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Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycles A. V. Dukov Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
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     § 1. IntroductionLet $\mathcal{M}$ be an infinitely smooth two-dimensional oriented manifold. We let $\operatorname{Vect}^{\infty}(\mathcal{M})$ denote the space of infinitely smooth vector fields on $\mathcal{M}$. Definition 1. A polycycle of a vector field is any finite directed graph $\Gamma$ satisfying the following conditions: Definition 2. The codimension of a polycycle $\gamma$ of a field $v_0$ is the codimension of the Banach submanifold in the space $\operatorname{Vect}^\infty(\mathcal{M})$ that is formed by the fields with polycycle $\gamma$ which are close to $v_0$. Assume that a field $v_0 \in \operatorname{Vect}^\infty(\mathcal{M})$ has a polycycle $\gamma$. Consider a $k$-parameter family $V=\{v_\delta\}$, $\delta\in B=(\mathbb{R}^k, 0)$, that perturbs $v_0$. Recall that a limit cycle is of multiplicity $m$ if the germ of the Poincaré map along this cycle (see § 2.1) has a fixed point of multiplicity $m$. Definition 3. We say that a limit cycle (of multiplicity $m$) appears in a family $V$ after a perturbation of a polycycle $\gamma$ of a field $v_0$ if there is a sequence of parameters $\{\delta_\alpha\}_{\alpha \in \mathbb{N}}$ tending to zero (to which the field $v_0$ corresponds) such that for any $\alpha$ the field $v_{\delta_\alpha}$ has a limit cycle $\operatorname{LC}(\delta_\alpha)$ (of multiplicity $m$) and the sequence of limit cycles $\operatorname{LC}(\delta_\alpha)$ tends to the polycycle $\gamma$ as $\delta_\alpha \to 0$ in the Hausdorff metric. Definition 4. Assume that a polycycle $\gamma$ of a field $v_0$ is perturbed in a finite-parameter family $V=\{v_\delta\}$, $\delta\in B=(\mathbb{R}^k, 0)$. Let $\mu$ be the smallest number for which there exist neighbourhoods $U$ and $W$ such that $\gamma \subset U$, $0 \in W \subset B$, and for any $\delta \in W$ the field $v_\delta$ has at most $\mu$ limit cycles in $U$. Then $\mu$ is called the cyclicity of the polycycle $\gamma$ in the family $V$. Note that the definition of cyclicity takes account of the limit cycles emerging not only from the whole of the polycycle $\gamma$ but also from smaller polycycles, that is, those that are subgraphs of $\gamma$ in the sense of Definition 1. Definition 5. A polycycle is called elementary if it is entirely formed of elementary singular points, that is, of singular points with at least one nonzero eigenvalue. The maximum cyclicity that a nontrivial (distinct from a singular point) elementary polycycle perturbed in a generic $k$-parameter family can have is denoted by $E(k)$ or $E(n,k)$, where $n$ is the number of singular points forming the polycycle. In the 1930s, Andronov and Leontovich [1] proved that $E(1)=1$. From the 1970s through 1993, as a result of the work of a number of researchers (see papers by Mourtada, [2] by Roussarie, Rousseau and Dumortier, [3] by Grozovskii, [4] by Roitenberg, and [5] by Trifonov), it was shown that $E(2)=2$. (See [2] for more detail on the history of investigations on the cyclicity of polycycles of codimension 1 and 2.) In 1997, Trifonov proved the equality $E(3)=3$ (see [5]). At the turn of the century, attempts were made to estimate the cyclicity of elementary polycycles for an arbitrary number of parameters $k$. In 1995 Ilyashenko and Yakovenko proved that $E(k)$ is finite for any $k$ (see [6]). In 2003 Kaloshin [7] obtained Slightly later, in 2010 Kaleda and Shchurov [8] proved the inequality where $C(n)=2^{5n^2+20n}$ and $n$ is the number of vertices of the polycycle.As can be seen from the above overview, the early estimates were sharp but concerned only polycycles of small codimension. The later estimates due to Kaloshin, Kaleda and Shchurov extend to an arbitrary number of parameters but are hardly sharp. This is explained by the fact that the problem of estimating the cyclicity is rather complicated. An idea to consider an obviously simpler problem arises in this connection: what is the maximum multiplicity of a limit cycle appearing after a perturbation of a polycycle in a finite-parameter family? Note that in this paper we do not deal with elementary polycycles but only with hyperbolic polycycles, that is, polycycles formed by hyperbolic saddles only. It turns out that this problem is easily solvable for an arbitrary number of parameters, and the estimate for multiplicity depends at most linearly on the number of saddles in the polycycle. Basic resultsAssume that a field $v_0$ contains a polycycle $\gamma$ formed by $n$ separatrix connections of hyperbolic saddles $S_1, \dots, S_n$ (some saddles can coincide). We denote the characteristic numbers of $S_1, \dots, S_n$ by $\lambda_1, \dots, \lambda_n$, respectively. (Recall that the characteristic number of a saddle is the absolute value of the ratio of the eigenvalues where the negative eigenvalue is in the numerator.) The main results of this work are the following two theorems. Theorem 1. For any natural number $n$ there exists a nontrivial polynomial $\mathcal{L}_n$ in $n$ variables such that for any field $v_0$ with a hyperbolic polycycle $\gamma$ whose saddles have characteristic numbers $\lambda_1, \dots, \lambda_n$ satisfying the inequality the following is true: when $v_0$ is perturbed in a $C^\infty$-smooth finite-parameter family, the multiplicity of any limit cycle emerging from $\gamma$ does not exceed $n$. As can be seen in § 2.6 below, it follows from the presence of a multiple limit cycle that some polynomial system of homogeneous equations whose coefficients depend on the characteristic numbers $\lambda_1, \dots, \lambda_n$ has a nontrivial solution. Looking ahead, we note that the polynomial $\mathcal{L}_n$ can be expressed in terms of the resultant of this polynomial system. In the case of a polycycle of small codimension, the polynomial $\mathcal{L}_n$ can be written out explicitly. To do this, we need to introduce several polynomials. For any natural number $n$ we let $\Lambda_n$ denote the following polynomial in the characteristic numbers $\lambda_1, \dots, \lambda_n$:
$$
\begin{equation*}
\Lambda_n(\lambda_1, \dots, \lambda_n)=\prod_{I\neq (0, \dots, 0)} (\lambda^I-1),
\end{equation*}
\notag
$$
where $I=(i_1, \dots, i_n)$ is a multi-index. Here $\lambda^I$ denotes the product $\lambda_1^{i_1}\dotsb\lambda_n^{i_n}$. For any $j=1, \dots, n$, the component $i_j \in \{0, 1\}$ of the multi-index indicates whether or not $\lambda_j$ is involved in the product $\lambda^I$. For example, $\Lambda_2(\lambda_1,\lambda_2)=({\lambda_1-1})({\lambda_2-1})(\lambda_1\lambda_2-1)$. In addition, we let $M(\lambda_1, \lambda_2, \lambda_3)$ denote the polynomial
$$
\begin{equation*}
M(\lambda_1, \lambda_2, \lambda_3)=4(\lambda_1\lambda_2\lambda_3 -1)- (\lambda_1-1)(\lambda_2-1)(\lambda_3-1).
