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Saddle connections A. V. Dukov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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    § 1. Introduction1.1. A lower bound for the cyclicity of polycyclesLet $\mathcal{M}$ be a $C^{r+1}$-smooth orientable 2-manifold, $r \geqslant 3$. Let $\operatorname{Vect}^r(\mathcal{M})$ denote the space of all $C^r$-smooth vector fields on $\mathcal{M}$. Definition 1. A hyperbolic polycycle of a vector field is a finite orientable graph $\gamma$ embedded in the manifold $\mathcal{M}$ and satisfying the following conditions: The question of the number of limit cycle born in the decomposition of polycycles is one of the most difficult problems in the theory of nonlocal bifurcations; it is a special case of the Hilbert–Arnold problem (see details in [7]). For upper bounds the reader can consult [10], [13], [15], [16] and [21]. In this paper we establish a lower bound for the number of limit cycles appearing in the process. Definition 2. The Poincaré map of a polycycle is the monodromy self-map of a transversal cross-section of the polycycle along the vector field. The polycycle is said to be monodromic if its Pointcaré map is well defined even for the unperturbed field (Figure 1). Definition 3. The characteristic number of a saddle is the modulus of the ratio of its eigenvalues, where the negative eigenvalue is in the denominator. Theorem 1. Let $\gamma$ be a monodromic hyperbolic polycycle of a field $v_0 \in \operatorname{Vect}^r(\mathbb{S}^2)$, $r \geqslant 3$, which is formed by $n$ distinct saddles and $n$ separatrix connections. Assume that the saddles in $\gamma$ can be numbered (not necessarily in the order of traversal) so that for each $i=1,\dots,n-1$ the characteristic numbers $\lambda_1,\dots,\lambda_n$ of saddle points satisfy Then the decomposition of $\gamma$ in a generic $C^r$-smooth $n$-parametric family $V$ gives birth to at least $n$ limit cycles. We show in § 2.3 (see Proposition 1) that vector fields $v$ close to $v_0$ that have a polycycle $\gamma(v)$ close to $\gamma(v_0)$ form a Banach submanifold of codimension $n$. By saying that the family $V$ is generic we mean that $V$ intersects this Banach submanifold transversally. Theorem 1 has an involved history. It was originally stated by Reyn [18] in 1980. For $n=2$ and $n=3$ he presented an algorithm of the proof, which is based on splitting off saddles (we also use this algorithm below). At each step $k=1,\dots,n$ of this algorithms one must verify two assertions:
Reyn proved assertion (2) while supposing that (1) is self-evident. He did not prove Theorem 1 for an arbitrary $n$: he wrote that the general case of Theorem 1 is analogous to the case $n=3$, but presented no accurate proof. In his Ph.D. thesis [17] of 1990 Mourtada proved (without stating this explicitly) that for each $n$ there exists a hyperbolic polycycle of $n$ saddles such that a perturbation of it produces at least $n$ limit cycles. His proof is problematic to check because his notation is quite involved, which perhaps makes sense for some other results in the thesis, but not for the relatively simple result of Theorem 1. Mourtada did not mention Reyn’s paper. In 2004 Han, Wu and Bi [12] proved a similar theorem. The only difference from Theorem 1 was that they proved the existence of a smooth family such that the decomposition of a polycycle in this family gives birth to $n$ cycles, but they did not show than the same holds for each generic family. Like Mourtada, they did not refer to Reyn. They did refer to Mourtada’s papers, although not to his thesis [17], where a similar result had been established. Thus, several authors proved independently analogues of Theorem 1. On the other hand each of these statements or its proof had some flaws. This prompted us to give a new proof of Theorem 1. 1.2. Definitions and notationIn fact, Theorem 1 is not our central result here. We are interested in possible connections between hyperbolic saddles appearing in perturbations of connections of the original field. A special case is the question of the birth of child cycles (see Definition 5 below) in perturbations of the original hyperbolic polycycles (as mentioned above, Reyn ignored this question in [18]). Let $v_0 \in \operatorname{Vect}^r(\mathcal{M})$ be a field with a few connections $\mathrm{SC}_i$, $i=1,\dots,n$, between hyperbolic saddles. We call them the basis system. We do not prohibit other degeneracies of $v_0$ (such as other saddle connections not included in the basis system). We draw a $C^r$-smooth transversal cross-section $\Gamma_i$ to each separatrix connection $\mathrm{SC}_i$, which is disjoint from the other connections in the basis system. Let $U_\varepsilon \subset \mathcal{M}$ denote some $\varepsilon$-neighbourhood of $\bigcup_{i=1}^n \mathrm{SC}_i$ (here we mean implicitly that saddle points belong to the connection between them). We assume that $\varepsilon$ is sufficiently small, so that
Throughout this paper we fix the field $v_0$, smoothness exponent $r$, basis system of connections $\{\mathrm{SC}_i\}_{i=1}^n$, cross-sections $\Gamma_i$, $i=1, \dots, n$, and neighbourhood $U_\varepsilon$. If we perturb $v_0$ slightly, then some connections in the basis system open up and new connections can form. Definition 4. We say that a separatrix connection of a field $v$ close to $v_0$ is interior for the basis system of saddle connections $\mathrm{SC}_i$, $i=1, \dots, n$, of the field $v_0$ if it lies fully in the neighbourhood $U_\varepsilon$ (Figure 2, b). Other connections of $v$ are said to be exterior. Although we will only deal with interior connections, we give an example of an exterior one. In Figure 3, a, the basis set consists of a single connection, which is a separatrix loop of the saddle $S_1$. After a small perturbation a connection shown by dashed line can appear. It is exterior to the basis system because the saddle point $S_2$ does not lie in an arbitrarily small neighbourhood $U_\varepsilon$ of the separatrix loop. If we add a separatrix loop of the saddle $S_2$ (Figure 3, b) to the basis system, then $S_2$ lies in the neighbourhood $U_\varepsilon$. Nonetheless, the connection shown by dashed line will still be exterior because it connects saddle points in different components of $U_\varepsilon$. We denote interior separatrix connections by $\mathrm{ISC}$ (from ‘interior saddle connection’): for instance, $\mathrm{ISC}_1$, $\mathrm{ISC}_2$ and so on. Clearly, an interior separatrix connection contains the same saddle points as basis connections do. Definition 5. We call a polycycle $\widetilde \gamma$ born in the family $V=\{v_\delta\}_\delta$, $\delta \in (\mathbb{R}^k,0)$, a child polycycle of a given polycycle $\gamma$ of $v_0$ if it tends to $\gamma$ as $\delta \to 0$ in the Hausdorff metric. This convergence in the Hausdorff metric shows that each child polycycle $\widetilde \gamma$ is formed by interior connections. 1.3. The main resultsThe set of fields in the space $\operatorname{Vect}^r(\mathcal{M})$, $r \geqslant 3$, that are close to $v_0$ and have the same set of connections $\{\mathrm{SC}_i\}_{i=1}^n$ forms a $C^{r-1}$-smooth Banach submanifold $\mathcal{X}_{\mathrm{SC}_1\dots \mathrm{SC}_n}$ of codimension $n$ (see § 2.3, where we also present the definition of a Banach submanifold). The main results in this paper are stated as three theorems. Theorem 2. Let $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$ be interior saddle connections no two of which intersect the same cross-section. Then each $C^1$-smooth manifold $V$ that intersects the set $\mathcal{X}_{\mathrm{SC}_1\dots \mathrm{SC}_n}$ transversally at a point $v_0$ also intersects transversally in a small neighbourhood of $v_0$ the Banach submanifold $\mathcal{X}_{\mathrm{ISC}_1\dots \mathrm{ISC}_l}$ corresponding to the system of connections $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$. This theorem illustrates the general principle: ‘transversality to a smaller object implies transversality to a larger one’. It is a simple consequence of Theorem 3. One of the central problems in bifurcation theory is the description of perturbations of a vector field. This means implicitly that we must construct the bifurcation diagrams of families of perturbations (that is, of the sets of values of parameters corresponding to degenerate fields). However, if the original field is strongly degenerate, then such a diagram can be so complex that it is difficult to describe it in one paper because of the amount of work required. The first solution suggesting itself is to consider only fields with simple degeneracies (for instance, see [1], [20], [5], [17], [11] and [19]). Another possible way out is to abandon the idea of a full analysis of the bifurcation and look only at some elements of the bifurcation diagram, but consider arbitrarily strongly degenerate vector fields. We take the second way in this paper and will only be interested in saddle connections. Definition 6. Assume that a field $v_0$ is perturbed in a family $V$ with parameter base $B$. Consider the set $X_{\mathrm{ISC}}\subset B$ of values of the parameters such that the corresponding