bifurcation curves we have $h \to -\infty$ as $f \to 0$. We have $F=\nu(r_1^2+r_2^2)$, and therefore the bifurcation curves $\Sigma_1$ and $\Sigma_{2,3}$ lie in the left half-plane of $\mathbb R^2(f,h)$ for $\nu<0$ (see Figure 2, (a)), and in the right half-plane for ${\nu>0}$ (see Figure 2, (b)). In view of (3.4) the quantities $A_i$ and $B_i$ are as follows:
$$
\begin{equation*}
\begin{gathered} \, A_1=\delta(\kappa+2), \qquad A_{2,3}=-\frac{\delta}{2} \bigl[ \kappa(\kappa+3) \pm (\kappa+1)\sqrt{\kappa(\kappa+4)}\, \bigr], \\ B_1=A_1\quad\text{and} \quad B_{2,3}=\frac{\delta}{2}\bigl[\kappa \mp \sqrt{\kappa(\kappa+4)}\, \bigr]. \end{gathered}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
C_1=\kappa(\kappa+4)\quad\text{and} \quad C_{2,3}=-\frac{C_1}{2}(\kappa+2)\bigl[ (C_1+1)(\kappa+2) \pm (\kappa+1)(\kappa+3)\sqrt{C_1}\, \bigr].
\end{equation*}
\notag
$$
For this set of parameters we have $C_1>0$, $C_2<0$ and $C_3<0$. Hence by Theorem 3 the critical circles $\Pi_1$ have the elliptic (stable) type, and $\Pi_2$ and $\Pi_3$ are of hyperbolic (unstable) type. Using explicit formulae (3.9) for the evaluation of eigenvalues of $\mathbf{G}_{Q_h^3}$ we can also find the indices of critical circles. For $\nu>0$ the circles $\Pi_1$ have index 0 (minimal critical circles), and for $\nu<0$ the index is 2 (maximal circles). Index 1 corresponds to the saddle singularities $\Pi_2$ and $\Pi_3$. 5.2. Energy levels of the reduced systemFor the above values of the parameters the level set $Q_{h_0}(f_0)$ is given by the equation
$$
\begin{equation*}
z(\varphi-z)(\varphi-2u\sqrt{\varphi-z}\, )^\kappa=h_\kappa,
\end{equation*}
\notag
$$
where $z=u^2+v^2$, $\varphi=f_0\operatorname{sgn}\nu$, $h_\kappa=|\kappa|^{\kappa+2}e^{2h_0/\kappa}$, and we have $\varphi > z > 0$ and $h_\kappa>0$. The level of the reduced energy $Q_{h_0}(f_0)$ can be parametrized as follows:
$$
\begin{equation*}
u(z)=\frac{N_1(z)}{2 \sqrt[\kappa]{D(z)}} \quad\text{and} \quad v(z)=\pm \frac{\sqrt{N_2(z)}}{2 \sqrt[\kappa]{D(z)}},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
N_1(z)=\varphi \sqrt[\kappa]{z(\varphi-z)}-\sqrt[\kappa]{h_\kappa}, \qquad N_2(z)=4z D(z)^{2/\kappa}-N_1^2(z)
\end{equation*}
\notag
$$
and Consider the map $\mathcal{W}(z)=( u(z), v(z) )$ and its domain
$$
\begin{equation*}
\mathcal{Z}=\{ z \mid N_2(z) \geqslant 0,\,0<z<\varphi \}.
\end{equation*}
\notag
$$
Note that the function $N_2(z)$ is continuous on the interval $z \in (0, \varphi)$ and assumes negative values on its endpoints, that is,
$$
\begin{equation*}
\lim_{z \to 0^+} N_2(z) < 0\quad\text{and} \quad \lim_{z \to \varphi^-} N_2(z) <0.
\end{equation*}
\notag
$$
Hence there exist $n \geqslant 0$ closed intervals
$$
\begin{equation*}
\zeta_i=[z_{(2i-1)}, z_{(2i)}] \subset (0, \varphi), \qquad i=1,\dots,n,
\end{equation*}
\notag
$$
such that $N_2(\zeta_i) \geqslant 0$. Let $\mathcal{Z}_0=\{ z \mid N_2(z)=0 \}$. Then
$$
\begin{equation*}
\mathcal{Z}=\mathcal{Z}_0 \cup \mathcal{Z}_\zeta\quad\text{and} \quad \mathcal{Z}_\zeta=\bigcup_i \zeta_i.
\end{equation*}
\notag
$$
The set $\mathcal{Z}$ is compact, and therefore so is $\mathcal{W}(\mathcal{Z})=Q_{h_0}(f_0)$ as the image of a compact set under a continuous map. Hence by Theorem 2 the regular level $Q_{h_0}(f_0)$ is a disjoint union of 1-dimensional Liouville tori (circles). 5.3. A complicated 3-atom and its splittingConsider a fixed value of the parameter $\kappa=1$ and the energy level $h_0=0$. In this case
$$
\begin{equation*}
u(z)=\frac{N_1^1(z)}{2z(\varphi-z)^{3/2}}\quad\text{and} \quad v(z)=\pm \frac{\sqrt{N_2^1(z)}}{2z(\varphi-z)^{3/2}},
\end{equation*}
\notag
$$
where and
$$
\begin{equation*}
N_2^1(z)=-4z^6+12z^5\varphi -13z^4\varphi^2+6z^3\varphi^3- z^2\varphi(\varphi^3+2)+2z\varphi^2-1.