\end{equation*}
\notag
$$
Theorem 2. For $n=1,2,3,4$, the role of the polynomial $\mathcal{L}_n$ in Theorem 1 can be played by the following polynomials: If the characteristic numbers are such that the polynomial $\mathcal{L}_n$ vanishes, then additional singularities can appear, namely, limit cycles of multiplicity higher than $n$. For example, it was shown in [4] that for $n=1$ and $\mathcal{L}_1(\lambda_1)=\lambda_1-1=0$, a double limit cycle appears when a separatrix loop is perturbed in a generic two-parameter family. The paper is divided into six sections. Our main task in § 2 is to reduce the problem of the presence of a multiple limit cycle to the problem of the presence of a solution of a certain polynomial system of equations. The rest of the paper uses algebraic methods which make it possible to study this polynomial system. Both theorems are proved using these methods. § 2. From vector fields to polynomials2.1. The correspondence maps of saddlesAssume that a finite-parameter family $V=\{v_\delta\}$, $\delta \in B=(\mathbb{R}^k, 0)$, perturbs a field $v_0$ containing a hyperbolic polycycle $\gamma$ with characteristic numbers $\lambda_1, \dots, \lambda_n$ of the corresponding saddles. We draw a $C^\infty$-smooth transversal to each separatrix connection of the polycycle $\gamma$ of the field $v_0$: for any $i=1, \dots, n$ the transversal to the connection between the saddles $S_i$ and $S_{i+1}$ is denoted by $\Gamma_i$ (Figure 1, a). We assume that the transversal $\Gamma_i$ is independent of the parameter $\delta$ for any $i=1, \dots, n$ and $\delta$ is chosen sufficiently small so that each $\Gamma_i$ remains transverse to the perturbed vector field $v_\delta$. Consider an arbitrary saddle $S_i$ and two neighbouring transversals $\Gamma_{i-1}$ and $\Gamma_i$ (we set $\Gamma_0=\Gamma_n$) in the unperturbed field $v_0$. We denote the point of intersection of the transversal $\Gamma_{i-1}$ and the incoming separatrix of the saddle $S_i$ by $s_i$ (derived from ‘stable’). We denote the point of intersection of the transversal $\Gamma_i$ and the outgoing separatrix of the saddle $S_i$ by $u_i$ (derived from ‘unstable’). Since the separatrices of the unperturbed field $v_0$ are unbroken, the points $u_{i-1}$ and $s_i$ on $\Gamma_{i-1}$ coincide for any $i=1, \dots, n$. For $i=1, \dots, n$ the saddle $S_i$ has exactly one hyperbolic sector bounded by parts of $\Gamma_{i-1}$ and $\Gamma_i$. We denote these parts by $\Gamma_{i-1}^-$ and $\Gamma_i^+$. Then the correspondence map $\Delta_i\colon \Gamma_{i-1}^- \to \Gamma_i^+$ of the saddle $S_i$ is defined for any $i=1, \dots, n$. Now consider the perturbed field $v_\delta$. By analogy, for any saddle $S_i=S_i(\delta)$ we introduce the points $s_i(\delta)$ and $u_i(\delta)$, the half-transversals $\Gamma_{i-1}^-(\delta)$ and $\Gamma_i^+(\delta)$, and the correspondence map $\Delta_i(\delta, \,\cdot\,)\colon \Gamma_{i-1}^-(\delta) \to \Gamma_i^+(\delta)$. Generally speaking, the points $s_i(\delta)$ and $u_{i-1}(\delta)$ on $\Gamma_{i-1}$ may not coincide, which would mean that the connection between $S_{i-1}(\delta)$ and $S_i(\delta)$ is broken. We consider an arbitrary Riemannian metric on the manifold $\mathcal{M}$. Then we can parameterize any smooth curve (choose a chart on it) by the natural parameter: the difference of the coordinates of any two points in this chart is equal in absolute value to the length of the interval of the curve between these points. We choose a natural parametrization (a chart) on the transversal $\Gamma_{i-1}$ so that the point $s_i(\delta)$ has coordinate $0$ and the coordinate of any point on $\Gamma_{i-1}^-(\delta)$ is positive. At the same time we choose a natural parametrization (a chart) on the transversal $\Gamma_i$ so that the point $u_i(\delta)$ has coordinate $0$ and the coordinate of any point on $\Gamma_i^+(\delta)$ is positive. Thus, we have chosen two charts on each transversal $\Gamma_i$. We denote the coordinate of $u_i(\delta)$ in the chart corresponding to the half-transversal $\Gamma_i^-(\delta)$ by $\tau_i(\delta)$. Then we can switch from the chart corresponding to $\Gamma_i^+(\delta)$ to the chart corresponding to $\Gamma_i^-(\delta)$ via the map If the hyperbolic sectors of $S_i(0)$ and $S_{i+1}(0)$ in the unperturbed field $v_0$, which are under consideration, are on one side of the common separatrix connection of these saddles, then we have the sign ‘$+$’ in (2) (Figure 2, a); if these are on opposite sides, then we have the sign ‘$-$’ (Figure 2, b).In the above coordinates on the half-transversals $\Gamma_{i-1}^-(\delta)$ and $\Gamma_i^+(\delta)$ the map $\Delta_i(\delta, \,\cdot\,)$ takes the form $\Delta_i(\delta, \,\cdot\,)\colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$. It is $C^\infty$-smooth with respect to $x$. In addition, all its derivatives with respect to $x$ depend continuously (in fact, smoothly) on the parameter $\delta$. For $i=1, \dots, n$ we introduce the map
$$
\begin{equation}
f_i(\delta, \,\cdot\,)\colon \Gamma_{i-1}^-(\delta) \to \Gamma_i, \qquad f_i(\delta, x)=\tau_i(\delta) \pm \Delta_i(\delta, x).
\end{equation}
\tag{3}
$$
It is the composition of the correspondence map $\Delta_i(\delta, \,\cdot\,)$ and (2). In addition, we consider the Poincaré map $\Delta(\delta, \,\cdot\,)\colon \Gamma_n^-(\delta) \to \Gamma_n$ of the polycycle in question (Figure 1, b). It can be represented as
$$
\begin{equation}
\Delta(\delta, \,\cdot\,)=f_n(\delta, \,\cdot\,) \circ \dots \circ f_1(\delta, \,\cdot\,),
\end{equation}
\tag{4}
$$
where the $f_i$, $i=1, \dots, n$, are specified by (3). 2.2. Equations for multiple limit cyclesAssume that for some $\delta$ the field $v_\delta$ has a limit cycle $\operatorname{LC}(\delta)$ born in a perturbation of the original polycycle $\gamma$. Assume that this cycle intersects the half-transversal $\Gamma_n^-(\delta)$ at a point with coordinate $x_0=x_0(\delta)$. Then the Poincaré map has a fixed point, that is, the pair $(\delta, x_0(\delta))$ is a solution of the equation If the limit cycle is of multiplicity at least $n+1$, then for $x=x_0(\delta)$ it is also true that andThroughout this paper the symbols $(\,\cdot\,)'$ and $(\,\cdot\,)^{(l)}$ are understood as derivatives with respect to $x$. We consider the function and the related system of equations
$$
\begin{equation}
\mathcal{D}^{(l)}(\delta, x)=0, \qquad l=0, \dots, n-1.
\end{equation}
\tag{9}
$$
Since the vector field is considered on an oriented manifold, the Poincaré map $\Delta(\delta, \,\cdot\,)$ is an orientation-preserving diffeomorphism defined on $\Gamma_n^-(\delta)$. Therefore, its derivative is always positive, which makes it possible to take the logarithm of $\Delta'$ in the definition of the function $\mathcal{D}$. Note that if $x=x_0(\delta)$ is a fixed point of $\Delta$ of multiplicity $n+1$, then $x_0(\delta)$ also satisfies (9). This follows from the fact that
$$
\begin{equation*}
\mathcal{D}(\delta, x)=\log \Delta'(\delta, x)=\log \bigl(1+ o((x-x_0)^{n-1})\bigr)=o((x-x_0)^{n-1})
\end{equation*}
\notag
$$
in a small neighbourhood of $x_0(\delta)$. 2.3. The general form of higher-order derivatives of the Poincaré mapThe previous subsection suggests that instead of the Poincaré map itself we can investigate the function $\mathcal{D}(\delta, x)$. It turns out that the derivatives of $\mathcal{D}(\delta, x)$ of an arbitrarily high order can be written in a convenient form. We introduce the notation
$$
\begin{equation}
F_i=f_i \circ \dots \circ f_0, \qquad f_0=\mathrm{id}, \quad i=0, \dots, n,
\end{equation}
\tag{10}
$$
and where the functions $f_i$ are defined by (3). In particular, $F_0(\delta, x)=x$ and ${Z_1(\delta, x)={1}/{x}}$. Throughout, the composition of two functions $g(\delta, x)$ and $h(\delta,x)$ is understood as $g \circ h(\delta,x)=g(\delta, h(\delta,x))$. We use the shorthand notation $g \circ h=g(h)$. In the new notation the equality $\mathcal{D}(\delta, x)=0$ is rewritten as
$$
\begin{equation}
\mathcal{D}(\delta, x)=\sum_{i=1}^n \log |f_i'(F_{i-1})|=0.