fields in the family have an interior separatrix connection $\mathrm{ISC}$. Then we call $X_{\mathrm{ISC}}$ the bifurcation set of the separatrix connection $\mathrm{ISC}$. By the bifurcation set of a system of connections $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$ we mean $X_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}=X_{\mathrm{ISC}_1} \cap \dots \cap X_{\mathrm{ISC}_l}$. Theorem 3 (theorem on smooth structure). Let $V$ be a $C^r$-smooth, $r\geqslant 3$, finite-parameter family of vector fields with parameter base $B=(\mathbb{R}^k, 0)$ in the space $\operatorname{Vect}^r(\mathcal{M})$ that intersects transversally the Banach submanifold $\mathcal{X}_{\mathrm{SC}_1\dots \mathrm{SC}_n}$. Let $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$ be interior separatrix connections no two of which intersect the same cross-section. Then the bifurcation set $X_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ is a $C^r$-smooth manifold of codimension $l$. Let us explain why in Theorems 2 and 3 we say that connections intersect cross-sections. By an interior connection $\mathrm{ISC}$ of a field $v$ close to $v$ we do not merely mean a connection between two particular saddle points but one that is also close to some particular basis connections of the original field $v_0$. That is, to define an interior connection $\mathrm{ISC}$ we must also indicate a ‘path’ it traverses. We write out all cross-sections intersecting an interior connection $\mathrm{ISC}$ of $v$: $\Gamma_{\sigma_1},\dots,\Gamma_{\sigma_s}$, $\sigma_i \in \{1, \dots, n\}$ (see Figure 2, b). Some of them can repeat. The order of cross-sections in this set is determined by moments of time on the $\mathrm{ISC}$. If the separatrix connection $\mathrm{ISC}$ is preserved (continues to be formed by the same separatrices) after a small perturbation of $v$, then it intersects the same ordered tuple of cross-sections. Another obvious statement is that a field $v$ close to $v_0$ has no two interior separatrix connections intersecting the same tuple of cross-sections. Both results follows from the definition of the neighbourhood $U_\varepsilon$. Thus, in what follows, to define consistently an interior separatrix connection we indicate an ordered tuple of cross-sections it intersects. We identify interior connections of distinct fields close to $v_0$ that intersect the same ordered tuple of cross-sections, and we distinguish connections intersecting distinct tuples, even when they connect the same saddle points. To each interior connection we assign the word $\sigma_1\dots\sigma_s$ in the finite alphabet $\{1,\dots,n\}$ that is formed by the indices of the cross-sections $\Gamma_{\sigma_1},\dots,\Gamma_{\sigma_s}$ it intersects. The order of characters in this word depends on the order of points of intersection of the connection with cross-sections. Since a separatrix connection can intersect the same cross-section several times, indices in a word can repeat. Below, in place of ‘the interior connection $\mathrm{ISC}$ corresponding to the word $\sigma_1\dots\sigma_s$’ we say ‘the interior connection $\sigma_1\dots\sigma_s$’, and we write $\mathrm{ISC}=\sigma_1\dots\sigma_s$. Let $A=\mathrm{ISC}_1,\dots,\mathrm{ISC}_l$ be a set of words in the alphabet $\{1,\dots,n\}$. Although words in this set can denote interior connections ($\mathrm{ISC}_i$), connections corresponding to these words are not necessarily born in perturbations of basis connections. The question is: what finite sets of words in the alphabet $\{1,\dots,n\}$ correspond to systems of interior connections appearing after perturbations of basis connections in a generic family? By the bifurcation set of a set of words $A$ we mean the bifurcation set of the corresponding system of interior connections and denote it by $X_A$. In particular, the above question on the birth of a system of connections in a family can also be stated as to whether the set $X_A$ is nonempty. In § 4.3 we introduce the class $\mathcal{A}$ of sets of words in the alphabet $\{1,\dots,n\}$ for which the following result holds. Theorem 4 (sufficient condition for the birth of a set of connections). Let $V$ be a $C^r$-smooth, $r\geqslant 3$, finite-parameter family of vector fields with parameter base $B=(\mathbb{R}^k, 0)$ in the space $\operatorname{Vect}^r(\mathcal{M})$ that intersects transversally the Banach submanifold $\mathcal{X}_{\mathrm{SC}_1\dots \mathrm{SC}_n}$. Then for each $A \in \mathcal{A}$ the bifurcation set $X_A$ is nonempty. The paper is organized as follows. In § 2 we introduce the main notation and, prove Theorem 2 using Theorem 3. The whole of § 3 is taken by the proof of Theorem 3, while § 4 is devoted to the description of the topological properties of the bifurcation sets of interior separatrix connections. In § 5 we prove Theorem 1. § 2. Preparatory steps2.1. Coordinates on cross-sectionsLet the field $v_0$ contain basis connections $\mathrm{SC}_1, \dots, \mathrm{SC}_n$, and let it be perturbed in a $C^r$-smooth $k$-parameter family $V=\{v_\delta\}$, $\delta \in B=(\mathbb{R}^k, 0)$, $k\geqslant n$. The reader will see below that the parameters $\delta_1, \dots, \delta_n$ are the ones destroying basis separatrix connections, while $\delta_{n+1}, \dots, \delta_k$ are some other parameters independent of the former. We assume that for each $i=1, \dots, n$ the transversal cross-section $\Gamma_i$ of the basis connection $\mathrm{SC}_i$ is independent of $\delta$, and $\delta$ is taken to be sufficiently small so that each $\Gamma_i$ remains transversal to the perturbed vector field $v_\delta$. Each basis connection $\mathrm{SC}_i$ is formed by two separatrices, an incoming and an outgoing one. We denote the point of intersection of $\Gamma_i$ with the incoming separatrix by $ s_i$ (from ‘stable’). We denote the point of intersection of $\Gamma_i$ with the outgoing separatrix by $u_i$ (from ‘unstable’). Since the separatrices of the unperturbed field are bound, for $i=1, \dots, n$ the points $u_i$ and $s_i$ on $\Gamma_i$ coincide. Assume that $v_0$ has two basis connections $\mathrm{SC}_{i-1}$ and $\mathrm{SC}_i$ sharing a saddle $S$: $\mathrm{SC}_{i-1}$ comes into $S$, while $\mathrm{SC}_i$ goes out of it. The saddle $S$ has a unique hyperbolic sector bounded by parts of $\Gamma_{i-1}$ and $\Gamma_i$. We denote these parts (hafl-sections) by $\Gamma_{i-1}^-$ and $\Gamma_i^+$, respectively. Now consider a perturbed field $v_\delta$. By analogy we define the points $s_i(\delta)$ and $u_i(\delta)$ and the hafl-sections $\Gamma_{i-1}^-(\delta)$ and $\Gamma_i^+(\delta)$ for the sector of the saddle ${S=S(\delta)}$ that is close to the sector of $S(v_0)$ considered above. Generally, the points $s_i(\delta)$ and $u_i(\delta)$ need not coincide on $\Gamma_i$, which means that the connection $\mathrm{SC}_i$ opens up. Consider some Riemannian metric on $\mathcal{M}$. Then we can introduce a natural parameter (fix a chart) on each smooth curve: the modulus of the difference between the coordinates of two points in this chart is equal to the length of the segment of the curve between these points. For $\Gamma_i$ let $\Gamma_i^-$ and $\Gamma_i^+$ be the two hafl-sections corresponding to the two sectors adjoining $\Gamma_i$. On $\Gamma_i$ we choose a natural parameter (a chart) so that $s_i(\delta)$ has coordinate 0 and the points on $\Gamma_i^-(\delta)$ have positive coordinates. We also choose another natural parameter (chart) on $\Gamma_i$ so that $u_i(\delta)$ has coordinate 0 and the points on $\Gamma_i^+(\delta)$ have positive coordinates. Since there are four sectors adjoining $\Gamma_i$ (two sectors for each saddle point on $\mathrm{SC}_i$), we have thus introduced up to four charts on $\Gamma_i$. Note that each phase curve of the perturbed field $v_\delta$ (which could even cross all four sectors in general) traverses only two sectors in a neighbourhood of its point of intersection with $\Gamma_i$, namely, sectors lying on different sides of $\Gamma_i$. Consider two sectors adjoining $\Gamma_i$ on different sides of it. We describe the transition from the chart corresponding to the hafl-section $\Gamma_i^+(\delta)$ (adjoining one of these sectors) to the one corresponding to the hafl-section $\Gamma_i^-(\delta)$ (adjoining the other sector). Let $\tau_i(\delta)$ be the coordinate of $u_i(\delta)$ in the chart corresponding to $\Gamma_i^-(\delta)$. In all results established in our paper the family $V$ under consideration intersects transversally the Banach submanifold $\mathcal{X}_{\mathrm{SC}_1 \dots \mathrm{SC}_n}$ corresponding to the basis connections, and therefore we can assume that $|\tau_i(\delta)|=|\delta_i|$. We also assume that for $\delta_i > 0$ the outgoing separatrix of $\mathrm{SC}_i$ lies to the right (in the direction of the vector field) of the incoming one, while for $\delta_i < 0$ this order is reversed. We call $\delta_i$ the splitting parameter of the separatrix connection $\mathrm{SC}_i$. Then the transition from the chart corresponding to $\Gamma_i^+(\delta)$ to the one corresponding to $\Gamma_i^-(\delta)$ is described by the map
$$
\begin{equation}
x \mapsto \epsilon \delta_i + \eta x, \qquad \epsilon, \eta=\pm 1.
