# Fractal Analysis of Hyperbolic Saddles
with Applications

Vlatko Crnković  University of Zagreb, Faculty of Electrical Engineering and
Computing, Department of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia;
Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek,
Belgium<EMAIL_ADDRESS>,  Renato Huzak  Hasselt University, Campus
Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium
<EMAIL_ADDRESS>and  Maja Resman  University of Zagreb, Faculty of
Science, Department of Mathematics, Horvatovac 102a, 10000 Zagreb, Croatia
<EMAIL_ADDRESS>
(Date: May 31, 2023)

###### Abstract.

In this paper we express the Minkowski dimension of spiral trajectories near
hyperbolic saddles and semi-hyperbolic singularities in terms of the Minkowski
dimension of intersections of such spirals with transversals near these
singularities. We apply these results to hyperbolic saddle-loops and
hyperbolic $2$-cycles to obtain upper bounds on the cyclicity of such limit
periodic sets.

###### Key words and phrases:

Minkowski dimension, saddle-loops, $2$-cycles, cyclicity

###### 2010 Mathematics Subject Classification:

37C10, 28A75, 37C27, 37C29

##  1\. Introduction

The Minkowski dimension is a fractal dimension that quantifies how the
Lebesgue measure of the $\delta$-neighborhood of a bounded set in
$\mathbb{R}^{N}$ behaves as $\delta\to 0$. There are several equivalent ways
of calculating this dimension but we mostly use the following one:

######  Definition 1 ([3]).

Let $G$ be a bounded set in $N$-dimensional Euclidean space $\mathbb{R}^{N}$.
Let

$G_{\delta}\colon=\\{p\in\mathbb{R}^{N}\colon\text{dist}(p,G)<\delta\\},\delta>0,$

be the $\delta$-neighborhood of $G$ and let $|G_{\delta}|$ be its Lebesgue
measure.
The upper and the lower Minkowski dimension of set $G$ are defined as the
limits

$\overline{\text{dim}}_{B}\,G=\limsup_{\delta\to
0}\left[N-\frac{\ln|G_{\delta}|}{\ln\delta}\right]\quad\text{and}\quad\underline{\text{dim}}_{B}\,G=\liminf_{\delta\to
0}\left[N-\frac{\ln|G_{\delta}|}{\ln\delta}\right]$

respectively. If these two values are equal, the common value is called the
Minkowski dimension of set $G$ and denoted by $\dim_{B}\,G$.

The Minkowski dimensions are preserved under bi-Lipschitz transformations,
even when the image and the original are not in the same ambient space. More
precisely, a transformation $\Psi:A\subset\mathbb{R}^{N}\to\mathbb{R}^{K}$ is
called bi-Lipschitz if there exist positive constants $m$ and $M$ such that,
for any $x,y\in A$,

$m||\Psi(x)-\Psi(y)||\leq||x-y||\leq M||\Psi(x)-\Psi(y)||.$

For a bi-Lipschitz map $\Psi:A\subset\mathbb{R}^{N}\to\mathbb{R}^{K}$ we have
that

$\underline{\text{dim}}_{B}A=\underline{\text{dim}}_{B}\Psi(A)\quad\text{and}\quad\overline{\text{dim}}_{B}{A}=\overline{\text{dim}}_{B}\Psi(A).$

All three Minkowski dimensions are monotone in the sense that, for $G\subseteq
H,\dim G\leq\dim H$, when both are defined. In addition, the Minkowski
dimension and the upper Minkowski dimension are finitely stable, meaning that
$\dim\,(G\cup H)=\max\\{\dim\,G,\dim\,H\\}$. For more on these and other
properties of the Minkowski dimension we refer the reader to [3, 8].

It is already known that the Minkowski dimension of spirals around weak foci
and limit cycles of planar analytic vector fields yields information on the
cyclicity of those limit periodic sets. First results of this type were
obtained in [10]. We state two main theorems that describe such connections.

######  Theorem 1 (Weak focus case, [10, 9]).

Let $\Gamma$ be a spiral trajectory of the system

$\begin{cases}\dot{r}=r(r^{2l}+\sum_{i=0}^{l-1}a_{i}r^{2i})\\\
\dot{\phi}=1\end{cases}$

near the origin. Then

  1. $(a)$

if $a_{0}\neq 0$, then $\dim_{B}\Gamma=1$.