\end{equation*}
\notag
$$
The zero set of the discriminant of $N_2^1(z)$ determines the critical values of the parameter of the additional integral: $f_1=\sqrt[3]{2}\operatorname{sgn}\nu$ and $f_{2,3}=3\operatorname{sgn}\nu$. In the first case the level set consists of a single fixed point with coordinates $(-2^{-1/3}, 0)$, and in the second it has the form $\mathbb S^1 \,\dot{\cup}\, \mathbb S^1 \,\dot{\cup}\, \mathbb S^1$, that is, it consists of two homoclinic orbits whose singular points are connected by a pair of heteroclinic orbits (the skeleton $D$; see [19]). Figure 3 shows the set $Q_{0}(f)$ for various values of the parameter $f$: the critical levels $(h_0,f_1)$ and $(h_0,f_{2,3})$ and the regular levels $2^{1/3} < f\operatorname{sgn}\nu < 3$ and $f\operatorname{sgn}\nu > 3$. Using Lemma 3 one can easily check that for $f\operatorname{sgn}\nu < \sqrt[3]{2}$ the level set $Q_{0}(f)$ is empty; for $\sqrt[3]{2} < f\operatorname{sgn}\nu < 3$ it contains one one-dimensional Liouville torus, and for $f\operatorname{sgn}\nu > 3$ it contains three tori. The bifurcations of one-dimensional Liouville tori corresponding to the bifurcation curves $\Sigma_1$ and $\Sigma_{2,3}$ are described, up to a leafwise homeomorphism, by 2-atoms, the simple elliptic atom $A$ and the complicated hyperbolic atom $D_1$, respectively. The singular leaf of system (2.3) in the inverse image of $\Sigma_1$ consists of a single critical circle $\Pi_1$ obtained as the Cartesian product of $\mathbb S^1$ and a point, and the inverse image of $\Sigma_{2,3}$ contains the Cartesian product of the skeleton $D$ and a circle. The bifurcations of two-dimensional Liouville tori in the original system with two degrees of freedom are described by the 3-atoms $A$ and $D_1$, respectively:
$$
\begin{equation*}
\varnothing \to \mathbb S^1 \to \mathbb T^2, \qquad \mathbb T^2\to\mathbb S^1 \times (\mathbb S^1 \,\dot{\cup}\, \mathbb S^1 \,\dot{\cup}\, \mathbb S^1 )\to3\mathbb T^2.
\end{equation*}
\notag
$$
In this case, $\Gamma_1/\Gamma_2=1$. Let us find the topology of the Liouville foliation under small perturbations of the parameters of the form $\Gamma_1/\Gamma_2=1+\varepsilon$. Then the critical common level surface of the integrals is unstable and splits into the two disjoint critical integral manifolds
$$
\begin{equation*}
\mathbb S^1 \times ( \mathbb S^1 \,\dot{\cup}\, \mathbb S^1 ) \cup \mathbb T^2\quad\text{and} \quad \mathbb S^1 \times ( \mathbb S^1 \,\dot{\cup}\, \mathbb S^1 ).
\end{equation*}
\notag
$$
The bifurcation curve $\Sigma_{2,3}$ splits into two bifurcation curves $\Sigma_2$ and $\Sigma_3$, that is, the saddle critical circles $\Pi_2$ and $\Pi_3$ diverge to distinct common level sets of the first integrals (see Figure 4). In Figures 2 and 4 thin solid bifurcation curves correspond to simple atom-bifurcations of type $A$ or $B$. Both atoms are stable under small integrable perturbations of the parameters of the system. Bold lines correspond to critical bifurcation curves such that for each point on these curves the singular leaf in the inverse image of the momentum map is not stable. In this case the unstable 3-atom $D_1$ is split under small integrable perturbations. Figure 5 shows level curves of the reduced Hamiltonian for different values of the parameter $f$ of the additional integral. Figure 5, (b), illustrates a bifurcation of one one-dimensional Liouville torus into three tori through the singular leaf $\mathbb S^1 \,\dot{\cup}\, \mathbb S^1 \,\dot{\cup}\, \mathbb S^1$, which contains two singular points. Figure 5, (a) and (c), shows the level set of the energy of the reduced system under a sufficiently small perturbation of the intensity parameters for $\varepsilon<0$ and $\varepsilon>0$, respectively. Figure 5 demonstrates the splitting of the complicated 2-atom $D_1$ into two simple atoms $B$. 5.4. Topology of isoenergy surfacesLet us find the topological types of isoenergy surfaces $Q_h^3$. For the parameters chosen the constant-energy surface $H=h_0$ is given by the equation
$$
\begin{equation*}
( x_1^2+y_1^2 )( x_2^2+y_2^2 ) \bigl[(x_1-x_2)^2+(y_1-y_2)^2\bigr]^\kappa=e^{2h_0/\kappa}.