\end{equation}
\tag{2}
$$
The signs $\epsilon$ and $\eta$ in (2) depend on whether the sectors in the unperturbed field $v_0$ lie on the same side (Figure 4, a) or on different sides (Figure 4, b) of the connection $\mathrm{SC}_i$, and also on the direction of the vector field (see details in § 3.3). 2.2. An equation for an interior saddle connectionGiven $\delta \in B$, let the field $v_\delta$ have an interior separatrix connection $\mathrm{ISC}=\sigma_1\dots \sigma_s$. It crosses the interiors of $s-1$ sectors (some of which can coincide) of saddles. For each $i=1, \dots, s-1$ the $i$th sector crossed is bounded by the hafl-sections $\Gamma_{\sigma_i}^-(\delta)$ and $\Gamma_{\sigma_{i+1}}^+(\delta)$. Then for each $i=2, \dots, s$ we have a correspondence map $\Delta_i(\delta, \,\cdot\,)\colon \Gamma_{\sigma_{i-1}}^-(\delta) \to \Gamma_{\sigma_i}^+(\delta)$ defined in the $(i-1)$st (in order) sector. The charts on $\Gamma_{\sigma_{i-1}}^-(\delta)$ and $\Gamma_{\sigma_i}^+(\delta)$ (see § 2.1) are defined so that $\Delta_i(\delta, \,\cdot\,)$ takes the form $\Delta_i(\delta, \,\cdot\,)\colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ in these coordinates. If the field is $C^r$-smooth, then this map is $C^r$-smooth in $x$ (a simple consequence of the theorem on straightening a vector field). Moreover, each of its $x$-derivatives depends continuously (in fact, $C^r$-smoothly) on $\delta$. The charts selected in this way are said to be natural for the sector under consideration. Lemma 1 ([8], Lemma 6). Let the $C^3$-smooth family $V=\{v_\delta\}_\delta$, $\delta \in B$, where $B= (\mathbb{R}^k, 0)$, perturb a vector field $v_0 \in \operatorname{Vect}^3(\mathbb{R}^2)$ with a hyperbolic saddle $S$. Then the correspondence map $\Delta_S(\delta, x)$ of $S(\delta)$ has the following properties in the natural charts on cross-sections: Both expressions tend to zero uniformly with respect to $\delta$ in a neighbourhood of zero in $B$. Consider the following functions:
$$
\begin{equation}
\begin{gathered} \, f_1(\delta, x)=\epsilon_1 \delta_{\sigma_1}, \qquad \epsilon_1=\pm 1; \\ f_i(\delta, \,\cdot\,)\colon \Gamma_{\sigma_{i-1}}^-(\delta) \to \Gamma_{\sigma_i}(\delta)\quad\text{and} \quad f_i(\delta, x)=\epsilon_i\delta_{\sigma_i} + \eta_i \Delta_i(\delta, x); \\ \epsilon_i, \eta_i=\pm 1, \qquad i=2, \dots, s. \end{gathered}
\end{equation}
\tag{4}
$$
For each $i=2,\dots, s$ the map $f_i(\delta, \,\cdot\,)$ is a composition of the correspondence map $\Delta_i(\delta, \,\cdot\,)$ and the attaching map (2). The signs $\epsilon_i$ and $\eta_i$ depend on the position of the $(i-1)$st and $i$th in order sectors with respect to the connection $\mathrm{SC}_i$ of the unperturbed field $v_0$ (see details in § 3.3). The bifurcation set $X_{\mathrm{ISC}}$ of the connection $\mathrm{ISC}$ has the equation Here and in what follows, by the composition $g \circ h$ of two functions $g(\delta, x)$ and $h(\delta, x)$ we mean $g \circ h(\delta, x)=g(\delta, h(\delta, x))$.2.3. Banach submanifolds. Plan of the proof of Theorem 2Definition 7. Consider a subset $\mathcal{X}$ of a normed space. We say that $\mathcal{X}$ is a $C^r$-smooth Banach submanifold of codimension $k \geqslant 0$, $r\geqslant 1$, if for each point $x\in \mathcal{X}$ there exist a neighbourhood $W$ of $x$ in the ambient space and a $C^r$-smooth function $\mathbf{F}\colon W\to \mathbb{R}^k$ such that: We show that close fields with a saddle connection form a smooth Banach submanifold. Lemma 2. Assume that a $C^r$-smooth vector field $v_0$, $r \geqslant 2$, on a 2-manifold $\mathcal{M}$ has a saddle connection $\mathrm{SC}$. Then the vector fields close to $v_0$ in $\operatorname{Vect}^r(\mathcal{M})$ which have a saddle connection close to $\mathrm{SC}$ form a $C^{r-1}$-smooth Banach submanifold $\mathcal{X}$. Moreover, for each point $x_0 \in \mathrm{SC}$ there exist an arbitrarily smooth neighbourhood $Ox_0$ of it and an vector field $h$ with support in this neighbourhood such that $d\mathbf{F}(v_0)[h] \neq 0$, were $\mathbf{F}$ is the functional defining $\mathcal{X}$. To prove1 Lemma 2 we need the following result due to Sotomayor [20]. Lemma 3 ([20], Lemma 4.3). Let $S$ be a hyperbolic saddle of a field $v_0 \in \operatorname{Vect}^r(\mathcal{M})$, $r \geqslant 1$. Then there exist a neighbourhood $W$ of $v_0$ and a neighbourhood $U$ of the saddle point $S$ such that the following results hold. 1. For each field $v \in W$ $U$ contains a unique singular point of $v$, which is a hyperbolic saddle $S(v)$. 2. Each of the separatrices $\gamma_1$, $\gamma_2$, $\gamma_3$, $\gamma_4$ of $S(v)$ intersects the boundary of $U$ at a unique point $y_i(v)$, $i=1,2,3,4$. 3. The boundary $\partial U$ is smooth; moreover, for each $i=1,\dots,4$ there exists an arc $C_i \ni y_i(v)$ of $\partial U \ni y_i(v)$ transversal to all fields $v \in W$ such that the map $y_i\colon W \to C_i$ is $C^{r-1}$-smooth.2 Proof of Lemma 2. The correspondence map along a trajectory tube not containing singular points in its closure depends $C^r$-smoothly on the vector field. This is because in a neighbourhood of each smooth point there exists a straightening chart, which depends smoothly on the field (see [2], Theorem 21.6). Hence on a smooth cross-section of the separatrix connection, for fields close to $v_0$ points of its intersection with separatrices of saddles forming the connection are well defined, and by Lemma 3 the splitting parameter $\mathbf{F}(v)$, equal to the difference between the coordinates of these points, is a $C^{r-1}$-smooth map. This functional vanishes when there is a saddle connection. It only remans to show that this map has a surjective differential.
By the theorem on straightening a vector field there exists a neighbourhood $Ox_0$ of $x_0$ such that in some coordinates $(x,y)$ in this neighbourhood the field has the form $v_0= \frac{\partial}{\partial x}$. Now we stay within this straightening coordinate chart. Consider an infinitely smooth field $h=\rho(x,y)\,\frac{\partial}{\partial y}$, where $\rho(x,y)$ is positive in $Ox_0$ and vanishes identically on the whole manifold $\mathcal{M}$ outside $Ox_0$. In this neighbourhood the field is orthogonal to $v_0$. Consider the one-parameter family of vector fields $v_0+\xi h$, where $\xi\in(\mathbb{R}, 0)$. In the straightening chart this field has the form
$$
\begin{equation}
\begin{cases} \dot{x}=1, \\ \dot{y}=\xi \rho(x,y). \end{cases}
\end{equation}
\tag{6}
$$
The phase curves $y_\xi(x)$ of this system depend on the parameter $\xi$, and for $\xi=0$ the phase curves have the equations $y_0(x)=\mathrm{const}$. Therefore,
$$
\begin{equation}
F(v_0+\xi h)=y_\xi(x)-y_0(x)=\xi \int_{-\infty}^x\rho(x,y_\xi)\,dx-y_0(x).
\end{equation}
\tag{7}
$$
Thus,
$$
\begin{equation*}
\frac{d}{d\xi} F(v_0+\xi h) =\int_{-\infty}^x\rho(x,y_\xi)\,dx+\xi\int_{-\infty}^x \frac{d}{d\xi}\rho(x,y_\xi)\,dx.
\end{equation*}
\notag
$$
By the theorem on the continuous dependence of the solution on a parameter the second term on the right tends to zero as $\xi\to 0$. Since $\rho$ is nonnegative and distinct from identical zero in $Ox_0$, we have $dF(v_0)[h]\neq 0$. This inequality also holds in the original coordinate chart.