  2. $(b)$

if $a_{0}=a_{1}=...=a_{k-1}=0,\ a_{k}\neq 0,\ k\geq 1$, then
$\dim_{B}\Gamma=\frac{4k}{2k+1}$.

######  Theorem 2 (Limit cycle case, [10, 9]).

Let the system

$\begin{cases}\dot{r}=r(r^{2l}+\sum_{i=0}^{l-1}a_{i}r^{2i})\\\
\dot{\phi}=1\end{cases}$

have a limit cycle $r=a$ of multiplicity $m,\ 1\leq m\leq l$. Let $\Gamma_{1}$
and $\Gamma_{2}$ be spiral trajectories of this system near the limit cycle
from outside or inside respectively. Then
$\dim_{B}\Gamma_{1}=\dim_{B}\Gamma_{2}=2-\frac{1}{m}$.

Due to the Flow-Box Theorem (see for instance [1, Theorem 1.12]), in order to
calculate the dimension of spiral trajectories near limit cycles, it is
sufficient to calculate the dimension of a sequence of points obtained by
intersecting any such spiral with a transversal to the limit cycle (i.e. of
the orbit of the first-return map on the transversal). For the case of foci,
there is a variant of the Flow-Box Theorem developped in [9], called _Flow-
Sector Theorem_ , that allows similar relation. For more details, see [9].

In addition, due to results from [2], there is a direct correspondence between
the multiplicity of a fixed point of a line diffeomorphism (the first return
map) and the Minkowski dimension of its orbit converging to the fixed point
and, as a consequence, with the cyclicity of a focus/limit cycle.

In this paper we deal with spiral trajectories near more complex limit
periodic sets: polycycles containing saddles and/or saddle-nodes. For a
hyperbolic saddle with eigenvalues $\lambda_{-}<0$ and $\lambda_{+}>0$, the
hyperbolicity ratio is the quantity $r=-\frac{\lambda_{-}}{\lambda_{+}}>0$. We
read an upper bound on cyclicity of those sets in some known cases from the
box dimension of their spiral trajectories. A better understanding of the
cyclicity of such limit periodic sets is crucial for tackling the Hilbert’s
$16$th problem. It is not possible to use the Flow-Box Theorem to calculate
the Minkowski dimension of spiral trajectories accumulating on such polycycles
(from within), because the theorem does not apply near singularities, so we
need a new method to calculate the Minkowski dimension of parts of the
trajectories near singular points. Our main results are stated in Theorem 3
and Theorem 4 of Section 2 which deal with neighborhoods of a hyperbolic
saddle and a semi-hyperbolic singularity respectively.

In Section 3 we apply Theorem 3 to a saddle-loop and find a relation between
the codimension of the saddle-loop (an upper bound on the cyclicity of the
loop) and the Minkowski dimension of its spiral trajectories. The Minkowski
dimension depends only on the codimension of the loop, but the correspondence
is $2$–$1$. For a more precise formulation, see Theorem 5.

Finally, in Section 4, we apply Theorem 3 to a hyperbolic $2$-cycle, and
compare to cyclicity results obtained in [6]. To summarize, for a non-resonant
hyperbolic $2$-cycle with ratios of hyperbolicity $r_{1}<1<r_{2}$ such that
$r_{1}r_{2}=1$ and $r_{1},r_{2}\not\in\mathbb{Q}$, the cyclicity of the
$2$-cycle is shown not to be greater than $3+(1+r_{1})\frac{d-1}{2-d}$, where
$d$ is the Minkowski dimension of any spiral trajectory near the $2$-cycle.

In the sequel we use two notions for the asymptotic behavior of functions as
$x\to 0$. For $f(x)$ and $g(x)$ two positive functions with $x\approx 0$ and
$x>0$, we write

$f(x)\simeq g(x),\ x\to 0,$

if there exist two positive constants $m$ and $M$ such that $mg(x)\leq
f(x)\leq Mg(x)$ for all $x$ sufficiently small. For $f(x)$ and $g(x)$ two
positive functions with $x\approx 0$ and $x>0$, we write

$f(x)\sim g(x),\ x\to 0,$

if

$\lim_{x\to 0+}\frac{f(x)}{g(x)}=1.$

##  2\. The main results

Let us explain the basic idea behind our method for calculating the Minkowski
dimension of spiral trajectories of a polycycle. Due to the finite stability
of the Minkowski dimension, in order to compute the dimension of a spiral
trajectory, we consider separately different parts of the spiral: parts near
the singular points and parts near the regular sides of the polycycle. The
Minkowski dimension of the entire spiral is the maximal dimension of its
constituting parts.