\end{equation*}
\notag
$$
We use the following coordinates: consider the polar radii $\rho_1$ and $\rho_2$ to the vortices; let $\varphi_1$ be the angle from $\rho_1$ to $\rho_2$, and let $\varphi_2$ be the angle between the $OX$-axis and the bisector of the angle $\varphi_1$. The explicit form of this transformation is as follows:
$$
\begin{equation}
\begin{gathered} \, x_1 =\rho_1\cos\biggl(\varphi_2-\frac{\varphi_1}{2}\biggr), \qquad y_1=\rho_1\sin\biggl(\varphi_2-\frac{\varphi_1}{2}\biggr), \\ x_2=\rho_2\cos\biggl(\varphi_2+\frac{\varphi_1}{2}\biggr), \qquad y_2=\rho_2\sin\biggl(\varphi_2+\frac{\varphi_1}{2}\biggr), \end{gathered}
\end{equation}
\tag{5.1}
$$
where $\rho_1, \rho_2 \in \mathbb R_{>0}$ and $\varphi_1,\varphi_2 \in \mathbb S^1$. We also note that $\rho_1 \neq \rho_2$ if $\varphi_1=0$ because the vortices cannot have the same position. The Jacobian of the map is distinct from zero, that is, the above transformation is nondegenerate for all admissible positions of vortices. In the new coordinates the isoenergy surface is given by the equation
$$
\begin{equation*}
\mathcal{R}_{\kappa} \colon \rho_1^2 \rho_2^2 ( \rho_1^2-2 \rho_1\rho_2\cos\varphi_1+\rho_2^2 )^\kappa=e^{2h_0/\kappa}
\end{equation*}
\notag
$$
for an arbitrary choice of the angle $\varphi_2$. So, $Q^3_{h_0} \cong \mathcal{R}_{\kappa} \times \mathbb S^1$. The two-dimensional manifold $\mathcal{R}_{\kappa}$ is a smooth regular orientable surface. In view of periodicity with respect to $\varphi_1$ this surface is diffeomorphic to a cylindric surface with puncture (Figure 6, (a)) or a thrice-punctured two-dimensional sphere ( Figure 6, (b)). For fixed $\varphi_1=\varphi_1^0$ we obtain an algebraic curve which is diffeomorphic to $\mathbb R$ for $\varphi_1^0 \in (0,2\pi)$ and consists of two disjoint open arcs for $\varphi_1^0=0$. Such a two-dimensional surface can conveniently be represented as an open two-dimensional disc $\mathbb D^2$ without two closed (smaller) discs $\overline{\mathbb D}^2$ (see Figure 6, (c)). Hence $Q_{h_0}^3$ is an open solid torus $\mathbb D^2\times\mathbb S^1$ without two (smaller) disjoint solid tori with boundary $\overline{\mathbb D}^2\times\mathbb S^1$. Thus, the three-dimensional surface of constant energy is an open manifold (without boundary) which is the interior of so-called ‘pants’, or a toric oriented saddle. The general pattern of the foliation of such a three-dimensional isoenergy surface by levels of the additional integral $F$ is shown in Figure 6, (d). Note that the noncompact three-dimensional manifold $Q^3_{h_0} \cong \mathcal{R}_{\kappa} \times \mathbb S^1$ carries the structure of a Seifert fibration in the sense that it falls into fibres of the form $\{*\} \times \mathbb S^1$ over the two-dimensional orientable surface $\mathcal{R}_\kappa$, and a neighbourhood of each fibre is trivial. § 6. Bifurcation of two Liouville tori into two tori6.1. The augmented bifurcation diagramIn this section we consider the system for the following values of the parameters:
$$
\begin{equation*}
\Gamma_1=\gamma, \qquad \Gamma_2=-1, \qquad \lambda_1=1, \qquad \lambda_2=-1,
\end{equation*}
\notag
$$
where $\gamma<\gamma^*$ for
$$
\begin{equation*}
\gamma^*=-\frac16 \Bigl( 4+\sqrt[3]{1261-57\sqrt{57}} +\sqrt[3]{1261+57\sqrt{57}} \,\Bigr) \approx -4.219136.
\end{equation*}
\notag
$$
Equation (3.2) has three admissible roots, which can be conveniently determined by the trigonometric Vieta formula
$$
\begin{equation}
\begin{aligned} \, k_1(\gamma) &=-\frac{2}{3}+\frac{2\sqrt{1-3\gamma}}{3} \sin\biggl(\frac{\pi}{6} + \frac13 \arccos\biggl[-\frac{18\gamma-25}{2(1-3\gamma)^{3/2}}\biggr]\biggr), \nonumber \\ k_2(\gamma) &=-\frac{2}{3}+\frac{2\sqrt{1-3\gamma}}{3} \sin\biggl(\frac13 \arcsin\biggl[-\frac{18\gamma-25}{2(1-3\gamma)^{3/2}}\biggr]\biggr), \\ k_3(\gamma) &=-\frac{2}{3}-\frac{2\sqrt{1-3\gamma}}{3} \cos\biggl(\frac13\arccos\biggl[-\frac{18\gamma-25}{2(1-3\gamma)^{3/2}}\biggr] \biggr); \nonumber \end{aligned}
\end{equation}
\tag{6.1}
$$
note that $k_1(\gamma) > k_2(\gamma) > k_3(\gamma)$ for $\gamma\in(-\infty,\gamma^*)$. The bifurcation diagram of the momentum map $\Sigma$ is the union of the images $\Sigma_i=\mathcal{F}(\Pi_i)$ of three critical circles $\Pi_i$, that is, By Theorem 4
$$
\begin{equation*}
\Sigma_i\colon h_i(f)=\frac{2\gamma+1}{2}\log(-f)+\frac{\eta_i(\gamma)}{2},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\eta_i(\gamma)=\log k_i^2(\gamma)+\gamma \log[k_i(\gamma)+1]^2 -(2\gamma+1)\log [k_i^2(\gamma)-\gamma].