Lemma 2 is proved. We can generalize the above result to the case of several connections. Proposition 1. Let $v_0 \in \operatorname{Vect}^r(\mathcal{M})$, $r \geqslant 2$, be a vector field with saddle connections $\mathrm{SC}_1,\dots,\mathrm{SC}_n$. Then the set of fields $v \in \operatorname{Vect}^r(\mathcal{M})$ close to $v_0$ and with saddle connections close to $\mathrm{SC}_1,\dots,\mathrm{SC}_n$ is a $C^{r-1}$-smooth Banach submanifold $\mathcal{\mathcal{X}}_{\mathrm{SC}_1 \dots \mathrm{SC}_n}$ of codimension $n$. Proof. By Lemma 2, for each separatrix connection $\mathrm{SC}_i$, $i=1,\dots,n$, there is a $C^{r-1}$-smooth functional $\mathbf{F}_i$ vanishing on all fields close to $v_0$ that have a saddle connection close to $\mathrm{SC}_i$. We can see from the proof of that lemma that $\mathbf{F}_i$ is the splitting parameter of $\mathrm{SC}_i$ on some $C^r$-smooth cross-section of this connection. Hence in a neighbourhood of $v_0$ the map $\mathbf{F}=(\mathbf{F}_1,\dots,\mathbf{F}_n)$ is a smooth map taking the value zero if and only if the field has $n$ saddle connections close to the original set of connection. Moreover, by Lemma 2, for each connection $\mathrm{SC}_i$ there exists a vector field $h_i$ with support in an arbitrarily small neighbourhood of a point on $\mathrm{SC}_i$ such that $\mathbf{F}_i(v_0)[h_i] \neq 0$. Since such neighbourhoods are arbitrarily small we can assume that they are disjoint and do not intersect other separatrix connections in the set. Thus, which means that the differential $d\mathbf{F}(v_0)$ is surjective.
The proof is complete. Remark 1. Bearing in mind the notation in § 2.1 we can assume that for each field $v_\delta$ in the generic family $V$ we have $\mathbf{F}_i(v_\delta) \equiv \delta_i$. Now we go over to the proof of Theorem 2. We need the following result. Proposition 2 ([6], Proposition 6). Let $W$ be an open subset of a Banach space, and let $\mathbf{F}\colon W \,{\to}\, \mathbb{R}^n$ be a map nondegenerate at all points in $W$. Let $V\colon {(\mathbb{R}^k,0) \,{\to}\, W}$, $k \geqslant n$, be a $C^1$-smooth family such that $\mathbf{F}(v_0)=0$ at some point $v_0\in W$ and $\mathbf{F}|_V$ is nondegenerate at $v_0$. Then $V$ intersects transversally the Banach submanifold $\{v \in W| \mathbf{F}(v)=0\}$ at $v_0$. Proof of Theorem 2. By Proposition 1, for each field $v$ in $V$ the set of vector fields in $\operatorname{Vect}^r(\mathcal{M})$ that are close to $v$ and have a set of connections $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$ is a Banach submanifold $\mathcal{X}_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$. By the definition of a Banach submanifold this set if a level surface of some map $\mathbf{F}$. On the other hand the family $V\colon {B \!\to\! \operatorname{Vect}^r(\mathcal{M})}$ is an embedding, and by Theorem 3 the bifurcation set $X_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ is a smooth manifold. Hence $\mathbf{F}|_V$ is nondegenerate. By Proposition 2 the family $V$ intersects the surface $\mathcal{X}_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ transversally.
The proof is complete. Thus Theorem 2 is proved, provided that Theorem 3 holds. Remark 2. In Theorems 1–4 we assume the smoothness exponent $r\geqslant 3$ of the fields because Lemmas 1 and 2 are used in their proofs. In all other arguments and constructions we only need $C^1$-smoothness. The smoothness assumptions in these two lemmas are perhaps excessive, so if one proves their analogues for fields that are just $C^1$-smooth, then all our theorems here will automatically be improved. § 3. Proof of Theorem 33.1. The plan of the proof of the theorem on a smooth structureFirst we look at the special case of Theorem 3 on a smooth structure, when the set of interior connections consists of a single connection, denoted by $\mathrm{ISC}$. We show that the bifurcation set $X_{\mathrm{ISC}}$ of the interior separatrix connection $\mathrm{ISC}$ is a $C^r$-smooth manifold. The interior separatrix connection is defined by equation (5). The following (flawed) strategy of the proof suggests itself: find the limit of $dF(\delta)$ as $\delta \to 0$. If this limit is finite and distinct from the zero linear functional (covector), then the theorem on a smooth structure is established for the connection $\mathrm{ISC}$. In fact, in this case the function $F(\delta)$ has a nontrivial differential for small $\delta$. Example. Unfortunately, this limit does not necessarily exist. Consider a connection $\mathrm{ISC}$ intersecting precisely two cross-sections $\Gamma_1$ and $\Gamma_2$. In particular, it goes close to a saddle point $S$ with correspondence map $\Delta_S(x)$. Any correspondence map of a hyperbolic saddle has a powerlike asymptotic behaviour with exponent $\lambda > 0$, where $\lambda$ is the characteristic number of $S$ (see Definition 3). In particular, for $\lambda(0) \notin \mathbb{Q}$ it is smoothly conjugate to the map $x \mapsto x^{\lambda(\delta)}$. We do not use this fact in our paper, so we refer the reader to [9] for details. Nevertheless, we can assume that the $\mathrm{ISC}$ has the equation $\delta_2 \pm \delta_1^\lambda=0$, and the differential of the left-hand side need not have a limit as $\delta_1, \delta_2 \to 0$ and $\lambda(\delta) \to 1$ (Figure 5). In view of the above example, instead of the limit of $dF(\delta)$ as $\delta \to 0$ we look for its set of limit points. For any $m$ covectors $u_1, \dots, u_m$ in the dual space $T^*_\delta B$ we set
$$
\begin{equation}
\langle u_1,\dots, u_m\rangle_+=\bigl\{a_1 u_1 +\dots + a_m u_m \mid \forall\, i \ a_i \geqslant 0,\,a_1 + \dots + a_m \neq 0 \bigr\}.
\end{equation}
\tag{8}
$$
Thus, the operation $\langle\dots\rangle_+$ assigns to several covectors a convex subset of $T^*_\delta B$. For each $i=1, \dots, s$, to the map where the functions $f_i$ are defined by (4), we assign the set $M_i$ defined recursively as follows:
$$
\begin{equation}
M_1=\langle \epsilon_1\, d\delta_{\sigma_1}\rangle_+, \qquad M_i=\langle \epsilon_i \,d\delta_{\sigma_i}, \eta_i M_{i-1}\rangle_+.
\end{equation}
\tag{10}
$$
The constants $\epsilon_j$ and $\eta_j$, $j=1, \dots, i$, are as in (4). Definition 8. We call the set $M=M_s\subset T_\delta^* B$ described above the restricting set for the function $F=F_s$ or for the connection $\mathrm{ISC}$ with equation (5). Lemma 4. For $i=1,\dots,s$ the restricting set for the function $F_i$ defined by (9) has the representation Furthermore, we consider the map
$$
\begin{equation}
\begin{gathered} \, \varphi_\delta\colon T^*_\delta B\setminus\{0\}\to \mathbf{Gr}(k-1, T_\delta B), \nonumber \\ \varphi_\delta\colon u \mapsto \operatorname{Ker}u, \end{gathered}
\end{equation}
\tag{11}
$$
where $\mathbf{Gr}(k-1, T_\delta B)$ is the Grassmannian manifold of all linear hyperplanes in the space $T_\delta B$. The value of a differential form $dF(\delta)$ at a point $\delta$ is a linear functional (a covector) in $T^*_\delta B$. Hence for each noncritical point $\delta$ the value $\varphi_\delta (dF(\delta))$ is the tangent space $T_\delta X_{\mathrm{ISC}}$ to the bifurcation set $X_{\mathrm{ISC}}$. The bifurcation set of the connection $\mathrm{ISC}$ and the restricting set for the function $F$ defining the connection are related by the following statement, proved in § 3.2. Proposition 3. Under the assumptions of Theorem 3 let the restricting set $M_i$ for the function $F_i$ in (9) contain the zero covector. Then for each small $\delta$ the differential $d F_i$ at $\delta$ is distinct from zero and the tangent space $\varphi_\delta(dF_i)$ tends to the set $\varphi_0(M_i)$ as $\delta \to 0$. Proof of Theorem 3. It follows from Lemma 4 that for $i=1, \dots, s$ the sets $M_i$ do not contain the covector zero, and therefore by Proposition 3 the bifurcation set $X_{\mathrm{ISC}}$ of the interior separatrix connection $\mathrm{ISC}$ is a smooth manifold.
Let $M_{\mathrm{ISC}_1}, \dots, M_{\mathrm{ISC}_l}$ be the restricting sets for the connections $\mathrm{ISC}_1,\dots, \mathrm{ISC}_l$. By assumption these connections intersect pairwise distinct cross-sections, so by Lemma 4 the sets $M_{\mathrm{ISC}_1}, \dots, M_{\mathrm{ISC}_l}$ are generated by distinct sets of covectors $d\delta_i$. Hence any covectors $u_j \in M_{\mathrm{ISC}_j}$, $j=1, \dots, l$, are linearly independent. For each small $\delta \in B$ let $\mathcal{T}_\delta \subset (\mathbf{Gr}(k-1, T_\delta B))^l$ denote the set of systems of $l$ transversally intersecting hyperplanes. The set $\mathcal{T}_\delta$ is open, and by the above it contains the set $\varphi_\delta(M_{\mathrm{ISC}_1}) \times \dots \times \varphi_\delta(M_{\mathrm{ISC}_l})$, where $\varphi_\delta$ is the map defined by (11). By Proposition 3 the system of tangent hyperplanes $(T_\delta X_{\mathrm{ISC}_1}, \dots, T_\delta X_{\mathrm{ISC}_l})$ tends to the (closed) set $\varphi_0(M_{\mathrm{ISC}_1}) \times \dots \times \varphi_0(M_{\mathrm{ISC}_l})$ as $\delta \to 0$, so if a point $\delta$ is sufficiently close to zero, then the tangent hyperplane $\delta X_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ lies in the open set $\mathcal{T}_0$. We have proved Theorem 3, provided that Lemma 4 and Proposition 3 hold. 3.2. Tangent hyperplanes to the bifurcation setIn this subsection we prove Proposition 3. Proof of Proposition 3. We use indiction on $i=1,\dots,s$.