The Flow-Box Theorem allows us to calculate the dimension of parts near the
regular sides of the polycycle, but we need a new tool to calculate the
dimension of the remaining parts. For any transversal to a regular side of a
polycycle, the points of intersection of the spiral with the transversal
define a sequence $(y_{n})_{n}$. The distance between consecutive points of
the sequence eventually starts to decrease. We take two transversals, one on
each side of the singular point and sufficiently close to the singular point,
in the domain where the saddle can be brought to a simpler normal form (see
the proofs of Theorems 3 and 4 in Section 2). Without loss of generality, in
the normal form coordinates we assume the transversals $\\{x=1\\}$ (vertical)
and $\\{y=1\\}$ (horizontal), and compute the dimension of the family of
curves passing through a given sequence of points on the entry transversal and
ending on exit transversal.

Note that planar saddles and saddle-nodes, unlike planar foci or complex
saddles, are not monodromic points, so that the first return map around the
saddle/saddle-node point is not well-defined before we close the connection
(as in the polycyle). Therefore, our first results in Theorems 3 and 4 concern
the box dimension of a union of disjoint local trajectories in the
neighborhood of the saddle/saddle-node singularity that correspond to (any)
prescribed sequence of points on the entry transversal. The expression for the
Minkowski dimension of spiral trajectories accumulating on a polycycle is
provided in Corollary 1.

Let $s$ be a hyperbolic saddle or a semi-hyperbolic singularity of an analytic
vector field. By $t_{S}$ and $t_{U}$ we denote the transversals to the stable
and the unstable manifold of $s$ in the saddle case, i.e. the transversals to
the stable (up to the time reversal preserving the geometry of the flow) and
the center manifold in the semi-hyperbolic case (in the saddle region). We
call $t_{S}$ also the _entry_ and $t_{U}$ the _exit_ transversal, due to
obvious reasons. Up to the reversal of the time, without loss of generality we
assume that the ratio of hyperbolicity of the saddle is greater than $1$ in
the hyperbolic saddle case. Up to the change of the axes, we additionally
assume that the stable, i.e. entry, transversal is the vertical transversal
$\\{x=1\\}$.

We take $(y_{n})_{n}$ to be any sequence on $t_{S}$ that converges
monotonically to the intersection of $t_{S}$ with the stable manifold, and
such that the distances between consecutive points $y_{n}$ decrease
monotonically (see Figure 1). Let $(x_{n})_{n}$ be the sequence of points
where the trajectories $(\Gamma_{n})_{n}$ of the vector field of the
saddle/saddle-node $s$ going through $(y_{n})_{n}$ intersect $t_{U}$. Under
the above assumptions it is not hard to see that
$\dim_{B}\,(y_{n})_{n}\geq\dim_{B}\,(x_{n})_{n}$. We show that the Minkowski
dimension of the union of trajectories $(\Gamma_{n})_{n}$ between the points
$y_{n}$ and $x_{n}$ is

$\dim_{B}(\cup_{n}\Gamma_{n})=1+\dim_{B}\,(y_{n})_{n}.$

######  Theorem 3 (Minkowski dimension of the hyperbolic saddle).

Let $s=0$ be a hyperbolic saddle of an analytic vector field with ratio of
hyperbolicity $\frac{1}{\alpha}\geq 1$:

$\begin{cases}x^{\prime}=-x+h.o.t.,\\\ y^{\prime}=\alpha y+h.o.t.\end{cases}$

Let $t_{S},\ t_{U}$ and $(y_{n})_{n\in\mathbb{N}}$ on $t_{S}$ be defined as
above. If the sequence $(y_{n})_{n\in\mathbb{N}}$ has Minkowski dimension,
$\dim_{B}\,(y_{n})_{n}$, then