\end{equation}
\tag{12}
$$
In addition, we introduce the notation
$$
\begin{equation}
\mu_{iq}(\delta, x)=y^q\frac{d^q}{dy^q}\log |f_i'(y)| \bigg|_{y=F_{i-1}(\delta, x)}, \qquad i=1,\dots,n, \quad q \in \mathbb{N}.
\end{equation}
\tag{13}
$$
In what follows we suppress the dependence on $x$ and $\delta$ and write $\mu_{iq}$ and $F_{i-1}$. Proposition 1. For any $l \in \mathbb{N}$ there exists a polynomial $P_{nl}$ with integer coefficients such that the $l$th derivative of the function $\mathcal{D}$ (see (8)) has the form The polynomial $P_{nl}$ is a homogeneous polynomial of degree $l$ in the variables $Z_1, \dots, Z_n$. Proof. We prove this assertion using induction on $l$.
The base of induction. For $l=1$ we derive from (12) that
$$
\begin{equation}
\mathcal{D}'=\sum_{i=1}^n \frac{d}{dx} \log |f_i'(F_{i-1})|=\sum_{i=1}^n y\frac{d}{dy} \log |f_i'(y)| \bigg|_{y=F_{i-1}} \frac{F_{i-1}'}{F_{i-1}}=\sum_{i=1}^n \mu_{i1} Z_i.
\end{equation}
\tag{15}
$$
We denote the resulting polynomial by $P_{n1}(\mu_{i1}, Z_i)$, $i=1,\dots,n$.
The step of induction. Assume that the assertion holds for some $l$. Then
$$
\begin{equation}
\mathcal{D}^{(l+1)}=\frac{d}{dx} P_{nl}(\mu_{iq},Z_i) =\sum_{\substack{ 1 \leqslant i \leqslant n\\ 1 \leqslant q \leqslant l}} \frac{\partial P_{nl}}{\partial \mu_{iq}} \mu_{iq}'+\sum_{i=1}^n \frac{\partial P_{nl}}{\partial Z_i} Z_i'.
\end{equation}
\tag{16}
$$
In view of the notation (11) and (13) the derivative of $\mu_{iq}$ has the form
$$
\begin{equation}
\mu_{iq}'=\biggl(q y^{q-1}\frac{d^q}{dy^q}+y^q\frac{d^{q+1}}{dy^{q+1}}\biggr)\log |f_i'(y)| \bigg|_{y=F_{i-1}}F_{i-1}'=(q\mu_{iq}+\mu_{i,q+1})Z_i.
\end{equation}
\tag{17}
$$
To calculate $Z_i'$ we find an expression for ${F_{i-1}''}/{F_{i-1}'}$. It follows from (10) and (13) that
$$
\begin{equation}
\begin{aligned} \, \frac{F_{i-1}''}{F_{i-1}'} &=(\log |F_{i-1}'|)'=\sum_{j=1}^{i-1}(\log |f_j'(F_{j-1})|)' \nonumber \\ &=\sum_{j=1}^{i-1}y\frac{d}{dy}\log |f_j'(F_{j-1}(y))|\bigg|_{y=F_{j-1}}\frac{F_{j-1}'}{F_{j-1}}=\sum_{j=1}^{i-1}\mu_{j1}Z_j. \end{aligned}
\end{equation}
\tag{18}
$$
Using (11) and (18) we obtain the derivative $Z_i'$ in the form
$$
\begin{equation}
Z_i'=\biggl(\frac{F_{i-1}'}{F_{i-1}} \biggr)'=\frac{F_{i-1}''}{F_{i-1}}- \frac{F_{i-1}'^2}{F_{i-1}^2}=\frac{F_{i-1}''}{F_{i-1}'}Z_i-Z_i^2=-Z_i^2+ Z_i\sum_{j=1}^{i-1}\mu_{j1}Z_j.
\end{equation}
\tag{19}
$$
Substituting (17) and (19) into (16) we infer that
$$
\begin{equation}
\mathcal{D}^{(l+1)}=\sum_{\substack{ 1 \leqslant i \leqslant n \\ 1 \leqslant q \leqslant l}} (q\mu_{iq}+ \mu_{i,q+1})Z_i \frac{\partial P_{nl}}{\partial \mu_{iq}} +\sum_{i=1}^n \biggl(-Z_i+ \sum_{j=1}^{i-1}\mu_{j1}Z_j\biggr) Z_i \frac{\partial P_{nl}}{\partial Z_i}.
\end{equation}
\tag{20}
$$
It follows from the induction assumption that the resulting expression is a homogeneous polynomial in $Z_i$ of degree $l+1$ with integer coefficients, which we denote by $P_{n,l+1}(\mu_{iq}, Z_i)$, where $i=1, \dots, n$ and $q=1, \dots, l+1$. Proposition 1 is proved. 2.4. Taking the limit as $\delta, x \to 0$. The O-symbolicsIn this and the next two subsections we consider the limiting properties of the derivatives of $\mathcal{D}$ involved in the system of equations (9). It turns out that the limit values of the derivatives of the function $\mathcal{D}$ as $\delta, x \to 0$ can be described by a homogeneous polynomial in the variables $Z_1, \dots, Z_n$ with coefficients depending only on the characteristic numbers $\lambda_1, \dots, \lambda_n$. To find the limit of the function $\mu_{iq}(\delta,x)$ specified by (13) as $\delta, x \to 0$, we need the following lemma. Lemma 1. Consider a $C^\infty$-smooth finite-parameter family $V=\{v_\delta\}$, $\delta \in (\mathbb{R}^k, 0)$, of $C^\infty$-smooth vector fields on a two-dimensional plane. Let $\Delta_S(\delta, x)$ be the correspondence map of a hyperbolic saddle $S(\delta)$ of $v_\delta$ with characteristic number $\lambda(\delta)$, $\lambda(0)=\lambda$. Then for any natural number $q$. To prove this lemma we introduce two useful classes of functions proposed by Trifonov [5]. Let $\lambda\in \mathbb{R}$ and $r \in \mathbb{N}$; assume that a function $f_\delta(x) \in C^r(\mathbb{R}, 0)$ depends continuously on the parameter $\delta$ in the $C^r(\mathbb{R}, 0)$-topology. 1. The function $f_\delta$ is said to be in the class $\widetilde{o}_r^\lambda$ if
$$
\begin{equation*}
\lim_{\delta,x \to 0} x^{m-\lambda}f_\delta^{(m)}(x)=0
\end{equation*}
\notag
$$
for any $m=0, \dots, r$. 2. The function $f_\delta$ is said to be in the class ${{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_r$ if $x^\varepsilon f_\delta(x)$ is in the class $\widetilde{o}_r^\lambda$ for any small $\varepsilon > 0$. Like in the case of $o(1)$ and $O(1)$, we use the ordinary equality symbol instead of the membership symbol, for example, $f_\delta(x)=x+\widetilde{o}^\lambda_r$ means that $f_\delta(x)-x \in \widetilde{o}_r^\lambda$. The classes $\widetilde{o}_r^\lambda$ and ${{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_r$ are a more convenient tools than the standard classes of functions $o(x^\lambda)$ and $O(x^\lambda)$, since they admit differentiation. A long list of their properties is presented in [5], § 2.2. We need only the following ones:
$$
\begin{equation}
\forall\, \lambda, \mu \quad {{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_r \cdot {{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\mu_r={{\underset{\widetilde{\,\,\,}}{\,O}}}{}^{\lambda+\mu}_r,
\end{equation}
\tag{22}
$$
$$
\begin{equation}
\forall\, \lambda \quad \bigl({{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_r\bigr)'={{\underset{\widetilde{\,\,\,}}{\,O}}}{}^{\lambda-1}_{r-1},
\end{equation}
\tag{23}
$$
$$
\begin{equation}
\forall\, \lambda > \mu > 0 \quad \forall\, f_\delta \in {{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_r \quad\Longrightarrow\quad f_\delta \in \widetilde{o}^\mu_r \quad\Longrightarrow \quad f_\delta \to 0 \quad\text{as } \delta,x \to 0,
\end{equation}
\tag{24}
$$
$$
\begin{equation}
\forall\, \lambda, \quad\forall\, g\in C^r(\mathbb{R},0) \quad g\bigl({{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_r\bigr)=g(0)+{{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_r,
\end{equation}
\tag{25}
$$
$$
\begin{equation}
\forall\, \lambda \quad x^\lambda={{\underset{\widetilde{\,\,\,}}{\,O}}}{}^\lambda_\infty.