The base of induction: $i=1$. Note that throughout the proof we denote the differential with respect to the parameter $\delta$ by $d$. From (4) we obtain
$$
\begin{equation*}
d F_1=d f_1=\epsilon_1\, d\delta_{\sigma_1} \in M_1=\langle \epsilon_1 d\delta_{\sigma_1}\rangle_+.
\end{equation*}
\notag
$$
The base of induction is established.
The step of induction. Assume that we have proved the result for some $i-1$. Then by (9) We prove that for each $i=1, \dots, s$ we have the recurrence relation
$$
\begin{equation}
d F_i=\epsilon_i \,d\delta_{\sigma_i} + \eta_i\,\frac{\partial \Delta_i}{\partial x}(F_{i-1})\,d F_{i-1} + o(1)
\end{equation}
\tag{13}
$$
as $\delta \to 0$. In fact, taking the differential of (12) we obtain
$$
\begin{equation}
d F_i=d f_i\big|_{F_{i-1}} + \frac{\partial f_i}{\partial x}(F_{i-1})\,d F_{i-1}.
\end{equation}
\tag{14}
$$
Here and below we write $\big|_{F_{i-1}}$ to indicate the substitution as the variable $x$.
Note that by property (3) of the correspondence map $\Delta_i$ of a saddle and by the definition (4) of $f_i$ we have
$$
\begin{equation}
d f_i\big|_{F_{i-1}}=\epsilon_i \,d\delta_{\sigma_i} + \eta_i \,d \Delta_i \big|_{F_{i-1}}= \epsilon_i\, d\delta_{\sigma_i} + o(1)
\end{equation}
\tag{15}
$$
as $\delta \to 0$. We substitute (15) into (14). Since (4) implies that $\frac{\partial f_i}{\partial x}=\eta_i\frac{\partial \Delta_i}{\partial x}$, this yields (13).
Consider an arbitrary sequence of values of $\delta(\alpha) \to 0$ such that the expression $\frac{\partial \Delta_i}{\partial x}(F_{i-1})$ has a (finite or infinite) limit and the hyperplane $\varphi_{\delta(\alpha)}(dF_{i-1})$ has a limit $\varphi_0 (u)$, where the covector $u$ belongs to the set $M_{i-1}$. This is possible by the induction hypothesis. We look at two cases.
In either case the argument of the limit function $\varphi_0$ is a nontrivial linear combination of the covectors $d\delta_{\sigma_i}$ and $u$. Since $u$ belongs to $M_{i-1}$ by the induction hypothesis, this linear combination belongs to $M_i$ by the definition of restricting sets (see (10)). By assumption the set $M_i$ does not contain the zero covector, so for each plane $\varphi_0(u)$, $u \in M_i$, and each sequence of values of the parameter $\delta(\alpha) \to 0$ such that the limit $\varphi_{\delta(\alpha)}(dF_i) \to \varphi_0(u)$ exists as $\delta(\alpha) \to 0$, there exists a sequence of scalars $c_\alpha \neq 0$ such that $c_\alpha\, dF_i\to u \neq 0$ as $\delta(\alpha) \to 0$. Hence for small $\delta$ the differential $dF_i(\delta)$ is distinct from zero. Proposition 3 is proved. 3.3. The lemma on restricting sets Proof of Lemma 4. Each basis saddle connection is formed by two separatrices, one going out of some saddle point and another coming in some (maybe the same) saddle point. Consider the following auxiliary result: if an interior separatrix connection $\mathrm{ISC}$ intersects the cross-section $\Gamma_{\sigma_i}$ to the right of the incoming separatrix of $\mathrm{SC}_{\sigma_i}$ (in the direction of the vector field), then the coefficient $\epsilon_i$ is $+1$, while if $\mathrm{ISC}$ intersects the cross-section to the left of this separatrix, then $\epsilon_i$ is $-1$.
Using induction on $i=1, \dots, s$ we prove this auxiliary result alongside the proof of Lemma 4. The base of induction. Let $i=1$. Then by (10)
$$
\begin{equation*}
M_1=\langle\epsilon_1 d\delta_{\sigma_1}\rangle_+=\epsilon_1 \langle d\delta_{\sigma_1}\rangle _+,
\end{equation*}
\notag
$$
which is the assertion of the lemma. Since $f_2(\delta, x)$ is only defined for $x>0$, by formulae (4) and (5) the quantity $f_1(\delta, 0)=\epsilon_1\delta_{\sigma_1}$ must be positive. However, $\delta_{\sigma_1}$ is the splitting parameter of the connection $ \mathrm{SC}_{\sigma_1}$, and by definition (see § 2.1) it is positive if and only if the outgoing separatrix of $\mathrm{SC}_{\sigma_1}$ lies to the right of the incoming one. Hence the auxiliary result also holds. The base of induction is established.
The step of induction. Assume that both the auxiliary result and the assertion of the lemma have been proved for $i-1$. Also assume that the splitting parameter $\delta_{\sigma_i}$ is nonzero, that is, the basis connection $\mathrm{SC}_{\sigma_i}$ is opened up. Furthermore, let $i \neq s$, which means that $\mathrm{ISC}$ does not coincide with the incoming separatrix of the basis connection $\mathrm{SC}_{\sigma_i}$ (opened up in the field $v_\delta$). We discuss the exceptional cases below, at the end of this proof. The step of induction follows because the manifold $\mathcal{M}$, which is the phase space here, is orientable, from formula (10), by looking at the six possible cases listed in Table 1 (we illustrate all these cases by Figure 6). Table 1.Calculating the sign of the restricting set $M_i$
Now we explain how Table 1 implies Lemma 4. In each of the six cases (see the field ‘Case’ in the table), using the appropriate figure (the field ‘Figure’) we establish the explicit form of the map
$$
\begin{equation*}
f_i(\delta, x)=\epsilon_i\,\delta_{\sigma_i} + \eta_i\,\Delta_i(\delta, x)
\end{equation*}
\notag
$$
(the field ‘$f_i(\delta, x)$’) and the position of the connection $\mathrm{ISC}$ with respect to the outgoing and incoming separatrices of the basis connection $\mathrm{SC}_{\sigma_i}$ (the fields ‘Was’ and ‘Now is’, respectively). From the formula for $f_i(\delta, x)$ we find the constants $\epsilon_i$ (the field ‘$\epsilon_i$’) and $\eta_i$ (the field ‘$\eta_i$’). By ‘the sign of the restricting set
$$
\begin{equation*}
M_{i-1}=\pm \langle d\delta_{\sigma_1},\dots, d\delta_{\sigma_{i-1}}\rangle _+\,'
\end{equation*}
\notag
$$
we mean the sign before the expression $\langle d\delta_{\sigma_1},\dots, d\delta_{\sigma_{i-1}}\rangle _+$. We can find it using the induction hypothesis: by the auxiliary result, if the entry of the field ‘Was’ is ‘to the right’, then we have a plus sign in the field ‘The sign of $M_{i-1}$’, and otherwise we have a minus sign. After we have filled in all fields in question in the table, we can find the set $M_i$. Using the recurrence formula (10) (the field ‘$M_i$’) we obtain that $M_i$ is equal to $\langle d\delta_{\sigma_1},\dots, d\delta_{\sigma_i}\rangle _+$ with some sign. This sign is put in the field ‘The sign of $M_i$’. Once we have found the sign of $M_i$, we verify that it is equal to the sign of $\epsilon_i$ and corresponds to the expression in the field ‘Now is’ (a plus sign corresponds to the value ‘to the right’ and a minus one to the value ‘to the left’). This verification proves the auxiliary result and Lemma 4 itself. It remains to consider exceptional cases. Let $\delta_{\sigma_i}=0$, that is, assume that the basis connection $\mathrm{SC}_{\sigma_i}$ is bound. Then the arguments are similar to cases 1 and 4 considered above (in the case when $\mathrm{ISC}$ lies to the right of $\mathrm{SC}_{\sigma_i}$), or to cases 3 and 6 (if its lies to the left). Let $i=s$, so that $\mathrm{ISC}$ coincides with the incoming separatrix of the basis connection $\mathrm{SC}_{\sigma_s}$. Then the argument is similar to cases 2 and 3 considered above (for $\delta_{\sigma_s}>0$), or to cases 4 and 5 (for $\delta_{\sigma_s} < 0$). Note that in each pair of cases the functions $f_s$ in the field ‘$f_i(\delta,x)$’ of the table differ by sign, which means generally speaking that the sets $M_s=M$ have different signs. Nevertheless, the function $f_s$ can have either of the representations because it does not matter whether we consider the equation $f_s(F_{s-1})=0$ or $-f_s(F_{s-1})=0$ for the connection $\mathrm{ISC}$. This completes the proof of Theorem 3. Remark 3. It follows from Lemma 4 that the restricting set $M$ of the function $F$ defined by (5) has in fact a very simple structure. Consider the particular case $s=k=n$, when the connection $\mathrm{ISC}=1\dots n$ appears in a perturbation of all $n$ basis connections in a generic $n$-parameter family. By Lemma 4 the restricting set $M$ for the function (5) defining $\mathrm{ISC}$ has the form $M= \pm\langle d\delta_1,\dots, d\delta_n\rangle _+$. Thus, in the linear space $T^*_\delta B$ with basis $\{d\delta_i\}_{i=1}^n$ the set $M$ is one of the $2^n$ (closed) coordinate orthants punctured at the origin. Note, moreover, that we have almost nowhere used the fact that the $\Delta_i(\delta,x)$, $i=1,\dots,s$, are the correspondence maps just for saddle points, apart from the properties provided by Lemma 1. Hence our result can be generalized to separatrix connections between arbitrary singular points. Proposition 4. Let $\mathcal{M}$ be an orientable 2-manifold, and let $v_0 \in \operatorname{Vect}^1(\mathcal{M})$ be a field with $s$ separatrix connections (not necessarily between saddles). Assume that in a perturbation in a $C^1$-smooth finite-parameter family $V= \{v_\delta\}_\delta$, $\delta \in (\mathbb{R}^k,0)$, a separatrix connection $\mathrm{ISC}$, with equation (5) (also see (4)) appears. For all $i=1,\dots,s$ let the correspondence maps $\Delta_i(\delta,x)$ of the $i$th singular point satisfy Then the differential $dF(\delta)$ is distinct from zero for sufficiently small $\delta$. § 4. Properties of bifurcation setsIn this section we describe the topological properties of the bifurcation sets of saddle connections. We also establish a sufficient condition for the birth of a prescribed family of separatrix connections. In all statements in this section (unless otherwise stated) we consider a field $v_0$ with a basis system of saddle connections $\mathrm{SC}_1, \dots, \mathrm{SC}_n$. The field $v_0$ is perturbed by a $C^r$-smooth, $r\geqslant 3$, family $V=\{v_\delta\}$ with parameter base $B=(\mathbb{R}^k, 0)$, $k \geqslant n$, such that $V$ intersects transversally the Banach submanifold $\mathcal{X}_{\mathrm{SC}_1 \dots \mathrm{SC}_n}$ corresponding to the basis connections. 4.1. The boundary of the bifurcation setDefinition 9. Let $X$ be a manifold embedded in Euclidean space. By the boundary of $X$ we mean the set $\overline{X}\setminus X$; we denote it by $\partial X$. Note that although we say about ‘boundaries’, the closure $\overline{X}_{\mathrm{ISC}}$ of the bifurcation set is not necessarily a manifold with boundary in general. Here we describe the structure of the boundary of the bifurcation set of an interior connection. Definition 10. By a sparkling separatrix connection we mean a separatrix connection intersecting some cross-section at least twice. We call all other connections nonsparkling. The reason for this name is as follows: if in the family $V$ an interior connection intersects a cross-section twice, then there are connections in the same family that intersect this connection an arbitrary number of times (we leave this without proof). The simplest example of a polycycle in whose decomposition we see sparkling connections is the ‘heart’ polycycle (see [5] for details). Definition 11. Assume that a field $v_\delta$ in the family $V$ has an nonsparkling separatrix connection $\mathrm{ISC}=\sigma_1\dots\sigma_s$. Then by a subdivision of $\mathrm{ISC}$ we mean a system of $l>1$ interior separatrix connections $\sigma_{m_j + 1}\dots\sigma_{m_{j+1}}$ (of some other fields in $V$, generally speaking), where $0=m_1<\dots<m_j<\dots<m_{l+1}=s$ (Figure 7). Proposition 5. Let $X$ be a connected component of the bifurcation set $X_{\mathrm{ISC}}$ of an nonsparkling separatrix connection $\mathrm{ISC}$. Then the boundary $\partial X$ lies in the disjoint union of the bifurcation sets of systems subdividing the connection $\mathrm{ISC}$. In fact, $\partial X$ is not just a subset of this union: it coincides with it. We prove this in § 4.4 below. Proof of Proposition 5. Recall equation (5) of the separatrix connection $\mathrm{ISC}$: If the function $F$ is defined at a point $\delta \in X$ close to zero, then by Theorem 3 the differential of $F$ is surjective at $\delta$. Hence the manifold $X$ can be extended to the boundary of a small neighbourhood of zero or to a point at which $F$ is not defined.
By (4) the functions $f_i(\delta, x)$ are only defined for small positive $x$. Hence for each point $\delta \in \partial X$ there exists a set of $l+1$ integers $1=m_1<\dots<m_j<\dots<m_{l+1}=s$ such that
$$
\begin{equation}
f_{m_{j+1}} \circ \dots \circ f_{m_j+1}(\delta, 0)=0.
\end{equation}
\tag{16}
$$
Here we mean by $f_{m_j+1}(\delta, 0)$ the limit as $x \to 0$ of the values $\epsilon_{m_j+1} \delta_{\sigma_{m_j+1}}$ of $f_{m_j+1}(\delta, x)$. The function $F$ is not defined at $\delta$ because we cannot apply the function $f_{m_{j+1}+1}$ to the expression (16): the point $x=0$ lies outside the domain of definition of this function.
Note that for each $j=1,\dots, l$ equation (16) defines the bifurcation set $X_{\mathrm{ISC}_j}$ of the interior separatrix connection $\mathrm{ISC}_j=\sigma_{m_j+1}\dots\sigma_{m_{j+1}}$. Thus, each point $\delta \in \partial X$ corresponds to a field $v_\delta$ with a system of interior saddle connections that is a subdivision of $\mathrm{ISC}$. The proof is complete. 4.2. Typical families of vector fieldsDefinition 12. Let $v \in V$ be a field with a finite system of (not necessarily basis) saddle connections. We say that $V$ is generic with respect to this system of connections if $V$ intersects transversally the Banach submanifold corresponding to this system (see Proposition 1). In particular, the family $V$ under consideration is generic with respect to the basis system of connections $\mathrm{SC}_1, \dots, \mathrm{SC}_n$. Proposition 6. Let $X_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ be the bifurcation set of a set of interior separatrix connections $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$ that intersect pairwise distinct cross-sections. Then each $C^1$-smooth manifold $L\subset B$ of dimension $m \geqslant l$ that intersects $X_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ transversally at some point $\delta_0$ defines an $m$-parameter family $\widetilde{V}$ with parameter base $\widetilde{B}=(L, \delta_0)$ which is generic with respect to $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$ . Proof. By Proposition 1 the Banach submanifold $\mathcal{X}_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ corresponding to $\mathrm{ISC}_1,\dots,\mathrm{ISC}_l$ is defined in a neighbourhood of $v_{\delta_0}$ by an equation $\mathbf{F}(v)=0$ (see Definition 7), where $\mathbf{F}$ is a smooth map to $\mathbb{R}^l$ with surjective differential. Since by Theorem 3 the bifurcation set $X_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ of the connections $\mathrm{ISC}_1, \dots, \mathrm{ISC}_l$ is a $C^r$-smooth manifold, the map $\mathbf{F}|_{\widetilde{V}}$ has a surjective differential. Hence, by Proposition 2 the family $\widetilde{V}$ intersects $\mathcal{X}_{\mathrm{ISC}_1 \dots \mathrm{ISC}_l}$ transversally.