$\dim_{B}\,\left(\cup_{n\in\mathbb{N}}\Gamma_{n}\right)=1+\dim_{B}(y_{n})_{n}.$

######  Theorem 4 (Minkowski dimension of the semi-hyperbolic singularity).

Let $s=0$ be a semi-hyperbolic singularity of an analytic vector field:

$\begin{cases}x^{\prime}=-x+h.o.t.,\\\ y^{\prime}=\alpha y^{m}+h.o.t.,\
\alpha>0,\ m\geq 2.\end{cases}$

Let $t_{S},\ t_{U}$ and $(y_{n})_{n\in\mathbb{N}}$ on $t_{S}$ be defined as
above. If the sequence $(y_{n})_{n\in\mathbb{N}}$ has the Minkowski dimension,
$\dim_{B}\,(y_{n})_{n}$, then

$\dim_{B}\,\left(\cup_{n\in\mathbb{N}}\Gamma_{n}\right)=1+\dim_{B}(y_{n})_{n}.$

In Propositions 1 and 2, to fix the ideas, we first prove the weaker versions
of Theorems 3 and 4 for linear saddles:

(1) $\begin{cases}\dot{x}=-x\\\ \dot{y}=\alpha y,\ \ 0<\alpha\leq
1,\end{cases},$

and the simplest semi-hyperbolic singularities

(2) $\begin{cases}\dot{x}=-x\\\ \dot{y}=\alpha y^{m},\ \ m\geq 2,\
\alpha>0.\end{cases}$

######  Proposition 1.

For a linear saddle (1), under notation of Theorem 3, it holds that:

$\dim_{B}\left(\cup_{n\in\mathbb{N}}\Gamma_{n}\right)=1+\dim_{B}(y_{n})_{n}.$

###### Proof.

Since bi-Lipschitz transformations do not change the Minkowski dimensions,
using a rescaling in $x$ and $y$ we may assume that $t_{S}=\\{x=1\\}$ and
$t_{U}=\\{y=1\\}$. The rescaled sequence on $t_{S}=\\{x=1\\}$ satisfies the
same assumptions as the original one, and we use the same notation
$(y_{n})_{n\in\mathbb{N}}$. We first present the standard computation of the
Minkowski dimension of the sequence $(y_{n})_{n\in\mathbb{N}}$ in one-
dimensional ambient space $\mathbb{R}$. Let us denote $Y:=\\{y_{n}\colon
n\in\mathbb{N}\\}$. For $\delta>0$ small enough, there is a unique critical
index $n_{\delta}$ such that $y_{n_{\delta}}-y_{n_{\delta}+1}<2\delta$ and
$y_{n}-y_{n+1}\geq 2\delta$, for all $n<n_{\delta}$. We now divide the
$\delta$-neighborhood $Y_{\delta}$ into two parts. The $\delta$-neighborhoods
of points $y_{1},y_{2},...,y_{n_{\delta}-1}$ do not intersect, and we call
their union the tail of $Y_{\delta}$ and denote it by $T_{\delta}$. On the
other hand, the $\delta$-neighborhood of the remainder of the sequence is the
interval $(-\delta,y_{n_{\delta}}+\delta)$, and it is refered to as the
nucleus of $Y_{\delta}$ and denoted by $N_{\delta}$ (see e.g.[8]). Note that
$N_{\delta}$ and $T_{\delta}$ are disjoint. The Lebesgue measure of
$Y_{\delta}$ is now equal to:

$|Y_{\delta}|=|N_{\delta}|+|T_{\delta}|=(y_{n_{\delta}}+2\delta)+(n_{\delta}-1)\cdot
2\delta=y_{n_{\delta}}+2\delta n_{\delta}.$

Since, by our assumptions, $Y$ has Minkowski dimension, by definition of
Minkowski dimension it holds that

$\dim_{B}Y=\lim_{\delta\to 0}\left[1-\frac{\ln(y_{n_{\delta}}+2\delta
n_{\delta})}{\ln\delta}\right].$

Let

$\Gamma:=\\{\Gamma_{n}\colon n\in\mathbb{N}\\}.$

Let us first consider the part of the set $\Gamma$ of trajectories in the
region $\\{\frac{1}{2}\leq x\leq 1\\}$. Since the Minkowski dimension of the
Cartesian product is the sum of Minkowski dimensions, in this region the Flow-
Box Theorem allows us to easily calculate the Minkowski dimension to be
$1+\dim_{B}Y$. Due to monotonicity of $\underline{\text{dim}}_{B}$, we now
have the lower bound