\end{equation}
\tag{26}
$$
It was proved in [5], § 3.1, that the correspondence map of a hyperbolic saddle has the following asymptotic behaviour. Proposition 2. Consider a $C^\infty$-smooth family $V\!=\!\{v_\delta\}$, $\delta\!\in\! (\mathbb{R}^k, 0)$, of $C^\infty$-smooth vector fields on a two-dimensional plane. Let $\Delta_S(\delta, x)$ be the correspondence map of a hyperbolic saddle $S(\delta)$ of $v_\delta$ with characteristic number $\lambda(\delta)$. Then where $C(\delta)$ and $\lambda(\delta)$ are $C^1$-smooth functions, $C(0) > 0$ and $r$ is an arbitrarily large natural number. Proof of Lemma 1. Using properties (22), (23) and (26) we differentiate (27):
$$
\begin{equation*}
\Delta_S'(\delta,x)=C(\delta)\lambda(\delta) x^{\lambda(\delta)-1}\bigl(1+{{\underset{\widetilde{\,\,\,}}{\,O}}}{}^1_{r-1}\bigr).
\end{equation*}
\notag
$$
Taking the logarithm of the above expression, from (25) we obtain
$$
\begin{equation}
\log\Delta_S'(\delta, x)=\log C(\delta)+\log \lambda(\delta)+(\lambda(\delta)-1)\log x+{{\underset{\widetilde{\,\,\,}}{\,O}}}{}^1_{r-1}.
\end{equation}
\tag{28}
$$
We differentiate the resulting relation $q$ times. From (23) we derive that
$$
\begin{equation*}
\frac{d^q}{dx^q}\log\Delta_S'(\delta, x)=(-1)^{q-1}(q-1)!\,(\lambda(\delta)-1)x^{-q}+ {{\underset{\widetilde{\,\,\,}}{\,O}}}{}^{1-q}_{r-q-1}.
\end{equation*}
\notag
$$
We multiply by $x^q$ and let $\delta$ and $ x$ tend to $0$. Using (22), (24) and (26), we arrive at the required equality. The lemma is proved. According to formula (3) for $f_i$, we have $|f_i'|=\Delta_i'$. Therefore, by Lemma 1, we have
$$
\begin{equation}
\lim_{\delta,x\to 0} \mu_{iq}(\delta,x)=(-1)^{q-1}(q-1)!\,(\lambda_i-1),
\end{equation}
\tag{29}
$$
where $\lambda_i$ is the characteristic number of the saddle $S_i$ of the unperturbed polycycle $\gamma$. 2.5. Taking the limit as $\delta, x \to 0$. Derivatives of the Poincaré mapWe let $\mu_{iq}^0$ denote the expression on the right-hand side of (29). Consider the polynomials
$$
\begin{equation}
Q_{nl}(Z_1, \dots, Z_n)=P_{nl}(\mu_{iq}^0, Z_i), \qquad i=1, \dots, n, \quad q=1, \dots, l,
\end{equation}
\tag{30}
$$
where the $P_{nl}$ are the polynomials from (14). Proposition 3. Under the assumptions of Proposition 1, for any natural numbers $n$ and $l$ the polynomials $Q_{nl}$ have the following properties. 1. $Q_{nl} (Z_1, \dots, Z_n)$ is a homogeneous polynomial of degree $l$. 2. $Q_{nl} (Z_1, \dots, Z_n) \in \mathbb{Z}[\lambda_1,\dots,\lambda_n][Z_1, \dots, Z_n]$. 3. The $Q_{nl}(Z_1, \dots, Z_n)$ are recursively defined by and where
$$
\begin{equation}
\mathfrak{D}_n=(Z_1, \dots, Z_n) \begin{pmatrix} -1 & \lambda_1-1 & \lambda_1-1 & \dots & \lambda_1-1 \\ 0 & -1 & \lambda_2-1 & \dots & \lambda_2-1 \\ \vdots & & \ddots & & \vdots \\ 0 & \dots & 0 & -1 & \lambda_{n-1}-1 \\ 0 & \dots & & 0 & -1 \end{pmatrix} \begin{pmatrix} Z_1\dfrac{\partial}{\partial Z_1} \\ \vdots \\ Z_n \dfrac{\partial}{\partial Z_n} \end{pmatrix}.
\end{equation}
\tag{33}
$$
Proof. Property 1 follows from Proposition 1, while property 2 follows from Lemma 1.
We prove property 3. From (29) we derive that
$$
\begin{equation*}
\mu_{i1}^0=\lambda_i-1\quad\text{and} \quad q\mu_{iq}^0+\mu_{i,q+1}^0=0
\end{equation*}
\notag
$$
for any $i=1,\dots,n$ and $q=1, \dots, l$. The first of these equalities and formula (15) for $P_{n1}$ imply (31). The second equality, along with the recurrence relation (20) for the polynomials $P_{nl}$ yield the recurrence relation
$$
\begin{equation}
Q_{n,l+1}(Z_1, \dots, Z_n)=\sum_{i=1}^n \biggl(-Z_i+ \sum_{j=1}^{i-1}(\lambda_i-1)Z_j\biggr)Z_i \frac{\partial Q_{nl}}{\partial Z_i}.
\end{equation}
\tag{34}
$$
It is straightforward to see that the right-hand side of (34) is the result of applying the operator (33) to the polynomial $Q_{nl}$. The proposition is proved. By property 2, $Q_{nl}$ can be treated as polynomials $Q_{nl}(\lambda, Z)$ in the $2n$ variables $\lambda=\lambda_1, \dots, \lambda_n$ and $Z=Z_1, \dots, Z_n$. Corollary 1. For any natural numbers $n$ and $l$ and any $j=1,\dots,n$, the polynomials $Q_{nl}$ have the property where $\lambda'=\lambda_1, \dots, \widehat{\lambda}_j, \dots, \lambda_n$ and $Z'=Z_1, \dots, \widehat{Z}_j, \dots, Z_n$. Here the symbol $\widehat{\phantom{a}}$ marks variables left out. Proof. We prove this assertion using induction on $l$. For $l=1$ the assertion obviously follows from formula (31) for $Q_{n1}$. Assume that the assertion holds for some $l$. Substituting in $z_j=0$, we see from (34) that the operator $\mathfrak{D}_n\big|_{z_j=0}$ turns to an operator similar to $\mathfrak{D}_{n-1}$ but acting on polynomials in the variables $z_1, \dots, \widehat{z}_j, \dots, z_n$, which yields the required assertion. As for substituting in $\lambda_j=1$, by the induction assumption the polynomial $Q_{nl}(\lambda, Z)\big|_{\lambda_j=0}$ depends on $Z_j$ only fictitiously. Therefore, ${\partial Q_{nl}}/{\partial z_j}=0$, and we deduce the required assertion from (34) again. 2.6. Taking the limit as $\delta, x \to 0$. Multiple limit cyclesSince both the polynomials $P_{nl}$ and $Q_{nl}$ are homogeneous in $Z_1, \dots, Z_n$ (see Propositions 1 and 3), we can regard them as defined on the projective space $\mathbb{R}P^{n-1}$. We denote points in $\mathbb{R}P^{n-1}$ by $Z=(Z_1 : \dots : Z_n)$. We consider the map
$$
\begin{equation}
\mathcal{Z}\colon (\delta, x) \mapsto \bigl(Z_1(\delta, x) : \dots : Z_n(\delta, x)\bigr),
\end{equation}
\tag{36}
$$
where the $Z_i(\delta,x)$ are specified by (11). By virtue of Proposition 1 the equations for a limit cycle of multiplicity $(n+1)$ take the form
$$
\begin{equation}
P_{nl}(\mu_{iq}(\delta, x), Z)=0, \quad l=1, \dots, n-1.