The proof is complete. 4.3. A special class of sets of wordsIn this subsection we define the class $\mathcal{A}$ of sets of words mentioned in Theorem 4. Recall that we code each interior separatrix connection $\mathrm{ISC}$ by the indices of cross-sections it intersects (see § 1). Definition 13. We say that a word $\sigma_1\dots\sigma_s$ in the alphabet $\{1,\dots,n\}$ is admissible if for each $i=1,\dots,s-1$ the basis connections $\mathrm{SC}_{\sigma_i}$ and $\mathrm{SC}_{\sigma_{i+1}}$ are formed by an incoming and an outgoing separatrix of the same saddle, respectively. Let $\mathcal{A}$ denote the class of finite sets $A$ of words in the alphabet $\{1,\dots,n\}$, that have the following properties:
If a system $A$ corresponds to a system of interior separatrix connections born in $V$, then the first condition means that all connections in this system are nonsparkling (see Definition 10), and the second condition means that no two connections in the system intersect the same cross-section. While the first two conditions reflect our deliberate simplification of the problem, the third condition is obviously necessary. We separate words in a set $A$ by vertical bars. For example, in Figure 7 the system of connections $A$ formed by the single connection $\mathrm{ISC}$ is denoted by ${A=12345}$, while the system $\widetilde{A}$ formed by $\mathrm{ISC}_1$, $\mathrm{ISC}_2$ and $\mathrm{ISC}_3$ is denoted by $\widetilde{A}=12|3|45$. By analogy with Definition 11, by a subdivision of a word $\sigma_1\dots\sigma_s$ we mean a set of words obtained from it by drawing some vertical bars; the resulting words can be written in an arbitrary order. We say that a system of words $\widetilde{A}$ is a subdivision of a set $A\in \mathcal{A}$ if $\widetilde{A}$ is obtained from $A$ by the replacement of at least one connection in $A$ by a subdivision of it. For instance, $\widetilde{A}=5|234|67|1$ is a subdivision of $A=1234|567$. If $A$ belongs to the class $\mathcal{A}$, then all of its subdivisions obviously belong to $\mathcal{A}$ too. If a set $\widetilde{A}$ is a subdivision of $A$, then we write $\widetilde{A} \prec A$. It is easy to see that the relation $\prec$ on the class $\mathcal{A}$ is transitive. We denote the subclass of $\mathcal{A}$ consisting of all sets of $l$ words by $\mathcal{A}^l$. It is clear that $\mathcal{A}=\bigsqcup_{l=1}^n \mathcal{A}^l$. We denote the bifurcation set of a set of words $A$, that is, the bifurcation set of the corresponding interior connections, by $X_A$. If no system of connections born in the family $V$ corresponds to the set of words $A$, then $X_A$ is an empty set. On the other hand, if $X_A \neq \varnothing$, where $A$ consists of $l$ words, then by Theorem 3 the set $X_A$ is a $C^r$-submanifold of codimension $l$. 4.4. A sufficient condition for the birth of a system of saddle connectionsIn this subsection we prove Theorem 4, that is, a result stating that by a perturbation of the field $v_0$ in an arbitrary family $V$ generic with respect to the basis connection, an arbitrary set $A \in \mathcal{A}$ can appear. Proposition 7. Let the field $v_0$ have precisely two basis saddle connections $\mathrm{SC}_1$ and $\mathrm{SC}_2$, which intersect the cross-sections $\Gamma_1$ and $\Gamma_2$, respectively. Moreover, let $\mathrm{SC}_1$ and $\mathrm{SC}_2$ be formed by an incoming and an outgoing separatrix of a saddle $S$, respectively. Let $V=\{v_\delta\}_\delta$, $\delta\in(\mathbb{R}^2,0)$, be a $C^r$-smooth family of $C^r$-smooth vector fields, $r \geqslant 3$, which is generic with respect to $\mathrm{SC}_1$ and $\mathrm{SC}_2$ (see Definition 12). Then the bifurcation set $X_{\mathrm{ISC}}$ of the interior separatrix connection $\mathrm{ISC}=12$ is a nonempty connected manifold of dimension $1$. Proof. Let $\Delta_S\colon \Gamma_1^- \to \Gamma_2^+$ be the correspondence map of the saddle $S$, which depends on the parameters $\delta_1$ and $\delta_2$. Then replacing $\delta_1$ by $-\delta_1$ and $\delta_2$ by $-\delta_2$ if necessary we can write equation (5) of the connection $\mathrm{ISC}$ as
$$
\begin{equation}
F(\delta_1, \delta_2)=\delta_2 - \Delta_S(\delta_1)=0.
\end{equation}
\tag{17}
$$
Note that, as $\delta_1 \to 0$ for $\delta_2>0$, the function $F$ tends to $\delta_2>0$ in view of (3). On the other hand, for $\delta_1 > 0$ and $\delta_2=0$ we have $F=-\Delta_S(\delta_1)<0$. Therefore, since $F$ is continuous, the zero set $\{F=0\}$ is nonempty and accumulates towards the origin.
By property (3) we have
$$
\begin{equation}
\frac{\partial F}{\partial \delta_2}=1 - \frac{\partial \Delta_S}{\partial \delta_2}(\delta_1) \to 1 \neq 0
\end{equation}
\tag{18}
$$
as $\delta_1, \delta_2 \to 0$, so by the implicit function theorem the set $\{F=0\}$ is a $C^r$-smooth one-dimensional manifold accumulating towards the origin. Since for sufficiently small $\delta_1$ and $\delta_2$ relation (18) implies that $F$ is monotone in $\delta_2$ for fixed $\delta_1$, the manifold $X_{\mathrm{ISC}}=\{ F=0 \}$ is connected.
The proof is complete. Corollary 1. Let $A \in \mathcal{A}^{n-1}$. Then the bifurcation set $X_A$ is nonempty and connected. Proof. Since the number of words in $A$ is just one less that the total number of symbols (indices of cross-sections) in the alphabet, and by the definition of $\mathcal{A}$ all characters in words must be distinct, there are two possible cases.
(1) Each word in $A$ consists of one symbol, that is, represents a basis connection. However, then the bifurcation set $X_A$ is a coordinate subspace of codimension $n-1$ of the parameter base. For example, if $A=1|2|\dots|n-1$, then
$$
\begin{equation*}
X_A=\{ \delta=(\delta_1, \dots, \delta_k) \in B \mid \delta_1=\dots=\delta_{n-1}=0\}.
\end{equation*}
\notag
$$
In this case it is obvious that the set $X_A$ is nonempty and connected.
(2) The system $A$ contains one word of length $2$, while all other words have length $1$. We can assume without loss of generality that $A=12|3|\dots|n$. Consider the two-dimensional subspace
$$
\begin{equation*}
L=\{ \delta=(\delta_1,\dots,\delta_k) \mid \delta_3=\dots=\delta_k=0\}.
\end{equation*}
\notag
$$
It is easy to see that $L \supset X_A$ and $L$ intersects the subspace $\{\delta=(\delta_1,\dots,\delta_k) \mid \delta_1=\delta_2=0\}$ transversally. Hence by Proposition 6 the family $\widetilde{V}$ with base $\widetilde{B}=(L, 0)$ is a generic 2-parameter family with respect to $\mathrm{SC}_1$ and $\mathrm{SC}_2$. Thus, by Proposition 7 the bifurcation set $X_A$ is nonempty and connected.
The proof is complete. Now we can turn to the proof of the sufficient condition for the birth of a system of connections. Proof of Theorem 4. We use induction on the strict order $\prec$.
The base of induction. Minimal elements of $\mathcal{A}$ with respect to the order $\prec$ are systems containing only words of length 1. Let $A$ be a set of $l$ words of length 1, for instance, $A=1|\dots|l$. It corresponds to a system of $l$ basis connections $\mathrm{SC}_1,\dots,\mathrm{SC}_l$. Then
$$
\begin{equation*}
X_A=\{ \delta=(\delta_1,\dots,\delta_k) \mid \delta_1=\dots=\delta_l=0 \}
\end{equation*}
\notag
$$
and the result is obvious.
The step of induction. Assume that we have proved the result for all subdivisions of a system $A\in \mathcal{A}^l$. Then consider a subdivision $\widetilde{A}\prec A$ with $l+1$ words. The system $\widetilde{A}$ contains precisely two words absent from $A$. By the induction assumption $X_{\widetilde A}$ is nonempty, so these two words correspond to interior connections appearing in the family $V$, which we denote by $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$. Moreover, as $A$ contains only admissible words (see Definition 13), the connections $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$ are formed by an incoming and an outgoing separatrix of some saddle, respectively. Consider an arbitrary $C^1$-smooth manifold $L$ of dimension $l+1$ that intersects $X_{\widetilde{A}}$ transversally at some point $\delta_0$. By Proposition 6 the family $\widetilde{V}$ with parameter base $\widetilde{B}= (L,\delta_0)$ is a generic $(l+1)$-parameter family with respect to $\widetilde{A}$. Since the set $A$ has cardinality $l$, which is one less than the number of basis connections in $\widetilde{V}$, $X_A \cap \widetilde{B}$ is nonempty by Corollary 1. That $X_A$ is smooth follows from Theorem 3. Theorem 4 is proved. 4.5. Corollaries to Theorem 4In this subsection we present a few consequences of the sufficient condition for the birth of systems of saddle connections. Since we proved in Theorem 4 that any set of words $A \in \mathcal{A}$ corresponds to a system of interior connections born in the family $V$, we can treat $A$ as a system of connections. Proof. The inclusion $\subset$ follows from Proposition 5. We prove the reverse inclusion using induction on the strict order $\prec$.