\end{equation*}
\notag
$$
The additional integral is negative. Hence the range of the momentum map has the boundary and the augmented bifurcation diagram has the form
$$
\begin{equation*}
\Sigma'=\Sigma_0 \cup \Sigma_1 \cup \Sigma_2 \cup \Sigma_3.
\end{equation*}
\notag
$$
Lemma 4. The images $\Sigma_2$ and $\Sigma_3$ of the critical circles $\Pi_2$ and $\Pi_3$ are equal if $\Gamma_1=\gamma^{**}$, $\Gamma_2=-1$, $\lambda_1=1$, $\lambda_2=-1$, where $\gamma^{**} \approx -13.552767614968$. Proof. Consider the coincidence condition (3.11) for $\Sigma_2$ and $\Sigma_3$:
$$
\begin{equation}
(2\gamma+1) \log \biggl[ \frac{\gamma-k_3^2(\gamma)}{\gamma-k_2^2(\gamma)} \biggr] +\gamma \log \biggl[ \frac{k_2(\gamma)+1}{k_3(\gamma)+1} \biggr]^2 + \log \biggl[ \frac{k_2(\gamma)}{k_3(\gamma)} \biggr]^2 =0.
\end{equation}
\tag{6.2}
$$
The values of $k_2(\gamma)$ and $k_3(\gamma)$ are determined by (6.1). Taking the first derivatives of $k_2(\gamma)$ and $k_3(\gamma)$ with respect to $\gamma$, it can be verified that for $\gamma<\gamma^*$ they are strictly monotonically increasing functions.
By monotonicity $k_2(\gamma) \in [k_2(-14),k_2(-13)]$ and $k_3(\gamma) \in [k_3(-14),k_3(-13)]$ for $\gamma \in [-14,-13]$. Using interval arithmetic, it can be shown that the derivative of the function (6.2) with respect to $\gamma$ does not change sign on this interval. Therefore, this function is strictly monotone for $\gamma \in [-14,-13]$. At the endpoints of this interval the function (6.2) has different signs. Hence, by Bolzano’s intermediate value theorem,
$$
\begin{equation*}
\exists!\,\gamma^{**}\in[-14,-13]\colon \quad \Sigma_2(\gamma^{**})=\Sigma_3(\gamma^{**}).
\end{equation*}
\notag
$$
Numerical calculations, with any prescribed accuracy, give us
$$
\begin{equation*}
\gamma^{**} \approx -13.552\,76761\,49683\,66593\,90987\,91141\,37627.
\end{equation*}
\notag
$$
This proves the lemma. Proceeding as in Lemma 4, it can be shown that
$$
\begin{equation*}
\Sigma_1(\gamma) \neq \Sigma_2(\gamma)\quad\text{and} \quad \Sigma_1(\gamma) \neq \Sigma_3(\gamma)\quad\text{for}\quad \gamma \in [-14,-13].
\end{equation*}
\notag
$$
So this interval contains only one value of the parameter $\gamma$ for which a complicated atom appears in the inverse image of a bifurcation curve (see Figure 7). From Theorem 3 it follows that for the parameters as above the stability type of the critical circles $\Pi_i$ depends on the sign of the polynomial which changes at the point
$$
\begin{equation*}
k^*=-\frac{1}{6}\Bigl( 2-\sqrt[3]{46-6\sqrt{57}}- \sqrt[3]{46+6\sqrt{57}} \, \Bigr) \approx 0.565198.