\end{equation}
\tag{39}
$$
Definition 6. A sequence of points $(\delta_\alpha, x_\alpha) \to 0$ in the space $B \times (\mathbb{R}_{>0}, 0)$ is said to correspond to a limit cycle (of multiplicity $m$) if for any $\alpha$ the field $v_{\delta_\alpha}$ contains a limit cycle (of multiplicity $m$) intersecting the half-transversal $\Gamma_n^-$ at the point with coordinate $x_\alpha$ (see § 2.1). Assume that there is a sequence $(\delta_\alpha, x_\alpha) \to 0$ corresponding to a limit cycle of multiplicity at least $n+1$. Then the system (37)–(39) has the solution $\delta_\alpha, x_\alpha, \mathcal{Z}(\delta_\alpha, x_\alpha)$. Proposition 4. Assume that a limit cycle of multiplicity at least $n+1$ is born in a perturbation of a polycycle $\gamma$ in a $C^\infty$-smooth family $V=\{v_\delta\}$, $\delta \in B= (\mathbb{R}^k,0)$. Let $\{(\delta_\alpha, x_\alpha)\}_{\alpha=1}^\infty$, $(\delta_\alpha, x_\alpha) \to 0$, be the corresponding sequence in the space $B \times (\mathbb{R}_{>0},0)$. Assume that the map $\mathcal{Z}$ defined by (36) tends to a point $\widetilde{Z} \in \mathbb{R}P^{n-1}$ along this sequence. Then $\widetilde{Z}$ satisfies the system of equations where the polynomials $Q_{nl}$ are specified by (31)–(33). Proposition 4 can be stated in a more general form, which can turn out to be useful. Proposition 5. Assume that a limit cycle of multiplicity at least $m+2$ is born in a perturbation of a polycycle $\gamma$ in a $C^\infty$-smooth family $V=\{v_\delta\}$, $\delta \in B=(\mathbb{R}^k,0)$. Let $\{(\delta_\alpha, x_\alpha)\}_{\alpha=1}^\infty$, $(\delta_\alpha, x_\alpha) \to 0$, be the corresponding sequence in ${B \times (\mathbb{R}_{>0},0)}$. Assume that the map $\mathcal{Z}$ defined by (36) tends to a point $\widetilde{Z} \in \mathbb{R}P^{n-1}$ along this sequence. Then $\widetilde{Z}$ satisfies the system of equations where the polynomials $Q_{nl}$ are specified by (31)–(33). This can be proved similarly to Proposition 4. Lemma 2. Assume that a field $v_0 \in \operatorname{Vect}^\infty(\mathcal{M})$ has a polycycle $\gamma$ formed by hyperbolic saddles $S_1, \dots, S_n$, $n \geqslant 2$, with characteristic numbers $\lambda_1, \dots, \lambda_n$ satisfying $\lambda_1 \cdots \lambda_n \neq 1$. Assume that a $C^\infty$-smooth family $V=\{v_\delta\}$ perturbs $v_0$. Let $C$ denote the set of pairs $(\delta, x)$ such that the field $v_\delta$ has a limit cycle of multiplicity at least $2$ passing through the point with coordinate $x$. Let Then $\mathfrak{Z} \subset \bigcup_{j=1}^n \mathbb{C}P^{n-2}_j$, where Proof. Assume that there is a point $\widetilde{Z}=(\widetilde{Z}_1 : \dots : \widetilde{Z}_n) \in \mathfrak{Z}$ such that for ${i=1, \dots, n}$ the coordinate $\widetilde{Z}_i$ is nonzero. By the definition of $\mathfrak{Z}$ there exists a sequence $(\delta_\alpha, x_\alpha) \to 0$ corresponding to a limit cycle of multiplicity at least two such that the map $\mathcal{Z}$ tends to $\widetilde{Z}$ along this sequence.
We prove by induction on $i=0, \dots, n$ that
$$
\begin{equation}
F_i(\delta_\alpha, x_\alpha)=x_\alpha^{\lambda_1(\delta_\alpha) \dotsb \lambda_i(\delta_\alpha)} * \quad \text{as } \delta_\alpha, x_\alpha \to 0,
\end{equation}
\tag{42}
$$
where the $F_i$ are defined by (10). Here and throughout the proof the symbol $*$ denotes multiplication by a function bounded away from zero and infinity.
The base $i=0$ of induction is obvious: $F_0(\delta_\alpha, x_\alpha)=x_\alpha$. For an arbitrary function $g(\delta, x)$ let $g\big|_{(\delta_\alpha, x_\alpha)}$ denote the value of the function at the point $(\delta_\alpha, x_\alpha)$. We let the assertion hold for $i-1$. Since we have assumed that the components $\widetilde{Z}_i$ of $\widetilde{Z}$, $i=1, \dots, n$, are nonzero, it is true that ${Z_{i+1}}/{Z_i}\big|_{(\delta_\alpha, x_\alpha)}=*$. On the other hand it follows from (11) that
$$
\begin{equation*}
\frac{Z_{i+1}}{Z_i}\bigg|_{(\delta_\alpha, x_\alpha)}=\frac{f_i'(F_{i-1})F_{i-1}}{F_i}\bigg|_{(\delta_\alpha, x_\alpha)}.
\end{equation*}
\notag
$$
Expressing $F_i$ from this equality we arrive at the relation
$$
\begin{equation}
F_i\big|_{(\delta_\alpha, x_\alpha)}=f_i'(F_{i-1}) F_{i-1}*\big|_{(\delta_\alpha, x_\alpha)} \quad \text{as } \delta_\alpha, x_\alpha \to 0.
\end{equation}
\tag{43}
$$
Formulae (3) and (28) imply that
$$
\begin{equation}
|f_i'(\delta, x)|=\Delta_i'(\delta, x)=x^{\lambda_i(\delta)-1} * \quad \text{as } \delta, x \to 0.
\end{equation}
\tag{44}
$$
Substituting this expression into (43) we obtain the recurrence relation $F_i(\delta_\alpha, x_\alpha)=F_{i-1}(\delta_\alpha, x_\alpha)^{\lambda_i(\delta_\alpha)}*$, which yields (42).
Using induction on $i=1, \dots, n$ again, we prove the formula
$$
\begin{equation}
F_i'(\delta_\alpha, x_\alpha)=x^{\lambda_1(\delta_\alpha)\dotsb\lambda_i(\delta_\alpha)-1}*.
\end{equation}
\tag{45}
$$
The base $i=1$ of induction follows from (44). We infer the step of induction from (44) and relation (42) already proved as follows:
$$
\begin{equation*}
F_i'(\delta_\alpha, x_\alpha)=f_n'(F_{i-1}) F_{i-1}' \big|_{(\delta_\alpha, x_\alpha)}=\bigl(x^{\lambda_1(\delta_\alpha) \dotsb\lambda_{i-1}(\delta_\alpha)}\bigr)^{\lambda_i(\delta_i)-1} x^{\lambda_1(\delta_\alpha)\dotsb\lambda_{i-1}(\delta_\alpha)-1}*.