The base of induction is trivial because maximal systems have no subdivisions, so their bifurcation sets are closed. The step of induction. Assume that we have proved the result for each subdivision of $A \in \mathcal{A}^l$. Then it suffices to prove that $X_{\widetilde{A}} \subset \partial X_A$ for each subdivision $\widetilde{A}\prec A$ consisting of $l+1$ connections. By Theorem 4 the manifold $X_{\widetilde{A}}$ is nonempty. Let $L$ be an arbitrary smooth manifold of dimension $l+1$ that intersects $X_{\widetilde{A}}$ transversally at some point $\delta_0$. Then by Proposition 6 it defines a generic family $\widetilde{V}$ with respect to $\widetilde{A}$ with $(l+1)$-dimensional base of parameters $\widetilde{B}=(L, \delta_0)$. By Theorem 4 the system of connections $A$ is born in $\widetilde{V}$. Hence the intersection $X_A \cap L$ is nonempty and accumulates towards $\delta_0$. Since we can draw a transversal manifold $L$ through an arbitrary point $\delta_0$ in $X_{\widetilde{A}}$, it follows that $X_{\widetilde{A}} \subset \partial X_A$. The proof is complete. Definition 14. Let the system of connections $A$ contain interior connections $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$ that are an incoming and an outgoing separatrices, respectively, of a saddle $S_i$. Consider the system of words $\widetilde{A}$ containing the same words as $A$, except that, in place of the words $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$, it contains the concatenation $\mathrm{ISC}_1+\mathrm{ISC}_2$. Then we say that the connections in $\widetilde{A}$ are obtained from $A$ by splitting off the saddle $S_i$. Lemma 5. Assume that a system $\widetilde{A}$ is obtained from $A \in \mathcal{A}$ by splitting off several saddles. Then $\widetilde{A} \in \mathcal{A}$. Proof. It is sufficient to prove the result in the case when one saddle is split off. Since the connections $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$ in Definition 14 are an incoming and an outgoing separatrix of the same saddle $S_i$, the word $\mathrm{ISC}_1\,{+}\,\mathrm{ISC}_2$ is admissible (see Definition 13). The number of occurrences of each symbol in the words in $\widetilde{A}$ remains the same (see § 4.3). Hence $\widetilde A \in \mathcal{A}$.
The lemma is proved. Let $\gamma$ be the hyperbolic polycycle (see Definition 1) formed by all basis connections $\mathrm{SC}_1,\dots,\mathrm{SC}_n$. It corresponds to the set of words $1|\dots|n$. Assume that $\gamma$ is perturbed in a $C^3$-smooth family $V= \{v_\delta\}_\delta$ which is generic with respect to these basis connections. Corollary 3. Each system of words $A \in \mathcal{A}$ containing the indices of all cross-sections corresponds to a child polycycle (see Definition 5) appearing in $V$. Proof. It is obvious that the set of words $A$ can be obtained from the basis system $1|\dots|n$ in finitely many steps by replacing some pairs of words by their concatenations. Assume that we replaced two words $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$ by their concatenation at some step. Since $A \in \mathcal{A}$ and the word $\mathrm{ISC}_1 + \mathrm{ISC}_2$ is a subword of a word in $A$, it is admissible (see Definition 13). Hence the connections $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$ are formed by an incoming and an outgoing separatrix of the same saddle. In other words, $A$ is obtained from the basis system by splitting off saddles (see Definition 14).
Since the set of words $A$ contains by assumption the indices of all cross-sections, as the parameters tend to zero the union of connections in the system tends to the polycycle $\gamma$ in the Hausdorff metric. Moreover, notice that when a saddle is split off, the order of traversal of the polycycle remains the same (see Definition 1). Hence $A$ corresponds to a child polycycle. By Lemma 5 and Theorem 4 this polycycle is born in the family $V$. The proof is complete. Now we prove a result which is converse to Corollary 3 is a certain sense. Let $\gamma$ be a hyperbolic polycycle of a vector field $v_0 \in \operatorname{Vect}^3(\mathbb{S}^2)$. We assume again that $\gamma$ is formed by all basis connections. Let the $C^3$-smooth finite-parameter family $V$ be generic with respect to the basis connections. Corollary 4. Assume that a child polycycle is born in the family $V$ (see Definition 5). Then it corresponds to a set of words $A \in \mathcal{A}$ involving the indices of all cross-sections. Proof. Assume that two connections $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$ in the child polycycle intersect the same cross-section. Let $B_1$ and $B_2$ denote the points of intersection. By Definition 1 the child polycycle is path connected. Hence some curve $\tau$ in it connects $B_1$ with $B_2$. However, these points lie on the same cross-section, so the closed curve $\tau \cup [B_1,B_2]$, where $[B_1,B_2]$ is the segment of the cross-section between these two points, form a so-called Bendixson bag. Since a 2-sphere is the phase space, one of the connections $\mathrm{ISC}_1$ and $\mathrm{ISC}_2$ (the one coming in the bag in direct or reverse time) cannot leave it. This contradicts the Euler property of the child polydisc (see Definition 1). In a similar way we prove that no connection in the child polycycle can be sparkling (see Definition 10). It is obvious that each word corresponding to a connection in the child polydisc is admissible. Hence each child polydisc corresponds to a set of words in the class $\mathcal{A}$. Moreover, since $\widetilde \gamma(\delta) \to \gamma$ as $\delta \to 0$ by definition, the system of connections of the child polycycle, as coded by a set of words, must involve the indices of all cross-sections.
The proof is complete. Remark 4. By contrast with the previous results, the phase space in Corollary 4 is a 2-sphere. This is because on another surface, when a polycycle is destroyed, some of the appearing child polycycles can traverse twice a neighbourhood of a basis connection. In Figure 8, a, we show a polycycle on a torus (represented as a rectangle with glued opposite sides) formed by four separatrix connections. Under a perturbation of it (Figure 8, b) a child polycycle can appear one of whose connections traverses twice a neighbourhood of an opened up basis connection.3 § 5. Lower bound for cyclicityNow we prove a lower bound for the cyclicity of a monodromic polycycle. Proof of Theorem 1. We split the saddle with index $n$ off $\gamma$. By Corollary 3 the resulting system corresponds to a child polycycle $\gamma_{n-1}$ born in the family $V$, and by Proposition 6 $V$ remains a generic family with respect to the connections in $\gamma_{n-1}$. Now we split the saddle with index $n-1$ off $\gamma_{n-1}$ and obtain for the same reasons a monodromic hyperbolic polycycle $\gamma_{n-2}$, and so on. As a result, we obtain a sequence of child polycycles $\{\gamma_{n-i}\}$ ($\gamma_n=\gamma$), where for each $i=0,\dots,n-1$ the child polycycle $\gamma_{n-i}$ is formed by $n-i$ saddles and is a child polycycle of $\gamma_{n-i+1}$ in the same family $V$ (by Proposition 6). In particular, the bifurcation sets of these child polycycles accumulate towards the origin in the base of parameters $B$.
Let $\Delta(\delta, x)$ denote the Poincaré map on a transversal cross-section $\Gamma$ of the polycycle and $P(\delta, x)$ denote the shift map $P(\delta,x)= \Delta(\delta,x) - x$. We assume that a chart on the cross-section is chosen so that the maps $\Delta(0,x)$ and $P(0,x)$ for the field $v_0$ are defined on the positive half-axis in this chart. The maps $\Delta(\delta,x)$ and $P(\delta,x)$ are common for $\gamma$ and its child polycycles. Claim. For $i=0,\dots,n$, for each $\varepsilon>0$ there exist $\delta \in B$, $|\delta| < \varepsilon$, and points $\xi_0 > \dots > \xi_i$ on $\Gamma$ such that For $i=n$, by the continuity of the Poincaré map $\Delta(\delta,x)$ it follows from this result that this map has $n$ fixed points on the cross-section $\Gamma$. In particular, $n$ limit cycles are born in a perturbation of $\gamma$ in $V$ . Proof of the claim. We use induction on $i=0,\dots,n$. The following result, due to Cherkas, is required for the proof.
Theorem 5 ([3]). Let $v_0 \in \operatorname{Vect}^1(\mathbb{R}^2)$ be a field with a monodromic hyperbolic polycycle with characteristic numbers $\lambda_1,\dots,\lambda_n$ of the saddles. Assume that ${\lambda_1\mkern-1.5mu\cdots\mkern-1.5mu\lambda_n \mkern-1mu\!>\!\mkern-1mu 1}$ ($\lambda_1\cdots\lambda_n < 1$). Then this polycycle is stable (respectively, unstable), that is, trajectories wind around it in direct (respectively, reverse) time. The base of induction $i=0$. Consider the field $v_0$. By (1) the product of all characteristic numbers of saddles in $\gamma$ is distinct from zero. We can assume without loss of generality that $\lambda_1\cdots\lambda_n > 1$. Hence $\gamma$ is stable by Theorem 5. In particular, there exists $\xi_0$ such that for each $x \leqslant \xi_0$ we have $\Delta(0,x) > x$. The induction step. Assume that we have proved the proposition for $i-1<n-1$. Since the polycycle $\gamma_{n-i}$ of some field $v_\delta$ is a child polycycle for $\gamma_{n-i+1}$ and is obtained from it by an arbitrarily small perturbation, we can assume that for each point $\xi_j$, $j=0,\dots,i-1$, the shift map $P$ takes values of the same sign at $\xi_j$ for these polycycles. By (1) and Theorem 5 the stability properties of $\gamma_{n-i}$ are opposite to those of $\gamma_{n-i+1}$. Hence there exists a point $\xi_i$ such that for each $x \leqslant \xi_i$ at which $P(\delta,x)$ is defined we have At the final step of induction, for $i=n$ the argument is similar, but in place of going over to the child polycycle $\gamma_{n-i}$ we use the fact that in the decomposition of a separatrix loop $\gamma_1$ for $\lambda_1 \neq 1$ a limit cycle is born. This well-known result is due to Andronov and Leontovich [1]. Thus we have proved the claim, and therefore Theorem 1 is proved.
Citation in format AMSBIB
\Bibitem{Duk24} |
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