\end{equation*}
\notag
$$
We have $k_1 > k^*$ on the interval $\gamma \in (-\infty, \gamma^*)$, and so in the inverse image of the bifurcation curve $\Sigma_1$ the rank 1 singularities are of elliptic type. The inverse images of $\Sigma_2$ and $\Sigma_3$ contain hyperbolic singularities, as $k_2 < k^*$ and $k_3 < k^*$ for ${\gamma \in (-\infty, \gamma^*)}$ (see Figure 8). Since $C_2<0$ and $C_3<0$, the indices of the critical circles $\Pi_2$ and $\Pi_3$ are equal to 1. The index of the circle $\Pi_1$ is 2, that is, the critical submanifolds in the inverse images of points on $\Sigma_1$ are maximal critical circles. 6.2. The compactness of levels of the first integralsConsider the reduced system (4.3) and an arbitrary energy level $h_0$. We find out whether the surface $Q_{h_0}(f_0)$ is compact. As in § 5.2, we use the explicit parametrization $z=u^2+v^2$. Let $h_e=-e^{-2h_0}$. Then
$$
\begin{equation*}
\mathcal{W}(z) \colon \begin{cases} u(z)=\dfrac{N_1(z)}{\sqrt{D(z)}}, \\ v(z)=\pm\sqrt{\dfrac{N_2(z)}{D(z)}}, \end{cases}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, N_1(z)=(\gamma+1)z^2+f_0 \gamma z-\gamma^2[ h_e(z+f_0) ]^{-1/\gamma}, \\ N_2(z)=z D(z)-N_1^2(z) \end{gathered}
\end{equation*}
\notag
$$
and Again, we consider the domain $\mathcal{Z}$ of the map $\mathcal{W}(z)$. Since $D(z)$ and $h_e(z+f_0)$ are positive everywhere, we are interested in $z \in (0, -f_0)$ such that $N_2(z) \geqslant 0$. On this interval the function $N_2(z)$ is continuous; it is negative at the endpoints of this interval. Hence $\mathcal{Z}$ is either empty or is a union of severa closed intervals and points. Hence $\mathcal{Z}$ is compact. Therefore, the energy levels $Q_{h_0}(f_0)=\mathcal{W}(\mathcal{Z})$ are also compact. By Theorem 5 the common level sets of the integrals of the original system (2.3) with two degrees of freedom are compact and diffeomorphic to the Cartesian product of leaves of the reduced system and the circle $\mathbb S^1$. Compactness also implies that a neighbourhood of the singular leaf in the inverse image of a point of the bifurcation curve $\Sigma_1$ is described by the 3-atom $A$, since the critical circles $\Pi_1$ are of elliptic type. To the circles $\Pi_2$ and $\Pi_3$ there correspond simple saddle atoms without stars, that is, 3-atoms $B$. In the inverse images of points on the curve $\Sigma_{2,3}$, for $\gamma=\gamma^{**}$ these saddle singularities lie on the same common level of the first integrals. 6.3. A small neighbourhood of a singular leafLet us determine the type of the bifurcation of Liouville tori in the inverse image of a point on the bifurcation curve $\Sigma_{2,3}$. The value of $\gamma^{**}$ is only known approximately, and so finding the type of a complicated atom is facilitated by considering its splittings under perturbations of the parameter $\gamma$. Theorem 6. A small neighbourhood of the singular leaf in the inverse images of points of the bifurcation curve $\Sigma_{2,3}(\gamma^{**})$ is equivalent to the 3-atom $D_2$. Proof. Using Lemma 3 we find the number of Liouville tori in the inverse images of regular values of the momentum map under integrable perturbations of the parameter $\gamma$ of the form $\gamma=\gamma^{**} \pm \varepsilon$, where $\varepsilon > 0$. For these values of the parameters and $h_0=0$ equation (4.6) assumes the form
Let $\gamma_1=-13$. Then the curves $\Sigma_i(\gamma_1)$ correspond to the critical values $(f_i^{(1)}, h_0)$, where
$$
\begin{equation*}
f_1^{(1)} \approx -5.628284, \qquad f_2^{(1)} \approx -9.810511\quad\text{and} \quad f_3^{(1)} \approx -10.185146.
\end{equation*}
\notag
$$
We choose arbitrary regular values in the chambers of the augmented bifurcation diagram on the left and right of the curves $\Sigma_2(\gamma_1)$ and $\Sigma_3(\gamma_1)$. For example,
$$
\begin{equation*}
f_a^{(1)}=-11, \qquad f_b^{(1)}=-10\quad\text{and} \quad f_c^{(1)}=-9.
\end{equation*}
\notag
$$
We also consider equation (6.3) for $\gamma=\gamma_1$:
$$
\begin{equation}
u^{26}\bigl( u-\sqrt{-13(u^2+f)} \, \bigr)^{26}+13^{26}(u^2+f)=0.
\end{equation}
\tag{6.4}
$$
For $f=f_a^{(1)}$ equation (6.4) has four roots. Therefore, to the chamber of the augmented bifurcation diagram on the left of $\Sigma_3(\gamma_1)$ there correspond two Liouville tori. In a similar way, to the point $(f_b^{(1)}, h_0)$ and the chamber of the diagram between $\Sigma_3(\gamma_1)$ and $\Sigma_2(\gamma_1)$ there corresponds one Liouville torus, and the inverse image of the point $(f_c^{(1)}, h_0)$ (which lies in the chamber of the diagram between $\Sigma_2(\gamma_1)$ and $\Sigma_1(\gamma_1)$) contains two tori (see Figure 7, (c)). So under a sufficiently slight perturbation of $\gamma$ the complicated atom breaks down, and the perturbed molecule assumes the form shown in Figure 9, (a).
The bifurcation curves $\Sigma_2(\gamma)$ and $\Sigma_3(\gamma)$ interchange after passing through the point $\gamma^{**}$ (see Figure 7). It is known (see [19]) that, in all, there are four elementary bifurcations that swap two neighbouring atoms $B$ on a molecule: $C_1$, $C_2$, $D_1$ and $D_2$. In addition, their splittings under integrable perturbations are known. The resulting splitting indicates that the required complicated atom is of type $C_2$ or $D_2$ and corresponds to a bifurcation of two Liouville tori into two. A different perturbation is required to determine unambiguously the required complicated atom. Consider an integrable perturbation of the form $\gamma=\gamma^{**}-\varepsilon$. Let $\gamma_2=-14$; in this case to the bifurcation curves $\Sigma_i(\gamma_2)$ there correspond the critical values of the additional integral
$$
\begin{equation*}
f_1^{(2)} \approx -5.900392, \qquad f_2^{(2)} \approx -10.725630\quad\text{and} \quad f_3^{(2)} \approx -10.419932.