$1+\dim_{B}Y\leq\underline{\text{dim}}_{B}\Gamma.$

Therefore, to prove the proposition, it suffices to show that

(3) $\overline{\dim}_{B}\Gamma\leq 1+\dim_{B}Y.$

It can be shown that there exists a positive constant $C>0$ such that, for any
$n\in\mathbb{N}$, the Lebesgue measure of $(\Gamma_{n})_{\delta}$ is bounded
from above by $C\delta$, as $\delta\to 0$. This bound is uniform both in
$\delta\in(0,\delta_{0})$, $\delta_{0}>0$, and in $n\in\mathbb{N}$. Therefore,
for a given $\delta>0$, we bound the Lebesgue measure of the
$\delta$-neighborhood of the union of trajectories
$\cup_{n<n_{\delta}}\Gamma_{n}$, arising from the tail of $Y_{\delta}$, from
above by

(4) $\left|\cup_{n<n_{\delta}}(\Gamma_{n})_{\delta}\right|\leq
C\delta(n_{\delta}-1).$

On the other hand, the Lebesgue measure of the $\delta$-neighborhood of the
union of trajectories $\cup_{n\geq n_{\delta}}\Gamma_{n}$, arising from the
nucleus of $Y_{\delta}$, is bounded from above by

(5) $\left|\cup_{n\geq n_{\delta}}(\Gamma_{n})_{\delta}\right|\leq
y_{n_{\delta}}+\int_{y_{n_{\delta}}}^{1}x(y)dy+D\delta,$

where $D$ is a universal positive constant and $y\mapsto x(y)$ is a function
whose graph is the curve $\Gamma_{n_{\delta}}$. For more details, see Figure
1.

Figure 1. $\delta$-neighborhood of $\Gamma$

To prove inequality (3), we consider separately two cases.

  1. (1)

Case: $0<\alpha<1$.
By direct integration of the simple vector field (1), we get

$x(y)=\left(\frac{y_{n_{\delta}}}{y}\right)^{\frac{1}{\alpha}},$

so

(6)
${}\int_{y_{n_{\delta}}}^{1}x(y)dy=\frac{\alpha}{1-\alpha}(y_{n_{\delta}}-y_{n_{\delta}}^{\frac{1}{\alpha}}).$

Therefore, using (4), (5) and (6), there exists a universal positive constant
$M>0$ such that

$|\Gamma_{\delta}|\leq M(y_{n_{\delta}}+2\delta n_{\delta}),\ \delta>0.$

Finally,

$\displaystyle\limsup_{\delta\to
0}\left[2-\frac{\ln|\Gamma_{\delta}|}{\ln\delta}\right]$
$\displaystyle\leq\limsup_{\delta\to 0}\left[2-\frac{\ln
M+\ln(y_{n_{\delta}}+2\delta n_{\delta})}{\ln\delta}\right]$
$\displaystyle=\limsup_{\delta\to 0}\left[2-\frac{\ln(y_{n_{\delta}}+2\delta
n_{\delta})}{\ln\delta}\right]=1+\dim_{B}Y.$

Therefore, $\overline{\dim}_{B}\Gamma\leq 1+\dim_{B}Y$.

  2. (2)

Case: $\alpha=1$.
Note that formula (6) cannot be used. However, similarly as in Case 1, by
direct integration we get:

$x(y)=\frac{y_{n_{\delta}}}{y},\ \
\int_{y_{n_{\delta}}}^{1}x(y)dy=y_{n_{\delta}}(-\ln y_{n_{\delta}}).$

Therefore, there exists $M>0$ such that: $|\Gamma_{\delta}|\leq
M(y_{n_{\delta}}(-\ln y_{n_{\delta}})+2\delta n_{\delta})$. Now, since both
$y_{n_{\delta}}\to 0$ and $\delta n_{\delta}\to 0$, as $\delta\to 0$, for
every small $\kappa>0$ there exists $\delta_{\kappa}$, such that, for every
$0<\delta<\delta_{\kappa}$, it holds that