\end{equation*}
\notag
$$
Since for any $\alpha$ the pair $(\delta_\alpha, x_\alpha)$ corresponds to a limit cycle of multiplicity at least two, $\Delta'(\delta_\alpha, x_\alpha)=F_n'(\delta_\alpha, x_\alpha)=1$ due to (6). Taking the logarithm of this equality and applying (45) we arrive at the relation Dividing by $\log x_\alpha$ and passing to the limit as $\alpha \to \infty$, we obtain $\lambda_1 \dotsb \lambda_n=1$, which contradicts the assumption. Hence, at least one of the coordinates of $\widetilde{Z}$ is zero. The lemma is proved. 2.7. Plan of the proof of Theorem 1We consider the case when $n=1$: the polycycle is formed by a single saddle with characteristic number $\lambda_1$, that is, it is a separatrix loop of the saddle $S_1$. Theorem 1 is well known in this case: it is the classical result by Andronov and Leontovich concerning the generation of a rough cycle from a separatrix loop (see [1] Ch. IX, § 29). Namely, the polynomial is $\mathcal{L}_1(\lambda_1)=\lambda_1-1$. We switch to the case when $n \geqslant 2$. Assume that a limit cycle of multiplicity $n+1$ appears in a family $V$. Then Proposition 4 yields that the system of homogeneous equations (40) on the projective space $\mathbb{R}P^{n-1}$ has at least one solution. In addition, assume that the characteristic numbers satisfy Then Lemma 2 implies that for some $j=1, \dots, n$ the system has a solution in the subspace $\mathbb{R}P^{n-2}_j$. By (35), for some $j=1, \dots, n$ the system of equations
$$
\begin{equation}
Q_{n-1,l}(\lambda_1, \dots, \widehat{\lambda}_j, \dots, \lambda_n,Z_1,\dots, \widehat{Z}_j, \dots, Z_n)=0, \qquad l=1, \dots, n-1,
\end{equation}
\tag{48}
$$
has a nontrivial real solution. Here we have again marked variables left out by $\widehat{\phantom{a}}$. Thus, for any natural numbers $n$ and $l$ the polynomial $Q_{nl}$ depends on $2(n-1)$ variables. We denote these by $\mu_1, \dots, \mu_{n-1}$ and $w_1, \dots, w_{n-1}$. We consider the system
$$
\begin{equation}
Q_{n-1,l}(\mu_1, \dots, \mu_{n-1}, w_1, \dots, w_{n-1})=0, \qquad l=1, \dots, n-1.
\end{equation}
\tag{49}
$$
For fixed values of $\mu_1, \dots, \mu_{n-1}$ we have a system of $n-1$ equations on the ${(n-2)}$-dimensional projective space. Therefore, there exists a polynomial $\mathcal{R}_{n-1}(\mu_1,\dots,\mu_{n-1})$ such that system (49) has a nontrivial (in general, complex) solution if and only if $\mathcal{R}_{n-1}(\mu_1, \dots, \mu_{n-1})$ is zero [9]. The polynomial $\mathcal{R}_{n-1}$ is called the resultant of the system of equations. The nontriviality of the resultant of (49) is implied by the following lemma. This lemma is proved in § 3. We consider the polynomial
$$
\begin{equation}
\mathcal{L}_n(\lambda_1, \dots, \lambda_n)=(\lambda_1\dotsb\lambda_n-1)\prod_{j=1}^n \mathcal{R}_{n-1}(\lambda_1, \dots, \widehat{\lambda}_j, \dots, \lambda_n).
\end{equation}
\tag{50}
$$
Assume that the characteristic numbers $\lambda_1, \dots, \lambda_n$ are such that the quantity $\mathcal{L}_n(\lambda_1, \dots, \lambda_n)$ is nonzero. Then the inequality (46) holds, and Lemma 2 is applicable. By this lemma there exists $j=1, \dots, n$ such that the system (48) is solvable. It follows that $\mathcal{R}_{n-1}(\lambda_1, \dots, \widehat{\lambda}_j, \dots, \lambda_n)=0$, which contradicts the inequality $\mathcal{L}_n(\lambda_1, \dots, \lambda_n) \neq 0$. Hence $\mathcal{L}_n$ is the required polynomial. Thus, Theorem 1 is proved apart from Lemma 3. Remark 1. For any family $V$ perturbing a polycycle $\gamma$ formed by saddles with characteristic numbers $\lambda_1, \dots, \lambda_n$, the inequality $\mathcal{L}_n(\lambda_1, \dots, \lambda_n) \neq 0$ is a generic condition on the original field $v_0$. In fact, assume that an unperturbed polycycle $\gamma$ of the original vector field $v_0$ is formed by saddles $S_1, \dots, S_n$ with characteristic numbers $\lambda_1, \dots, \lambda_n$, respectively, where some saddles can coincide. If no two saddles coincide, then the characteristic numbers are independent quantities assuming arbitrary positive values. Since it follows from Lemma 3 that the polynomial $\mathcal{L}_n$ is not identically equal to zero, the set of values of characteristic numbers satisfying (1) is open and everywhere dense in $\mathbb{R}_{>0}^n$. Assume that some of the saddles coincide. Then their characteristic numbers are the same. However, we see from Lemma 3 again that the resultant $\mathcal{R}_{n-1}$ of (47) is nontrivial. Therefore, (1) is a generic condition. § 3. Nontriviality of the resultantIn this section we prove Lemma 3, which completes the proof of Theorem 1. Proof of Lemma 3. We prove the lemma by contradiction. Let the polynomial $\mathcal{R}_{n-1}(\mu)$ be identically equal to zero. According to (31), after the substitution $\mu_1=\dots=\mu_{n-1}=\mu$ the polynomial $Q_{n-1,1}$ takes the form
$$
\begin{equation*}
Q_{n-1,1}(w_1, \dots, w_{n-1})=(\mu-1) (w_1+\dots+w_{n-1}).
\end{equation*}
\notag
$$
Since the operator $\mathfrak{D}_{n-1}$ generating $Q_{n-1,l}$ is linear (see (33)), each polynomial $Q_{n-1,l}$ can be reduced by $\mu-1$.
We let $\mu$ tend to one. By (33), the operator $\mathfrak{D}_{n-1}=\mathfrak{D}_{n-1}(\mu, w_1, \dots, w_{n-1})$ turns in the limit to
$$
\begin{equation*}
\overline{\mathfrak{D}}_{n-1}=- \biggl( w_1^2\frac{\partial}{\partial w_1}+\dots+ w_{n-1}^2\frac{\partial}{\partial w_{n-1}}\biggr).
\end{equation*}
\notag
$$
Therefore, when we divide by $\mu-1$ and pass to the limit as $\mu \to 1$, the polynomials $({1}/(\mu-1))Q_{n-1,l}$ turn to the polynomials
$$
\begin{equation*}
\overline{Q}_{n-1,1}=\frac{1}{\mu-1} Q_{n-1,1}=w_1+\dots+w_{n-1}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \overline{Q}_{n-1,l} &=\overline{\mathfrak{D}}_{n-1}^{\,l-1} \overline{Q}_{n-1,1} =\overline{\mathfrak{D}}_{n-1}^{\,l-1} (w_1+\dots+w_{n-1}) \\ &=(-1)^{l-1} (l-1)!\,(w_1^l+\dots+w_{n-1}^l), \qquad l=1, \dots, n-1. \end{aligned}
\end{equation*}
\notag
$$
It follows from our assumption and the property of the resultant that the system of equations (49) has at least one solution in $\mathbb{C}P^{n-2}$ for any $\mu \neq 1$, which we denote by $w(\mu)$. Since the projective space $\mathbb{C}P^{n-2}$ is compact, there exists a (not necessarily unique) limit point $w(1) \in \mathbb{C}P^{n-2}$ to which the points $w(\mu)$ accumulate as $\mu \to 1$. Since the polynomials $({1}/(\mu-1)) Q_{n-1,l}$ depend continuously on $\mu$, the polynomial $\overline{Q}_{nl}$ is zero at the point $w(1)$ for any $l=1,\dots,n-1$. Thus, to obtain a contradiction, it suffices to show that the system of symmetric polynomials
$$
\begin{equation}
p_l(w_1, \dots, w_{n-1})=w_1^l+\dots+w_{n-1}^l=0, \qquad l=1, \dots, n-1,
\end{equation}
\tag{51}
$$
has no solution in $\mathbb{C}P^{n-2}$. The absence of nontrivial complex solutions of (51) is a well-known fact. We prove it for completeness.