\end{equation*}
\notag
$$
Again, we choose regular values in the chambers of the augmented bifurcation diagram on the left and right of $\Sigma_2(\gamma_2)$ and $\Sigma_3(\gamma_2)$:
$$
\begin{equation*}
f_a^{(2)}=-11, \qquad f_b^{(2)}=-10.5\quad\text{and} \quad f_c^{(2)}=-10.
\end{equation*}
\notag
$$
The equation
$$
\begin{equation*}
u^{28}\bigl( u-\sqrt{-14(u^2+f)} \bigr)^{28}+14^{28}(u^2+f)=0
\end{equation*}
\notag
$$
has four roots for $f=f_a^{(2)}$ or $f=f_c^{(2)}$ and six roots for $f=f_b^{(2)}$. So two Liouville tori correspond to the chamber of the bifurcation diagram on the left of $\Sigma_2(\gamma_2)$; the inverse images of points between the curves $\Sigma_2(\gamma_2)$ and $\Sigma_3(\gamma_2)$ contain three Liouville tori each; and the inverse images of points between $\Sigma_3(\gamma_2)$ and $\Sigma_1(\gamma_2)$ contain two tori each (see Figure 7, (a)). The required splitting is shown in Figure 9, (c). As a result, we have two different perturbed molecules; from them one can unambiguously determine that the complicated atom in the inverse image of the bifurcation curve $\Sigma_{2,3}(\gamma^{**})$ corresponds to the 3-atom $D_2$. This proves Theorem 6. Figure 10, which shows the level sets of the reduced Hamiltonian for various values of the parameter $f$ of the additional integral, demonstrates a bifurcation of one-dimensional Liouville tori in the reduced system. It is clearly seen that the singular leaf contains two homoclinic orbits and two heteroclinic orbits (as in the case of the atom $D_1$), which distinguishes substantially the 2-atom $D_2$ from the atom $C_2$, whose singular leaf consists only of heteroclinic trajectories. Figure 11 shows the canonical representation of the 2-atom $D_2$, which is leafwise homeomorphic to the level curves in Figure 10.  Figure 11.Leaves homeomorphic to the level curves (a) in Figure 10, (a); (b) in Figure 10, (b); (c) in Figure 10, (c); and (d) leaves constituting a small neighbourhood of the singular leaf in the form of the 2-atom $D_2$. Note that the set of parameters for which such a bifurcation of two Liouville tori into two tori can occur is not exhausted by the above values (as a simple example, one can consider the parameters $\Gamma_1=1$ and $\Gamma_2=-\gamma^{**}$ with the same polarities or similar values for $\lambda_1=-1$ and $\lambda_2=1$). In general, as in the case of the 3-atom $D_1$, the set of such parameters in this problem is infinite. However, the set of these parameters is more involved than the one for $D_1$, which complicates the analysis. 6.4. An isoenergy manifoldFor a more complete description of the topology of the Liouville foliation in the case when $\Gamma_1=\gamma$, $\Gamma_2=-1$ and $\lambda_1=-\lambda_2=1$ we need to find the topological type of the isoenergy surface $Q_h^3$ for the values in question of the parameters of the system. As in § 5.4, it is convenient to change to the coordinates (5.1). In this case
$$
\begin{equation*}
Q_{h_0}^3 \cong \mathcal{R}_{\gamma} \times \mathbb S^1, \qquad \mathcal{R}_{\gamma} \colon \rho_1^{2\gamma} \rho_2^2 ( \rho_1^2-2\rho_1\rho_2\cos\varphi_1+\rho_2^2 )^{\gamma}=e^{2h_0}.
\end{equation*}
\notag
$$
The smooth regular orientable surface $\mathcal{R}_{\gamma}$ is equivalent to the above surface $\mathcal{R}_{\kappa}$, is a two-dimensional sphere with three punctures, or an open two-dimensional disc without two disjoint closed discs. Again, we consider $\mathcal{R}_{\gamma}$ for various values of $\varphi_1=\varphi_1^0$. If $\varphi_1^0=0$, then the corresponding algebraic curve has three asymptotes: $\rho_1=0$, $\rho_2=0$ and $\rho_1=\rho_2$, which arise naturally from the condition that the vortices do not coincide. In this case the resulting one-dimensional manifold consists of two connected components diffeomorphic to $\mathbb R$. If $\varphi_1^0 \in (0,2\pi)$, then there is no asymptote $\rho_1=\rho_2$, and the manifolds $\mathcal{R}_{\gamma}|_{\varphi_1=\varphi_1^0}$ are open one-dimensional intervals. So the isoenergy surface is similar to the one considered above: the three-dimensional noncompact manifold $Q_{h_0}^3$ is the interior of a toric oriented saddle (pants). Figure 12 shows the image, up to rough Liouville equivalence, of the foliation of the three-dimensional surface of constant energy by levels of the additional integral $F$. As in the case considered in § 5.4, we obtain an $\mathbb S^1$-fibration of the three-dimensional isoenergy manifold over the orientable noncompact surface $\mathcal{R}_{\gamma}$. In addition, this circle fibration is of special form: in a neighbourhood of each leaf it is a trivially fibred solid torus. 