Consider the symmetric polynomials
$$
\begin{equation}
\begin{gathered} \, \sigma_0 (w_1, \dots, w_{n-1})=1, \nonumber \\ \sigma_l (w_1, \dots, w_{n-1})=\sum_{1\leqslant i_1 < \dots < i_l\leqslant n-1} w_{i_1} \cdots w_{i_l}, \qquad l=1, \dots, n-1. \end{gathered}
\end{equation}
\tag{52}
$$
As is known, symmetric polynomials can be expressed in terms of other symmetric polynomials. In particular, the polynomials $p_l$ and $\sigma_l$ are related by the Newton identity (see [10], § 11.1)
$$
\begin{equation*}
l \sigma_l=\sum _{i=1}^l (-1)^{i-1} \sigma_{l-i} p_i, \qquad l=1, \dots, n-1.
\end{equation*}
\notag
$$
Hence it follows from (51) that Note that if system (53) had at least one nontrivial solution $(w_1, \dots, w_{n-1})$, then, by Vieta’s theorem, the polynomial would have at least one nonzero root. However, this is not true, which is a contradiction. Consequently, the polynomial $\mathcal{R}_{n-1}(\mu)$ is nontrivial. The lemma is proved. The proof of Theorem 1 immediately follows from this lemma (see § 2.7). § 4. Proof of Theorem 2The case $n=1$ was considered in the proof of Theorem 1. We have
$$
\begin{equation*}
\mathcal{L}_1(\lambda_1)=\lambda_1-1=\Lambda_1(\lambda_1).
\end{equation*}
\notag
$$
For $n=2,3,4$, the plan of the proof is as follows. First we find the resultant $\mathcal{R}_{n-1}$ by solving the system (49) directly. The required polynomial $\mathcal{L}_n$ will be expressed in terms of $\mathcal{R}_{n-1}$ via (50). The case $n=2$. According to (31), in this case system (49) consists of the single equation $(\mu_1-1)w_1=0$, which has a solution $w_1 \neq 0$ if and only if the polynomial is zero. We derive from (50) that
$$
\begin{equation*}
\mathcal{L}_2(\lambda_1, \lambda_2)=(\lambda_1\lambda_2-1)(\lambda_1-1)(\lambda_2-1)=\Lambda_1(\lambda_1, \lambda_2).
\end{equation*}
\notag
$$
The case $n=3$. Using the recurrence relation (32) we conclude that system (49) in this case has the form
$$
\begin{equation}
Q_{22}(w_1,w_2)=-(\mu_1-1)w_1^2+(\mu_1-1)(\mu_2-1)w_1w_2-(\mu_2-1)w_2^2=0.
\end{equation}
\tag{56}
$$
We consider the linear combination
$$
\begin{equation}
Q_{22}(w_1, w_2)+(w_1+w_2)Q_{21}(w_1, w_2)=(\mu_1\mu_2-1)w_1w_2=0.
\end{equation}
\tag{57}
$$
System (55), (57) is obviously equivalent to (55), (56). Note that system (55), (57) has a nontrivial solution if and only if the polynomial
$$
\begin{equation}
\mathcal{R}_2(\mu_1, \mu_2)=(\mu_1\mu_2-1)(\mu_1-1)(\mu_2-1)
\end{equation}
\tag{58}
$$
is zero. It follows from (50) that
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{L}_3(\lambda_1, \lambda_2, \lambda_3)=(\lambda_1\lambda_2\lambda_3-1)(\lambda_1\lambda_2-1) (\lambda_1\lambda_3-1)(\lambda_2\lambda_3-1) \\&\quad\qquad\times(\lambda_1-1)(\lambda_2-1)(\lambda_3-1) =\Lambda_3(\lambda_1, \lambda_2, \lambda_3). \end{aligned}
\end{equation*}
\notag
$$
The case $n=4$. To write system (48) for $n=4$ we calculate the polynomials $Q_{31}$, $Q_{32}$ and $Q_{33}$. According to (31), we have By (33) the operator $\mathfrak{D}_3$ has the form
$$
\begin{equation*}
\mathfrak{D}_3=(w_1, w_2, w_3) \begin{pmatrix} -1 & \mu_1-1 & \mu_1-1 \\ 0 & -1 & \mu_2-1 \\ 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} w_1 \, \dfrac{\partial}{\partial w_1} \\ w_2 \, \dfrac{\partial}{\partial w_2} \\ w_3 \, \dfrac{\partial}{\partial w_3} \end{pmatrix}.
\end{equation*}
\notag
$$
We infer from the recurrence formula (34) that
$$
\begin{equation*}
Q_{32}=\mathfrak{D}_3 Q_{31}=- \sum _{i=1}^3 (\mu_i-1) w_i^2+\sum _{\substack{i,j=1 \\ i<j}}^3 (\mu_i-1)(\mu_j-1) w_i w_j
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
Q_{33}=\mathfrak{D}_3 Q_{32}=- 2\sum _{i=1}^3 (\mu_i-1) w_i \mathfrak{D}_3 w_i\,{+}\sum _{\substack{i,j=1 \\ i<j}}^3 (\mu_i-1)(\mu_j-1) (w_i \mathfrak{D}_3 w_j+w_j \mathfrak{D}_3 w_i).
\end{equation*}
\notag
$$
We consider the ideal formed by the polynomials $Q_{31}$, $Q_{32}$, and $Q_{33}$ in the ring $\mathbb{Z}[\mu_1, \mu_2, \mu_3, w_1, w_2, w_3]$ and simplify its generators. We set
$$
\begin{equation*}
\begin{gathered} \, \widetilde{Q}_{31}=Q_{31}=\sum _{i=1}^3 (\mu_i-1)w_i, \\ \widetilde{Q}_{32}=Q_{32}+(w_1+w_2+w_3)Q_{31} =\sum _{\substack{i,j=1 \\ i<j}}^3 (\mu_i \mu_j-1) w_i w_j, \\ \widetilde{Q}_{33}=\mathfrak{D}_3 \widetilde{Q}_{32} =Q_{33}+(w_1+w_2+w_3)Q_{32}+Q_{31} \mathfrak{D}_3 (w_1+ w_2+w_3). \end{gathered}
\end{equation*}
\notag
$$
We replace the generator $\widetilde{Q}_{33}$ by an even simpler one, which has the form
$$
\begin{equation*}
\widehat{Q}_{33}=\widetilde{Q}_{33}-\bigl(-w_1+(\mu_3-2)w_2+(2\mu_3-3)w_3\bigr)\widetilde{Q}_{32}+ (\mu_2\mu_3-1)\widetilde{Q}_{31}.
\end{equation*}
\notag
$$
A direct calculation, which we omit, shows that where
$$
\begin{equation*}
L(w_2, w_3)=w_2(\mu_1\mu_2\mu_3+\mu_1\mu_2+\mu_1\mu_3-\mu_1-2)-w_3(\mu_1\mu_3-1)(\mu_3-1).
\end{equation*}
\notag
$$
Remark 2. We can assume that all three variables $w_1$, $w_2$ and $w_3$ are nonzero. In fact, if any of them is zero, then, by property (35) the original system of three equations formed by $Q_{31}$, $Q_{32}$ and $Q_{33}$ turns to a system formed by the two polynomials $Q_{21}$ and $Q_{22}$ of two other variables. Therefore, to find the resultant $\mathcal{R}_3$ we must multiply the quantity derived based on the assumption that these variables are nonzero by the polynomials $\mathcal{R}_2$ of all possible pairs of variables $\mu_1, \mu_2$, $ \mu_3$. Consider the system formed by the polynomials $\widetilde{Q}_{31}$, $\widetilde{Q}_{32}$, and $L$ equated to zero. Using that $L$ is linear, we eliminate the variable $w_3$. We obtain the system of two equations
$$
\begin{equation}
\qquad\qquad +w_2(\mu_3-1)(2\mu_1\mu_2\mu_3+\mu_1\mu_2-\mu_1 -\mu_2-1)=0,
\end{equation}
\tag{59}
$$
$$
\begin{equation}
w_2 \bigl(w_1(\mu_1\mu_3-1)(2\mu_1\mu_2\mu_3+\mu_1\mu_3 -\mu_1-\mu_3-1) \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad +w_2(\mu_2\mu_3-1)(\mu_1\mu_2\mu_3+\mu_1\mu_2+\mu_1\mu_3-\mu_1 -2)\bigr)=0.