6.5. Degenerate singularityConsider system (2.3) for the parameters $\Gamma_1=\gamma^*$, $\Gamma_2=-1$ and $\lambda_1=-\lambda_2=1$, and in the case when equation (3.2) has a multiple root $k_{1,2}=k^*$ (the arguments for the intensities $\Gamma_1=1$ and $\Gamma_2=|\gamma^*|$ are similar). Since $C_{1,2}$ vanishes (see § 6.1), it follows that the Hessian of the function $F|_{Q_h^3}$ has two zero eigenvalues at all points of rank $1$ corresponding to the root $k_{1,2}$. Therefore, the inverse images of points on the curve $\Sigma_{1,2}$ contain degenerate singularities. Figure 13 presents an example of level curves of the reduced Hamiltonian in a small neighbourhood of this singularity. In particular, this figure shows how a one-dimensional Liouville torus diffeomorphic to $\mathbb S^1$ bifurcates into another smooth one-dimensional torus after passing through the singular leaf, which consists of a stationary point and a homoclinic orbit. For the above values of the parameters in the original system (2.3) with Hamiltonian (2.4) we have a bifurcation of one regular two-dimensional Liouville torus into another through a singular leaf homeomorphic to $\mathbb T^2$. This leaf ceases to be smooth at degenerate critical points, which make up a smooth one-dimensional circle (see Figure 1, (a), in [44]). Under small perturbations of the form $\gamma=\gamma^*+\varepsilon$, where $\varepsilon>0$, the critical manifold $\mathcal{F}^{-1}(\Sigma_{1,2})$ disappears because $k_1$ and $k_2$ assume inadmissible values (with nonzero imaginary part). In the case $\gamma=\gamma^*-\varepsilon$ the degenerate singularity splits into two nondegenerate singularities (an elliptic and a hyperbolic one; see Figure 14). Therefore, the critical surface in the inverse image of $\Sigma_{1,2}$ is not stable under small perturbations in the class of integrable systems. Thus, despite some similarities, the semilocal singularity is not a cusp singularity in the sense of [45]. Let us verify this claim by testing the type of points that make up the critical circle on the singular leaf. We use Definition 2.1 from [45], § 2, as a criterion for their parabolicity, and consider the one-parameter family of critical circles
$$
\begin{equation*}
\Pi^*(s) \colon \begin{cases} x_1=- \cos\theta_2s, \quad x_2=\cos\theta_2k^*s, \\ y_1=- \sin\theta_2s, \quad y_2=\sin\theta_2k^*s, \end{cases} \qquad \theta_2 \in [0,2\pi),
\end{equation*}
\notag
$$
with parameter $s > 0$. The circles $\Pi^*(s)$ consist of degenerate critical points of rank $1$ of system (2.3). Consider a critical point $p \in \Pi^*(s)$. Since $p \in \mathcal{C}^1$, there exists a unique factor defined by the expression (3.6) and such that $dH(p)+\lambda^* dF(p)=0$. Note that $dF(p) \neq 0$. Let $H_f=H|_{F=f}$ be the restriction of $H$ to the surface of level $F(p)=f$, and let $v^3 H_f(p)$ be the third derivative of $H_f$ along the vector $v$ at the point $p$. Hence by the definition from [45] the following conditions should be satisfied:
In the polar coordinates the level of the additional integral $F=f$ is given by the equation $\gamma^* \rho_1^2-\rho_2^2=f$, from which one easily finds that Hence $H_f(\rho_2,\theta_1,\theta_2)= H(\rho_1(\rho_2,f),\rho_2,\theta_1,\theta_2)$.The points $p \in \Pi^*(s)$ lie in the inverse images of the points $(f_s, h_s) \in \Sigma_{1,2}$, where
$$
\begin{equation*}
f_s=s^2[ \gamma^*-(k^*)]^2 \quad\text{and} \quad h_s=(2\gamma^*+1)\log s+\gamma^*\log(k^*+1)+\log k^*.
\end{equation*}
\notag
$$
In the polar coordinates we have $p(\rho_1,\rho_2,\theta_1,\theta_2)=(s,k^*s,\pi+\theta_2,\theta_2)$. A direct substitution of the degenerate points into the second differential shows that
$$
\begin{equation*}
d^2H_f(p)= a_1 \begin{pmatrix} 0 & 0 & 0 \\ 0 &1 & -1 \\ 0 & -1 &1 \end{pmatrix},
\end{equation*}
\notag
$$
where $a_1 \approx 0.973387$ is the unique real root of the equation It is clear that $\operatorname{rank} d^2H_f(p)=1$, and therefore condition (1) is fulfilled. We choose a vector $v_0=(1,0,0)$ from the kernel of the second differential of $H_f$ at the point $p$. The third derivative of $H_f$ along $v_0$ at $p$ is where $a_2 \approx 9.668038$ is the real root of the cubic equation So condition (2) in the criterion also holds.Now let us verify condition (3) in the criterion. Consider the second differential of the linear combination $H+\lambda^* F$ with respect to the variables $(x_1,y_1,x_2,y_2)$. After the substitution of the point $p$ the square matrix assumes the form
$$
\begin{equation*}
\frac{b_0}{s^2} \begin{pmatrix} b_1(\cos\psi+b_3) & b_1 \sin\psi & \cos\psi & \sin\psi \\ b_1\sin\psi & -b_1(\cos\psi-b_3) & \sin\psi & -\cos\psi \\ \cos\psi & \sin\psi & b_2(\cos\psi-b_3) & b_2\sin\psi \\ \sin\psi & -\cos\psi & b_2\sin\psi & -b_2(\cos\psi+b_3) \end{pmatrix},
\end{equation*}
\notag
$$
where $\psi=2\theta_2$, and the constants
$$
\begin{equation*}
b_0 \approx -1.722206, \quad b_1 \approx -3.449844, \quad b_2 \approx 0.817666\quad\text{and} \quad b_3 \approx 1.163833
\end{equation*}
\notag
$$
are the real roots of the cubic equations
$$
\begin{equation*}
b_0^3+5 b_0^2+19 b_0+23=0, \qquad 4 b_1^3+20 b_1^2+24 b_1+9=0,
\end{equation*}
\notag
$$
$$
\begin{equation*}
23 b_2^3+13 b_2^2-15 b_2-9=0, \qquad 9 b_3^3-15 b_3^2+13 b_3-9=0.
\end{equation*}
\notag
$$
It is easily checked that $\operatorname{rank} d^2(H+\lambda^* F)(p)=2$, that is, the degenerate local singularities do not satisfy the parabolicity criterion introduced in [45]. Note that the above criterion is particularly suitable as a practical means and has been applied several times in the study of degenerate singularities of other integrable systems of Hamiltonian mechanics (see, for example, the recent paper [46] on parabolic singularities of multiparameter family of integrable Joukowsky systems). A remarkable property of the degenerate bifurcation under consideration is that it appears in the inverse images of all points of the smooth bifurcation curve $\Sigma_{1,2}$, rather than in the inverse image of a singular point of the bifurcation diagram, as frequently occurs in the case of cusp singularities. § 7. ConclusionsAn analysis of the topology of the Liouville foliation in the generalized problem of three vortices with constraint has been carried out. For arbitrary values of the parameters of the system an explicit form of the bifurcation diagram of the momentum map has been found and a parametrization of the critical submanifolds lying in the inverse images of points of the bifurcation curves was obtained. The stability of critical motions has been examined, and explicit formulae for the indices of the critical circles have been derived. The study of bifurcations of integral manifolds has significantly been simplified by reducing the original system with two degrees of freedom to an integrable Hamiltonian system with one degree of freedom; in this way it has also enabled us to demonstrate graphically some bifurcations of Liouville tori. In particular, we have considered two bifurcations of two-dimensional tori which proceed through the singular leaf $\mathbb S^1 \times (\mathbb S^1 \,\dot{\cup}\, \mathbb S^1 \,\dot{\cup}\, \mathbb S^1)$. In § 5 we considered a neighbourhood of the singular leaf of the Liouville foliation corresponding to the 3-atom $D_1$ (a bifurcation of one Liouville torus into three tori). It is worth pointing out that this bifurcation occurs not only in the generalized model of three magnetic vortices with a constraint, but also in a particular case of the model, namely, the problem of motion of vortices in a perfect fluid, in which the polarities have equal parameters. In § 6 a much rarer bifurcation of Liouville tori was considered: for this bifurcation a neighbourhood of the singular leaf is described by the 3-atom $D_2$. In the case under study the appearance of such a bifurcation is related to the possibility of choosing opposite vortex polarities, which makes the generalized system more interesting from the topological point of view. Despite the difficulty in finding the precise values of the parameters for which the complicated atom $D_2$ appears in the inverse images of points on the bifurcation curve, it proved possible to verify analytically the presence of this bifurcation in a particular case by splitting the complicated atom under integrable perturbations of the parameters. In §§ 5.4 and 6.4 we studied the topology of three-dimensional isoenergy surfaces $Q_h^3$ for the corresponding values of the parameters of the system. In both cases the three-dimensional manifolds were shown to be noncompact and diffeomorphic to the set of interior points of so-called ‘pants’ (a toric oriented saddle). We also showed explicitly the foliation of the isoenergy surface by the levels of the additional integral, both when a $D_1$-type bifurcation is present and in the case of the 3-atom $D_2$. In § 6.5 we examined the values of the parameters of the system for which the inverse image of each point of some bifurcation curve contains a singular leaf homeomorphic to a two-dimensional torus all of whose critical points are degenerate and form a connected compact submanifold. This leaf was shown to be unstable under small integrable perturbations of the parameters of the system. It was verified that the points of its critical circle do not satisfy the parabolicity criterion introduced by Bolsinov, Guglielmi and Kudryavtseva in [45]. So the degenerate bifurcation of Liouville tori, which we found, was shown to be inequivalent to the cusp semilocal singularity defined in [45]. The main task for future research is to analyze the topology of the Liouville foliation for the whole set of parameters and to construct an atlas of bifurcation diagrams, that is, a partition of the plane of the intensity parameters (with fixed values of the vortex polarities) into regions in which the bifurcation diagram has the same form. For a nonzero total vortex moment, to each point of this atlas there corresponds a bifurcation diagram which can be described by a single rough molecule. However, special attention should be paid to subsets of values of the parameters for which the system involves various noncritical, noncompact or degenerate bifurcations. In addition, the topology of the system for a set of parameters for which the total vortex moment is zero requires a separate study. AcknowledgementThe author is deeply grateful to Prof. P. E. Ryabov for posing the problem and paying constant attention to this work. 
Citation in format AMSBIB
\Bibitem{Pal24} |
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