\end{equation}
\tag{60}
$$
In view of Remark 2 we divide the second equation by $w_2$, thus arriving at a linear system. This linear system has a nontrivial solution if and only if its determinant is zero. This determinant has the form
$$
\begin{equation*}
(\mu_3-1)(\mu_1\mu_3-1)(\mu_1\mu_2\mu_3-1) \bigl(4(\mu_1\mu_2\mu_3-1)-(\mu_1-1)(\mu_2-1)(\mu_3-1)\bigr).
\end{equation*}
\notag
$$
By Remark 2 this polynomial must be multiplied by the resultants $\mathcal{R}_2(\mu_1, \mu_2)$, $\mathcal{R}_2(\mu_1, \mu_3)$ and $\mathcal{R}_2(\mu_2, \mu_3)$ obtained before (see (58)). The resulting polynomial has the form
$$
\begin{equation*}
\begin{aligned} \, &R^*(\mu_1, \mu_2,\mu_3) =(\mu_1\mu_2\mu_3-1)(\mu_1\mu_2-1)(\mu_1\mu_3-1)^2(\mu_2\mu_3-1) \\ &\qquad\times (\mu_1-1)^2(\mu_2-1)^2(\mu_3-1)^3\bigl(4(\mu_1\mu_2\mu_3-1)-(\mu_1-1)(\mu_2-1)(\mu_3-1)\bigr). \end{aligned}
\end{equation*}
\notag
$$
If the original system formed by the polynomials $Q_{31}$, $Q_{32}$ and $Q_{33}$ has a nontrivial solution for some $\mu_1,\mu_2$ and $\mu_3$, then $R^*$ is zero. Hence $R^*$ is divisible by the resultant of this system. Since we are interested not so much in the resultant itself but rather in its set of zeros, we can consider the following polynomial instead of $R^*$:
$$
\begin{equation}
\begin{aligned} \, \notag &\mathcal{R}_3(\mu_1, \mu_2,\mu_3)=(\mu_1\mu_2\mu_3- 1)(\mu_1\mu_2-1)(\mu_1\mu_3-1)(\mu_2\mu_3-1) \\ &\qquad\times (\mu_1-1)(\mu_2-1)(\mu_3-1)\bigl(4(\mu_1\mu_2\mu_3-1)-(\mu_1-1)(\mu_2-1)(\mu_3-1)\bigr). \end{aligned}
\end{equation}
\tag{61}
$$
Nevertheless, computer calculations show that it is the polynomial (61) that is the resultant of the system formed by $Q_{31}$, $Q_{32}$ and $Q_{32}$. We substitute $\mathcal{R}_3$ into (50) and obtain
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{L}_4(\lambda_1,\lambda_2,\lambda_3,\lambda_4) =\Lambda_4(\lambda_1,\lambda_2,\lambda_3,\lambda_4) \\ &\qquad\qquad \times \bigl(4(\lambda_1\lambda_2\lambda_3-1)-(\lambda_1-1)(\lambda_2-1)(\lambda_3-1)\bigr) \\ &\qquad\qquad \times\bigl(4(\lambda_1\lambda_2\lambda_4-1)-(\lambda_1-1)(\lambda_2-1)(\lambda_4-1)\bigr) \\ &\qquad\qquad \times \bigl(4(\lambda_1\lambda_3\lambda_4-1)-(\lambda_1-1)(\lambda_3-1)(\lambda_4-1)\bigr) \\ &\qquad\qquad \times \bigl(4(\lambda_2\lambda_3\lambda_4-1)-(\lambda_2-1)(\lambda_3-1)(\lambda_4-1)\bigr), \end{aligned}
\end{equation*}
\notag
$$
which coincides with the assertion of the theorem.
§ 5. Multiple fixed points on the real lineNote that most of the paper has not appealed to the fact that the function $\Delta$ is the Poincaré map of a polycycle. In fact, we have sought fixed points of a function of a certain form defined on an interval. This makes it possible to reformulate the result in terms of functions on the real line. Let $f_i\colon \mathbb{R}_{>0} \to \mathbb{R}$, $i=1, \dots, n$, be $C^r$-smooth functions on the real half-line, $r\geqslant n$. Assume that the $f_i$ depend continuously on the parameter $\delta$ ranging over an arbitrary topological space $B$ with distinguished point $0$. Assume that there are positive numbers $\lambda_1, \dots, \lambda_n$ such that and
$$
\begin{equation}
\lim_{\delta, x \to 0} x^q \frac{\partial^q}{\partial x^q}\log|f_i'(x)|=(-1)^{q-1}(q-1)!\,(\lambda_i-1), \qquad q=1, \dots, r-1.
\end{equation}
\tag{62}
$$
In other words, the functions $f_i$ behave like power functions with exponents $\lambda_i$ (see § 2.4 for more detail). By analogy with (10), (11) and (36), we introduce the notation
$$
\begin{equation*}
\begin{gathered} \, F_i=f_i \circ \dots \circ f_0, \qquad f_0=\mathrm{id}, \\ Z_i=\frac{F_{i-1}'}{F_{i-1}}, \qquad i=1,\dots,n. \\ \mathcal{Z}\colon (\delta, x) \mapsto \bigl(Z_1(\delta, x) : \dots : Z_n(\delta, x)\bigr). \end{gathered}
\end{equation*}
\notag
$$
Consider the function The following theorems hold.Theorem 3. There exists a nonzero polynomial $\mathcal{L}_n\in\mathbb{Z}[\lambda_1, \dots, \lambda_n]$ such that for any numbers $\lambda_1, \dots, \lambda_n$ satisfying each fixed point of $\Delta$ that is close to zero for $\delta \to 0$ has multiplicity at most $n$. Theorem 4. Let $\mathcal{F}$ be the set of all pairs $(\delta, x)$ corresponding to fixed points of the function $\Delta$ of multiplicity $m+2\leqslant r$. Then any limit point $Z$ (as $\delta, x \to 0$) of the function $\mathcal{Z}\big|_{\mathcal{F}}$ satisfies the system of equations where the polynomials $Q_{nl}$ are defined by (31)–(33). Theorem 5. For $n=1,2,3,4$, the following polynomials satisfy the assumptions of Theorem 3: These theorems can be proved by repeating verbatim the proofs of Theorem 1, Proposition 5,and Theorem 2, respectively. The only difference is that assumption (62) is used instead of Lemma 1. In addition, all the three theorems are true for functions $f_i$ of finite smoothness, since the only place in their proofs where we use the infinite smoothness of vector fields is Lemma 1, which is replaced by (62). § 6. Open problemsWe began this paper with describing the available results on the cyclicity of polycycles. Can the investigation of multiple limit cycles help one in estimates for cyclicity? Yes, but only in estimates from below. This can be formulated as the following two conjectures. Let $\gamma$ be a hyperbolic polycycle of a field $v_0$ on a two-dimensional oriented manifold that is formed by $n$ saddles (some of which can coincide) with characteristic numbers $\lambda_1, \dots, \lambda_n$. Conjecture 1. There is an open subset $ U $ of $ \mathbb{R}^n_{>0}$ such that for any set of characteristic numbers $(\lambda_1, \dots, \lambda_n) \in U$ a limit cycle of multiplicity $n$ ($n$ limit cycles) is born in a generic $n$-parameter family perturbing the polycycle $\gamma$. Conjecture 2. There is an open (in the induced topology) subset $W$ of the surface $\{(\lambda_1, \dots, \lambda_n) \in \mathbb{R}^n_{>0} \mid \lambda_1\dotsb\lambda_n=1\}$ such that for any set of characteristic numbers $(\lambda_1, \dots, \lambda_n) \in W$ a limit cycle of multiplicity $n+1$ ($n+1$ limit cycles) is born in a generic $(n+1)$-parameter family perturbing the polycycle $\gamma$. 
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\Bibitem{Duk23} |
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