$y_{n_{\delta}}(-\ln y_{n_{\delta}})<y_{n_{\delta}}^{1-\kappa},\ 2\delta
n_{\delta}<(2\delta n_{\delta})^{1-\kappa},\ \delta<\delta_{\kappa}.$

Therefore,

$|\Gamma_{\delta}|\leq 2M\max\\{y_{n_{\delta}},2\delta
n_{\delta}\\}^{1-\kappa},\ \delta<\delta_{\kappa}.$

Now, for every $\kappa>0$,

$\displaystyle\limsup_{\delta\to 0}$
$\displaystyle\left[2-\frac{\ln|\Gamma_{\delta}|}{\ln\delta}\right]\leq\limsup_{\delta\to
0}\left[2-\frac{\ln(2M)+(1-\kappa)\ln\max\\{y_{n_{\delta}},2\delta
n_{\delta}\\}}{\ln\delta}\right]$ $\displaystyle\leq\limsup_{\delta\to
0}\left[2-(1-\kappa)\frac{\ln(y_{n_{\delta}}+2\delta
n_{\delta})}{\ln\delta}\right]=1+\kappa+(1-\kappa)\dim_{B}Y.$

Letting $\kappa\to 0$, we get

$\limsup_{\delta\to
0}\left[2-\frac{\ln|\Gamma_{\delta}|}{\ln\delta}\right]\leq 1+\dim_{B}Y.$

Finally, since

$1+\dim_{B}Y\leq\underline{\text{dim}}_{B}\Gamma\leq\overline{\text{dim}}_{B}\Gamma\leq
1+\dim_{B}Y,$

we conclude that

$\dim_{B}\Gamma=1+\dim_{B}Y.$

∎

######  Proposition 2.

For a semi-hyperbolic singularity (2), under notation of Theorem 4, it holds
that:

$\dim_{B}\left(\cup_{n\in\mathbb{N}}\Gamma_{n}\right)=1+\dim_{B}(y_{n})_{n}.$

###### Proof.

We use a similar nucleus-tail approach from the proof of Lemma 1. Again,
rescaling $x$ and $y$, we assume $t_{S}=\\{x=1\\}$ and $t_{U}=\\{y=1\\}$. This
changes the constant $\alpha$ in (2), but $\alpha$ remains positive. Again we
denote $Y:=\\{y_{n}\colon n\in\mathbb{N}\\}$ and $\Gamma=\\{\Gamma_{n}\colon
n\in\mathbb{N}\\}$. As in the proof of Lemma 1,
$\underline{\dim}_{B}\Gamma\geq 1+\dim_{B}Y$. Again, there exist uniform
positive constants $C$ and $D$ such that same bounds (4) and (5) hold. Let us
show that the integral in (5) satisfies

(7) $\int_{y_{n_{\delta}}}^{1}x(y)dy=o(y_{n_{\delta}}),\ \delta\to 0,$

where $x(y)=\exp\left(\frac{y^{1-m}-y_{n_{\delta}}^{1-m}}{\alpha(m-1)}\right)$
is the function whose graph is the trajectory $\Gamma_{n_{\delta}}$ through
$(1,y_{n_{\delta}})$. To prove this it suffices to show that the function

$y\mapsto\int_{y}^{1}\exp\left(\frac{t^{1-m}-y^{1-m}}{\alpha(m-1)}\right)dt$

is $o(y)$, as $y\to 0$. Indeed, we have that

$\displaystyle\lim_{y\to
0}\frac{\int_{y}^{1}\exp\left(\frac{t^{1-m}-y^{1-m}}{\alpha(m-1)}\right)dt}{y}$
$\displaystyle=\lim_{y\to
0}\frac{\int_{y}^{1}\exp\left(\frac{t^{1-m}}{\alpha(m-1)}\right)dt}{y\exp\left(\frac{y^{1-m}}{\alpha(m-1)}\right)}=$
$\displaystyle=\lim_{y\to
0}\frac{-\exp\left(\frac{y^{1-m}}{\alpha(m-1)}\right)}{(1-\frac{y^{1-m}}{\alpha})\exp\left(\frac{y^{1-m}}{\alpha(m-1)}\right)}=0,$

where we used the L’Hospital’s rule in the second step. Using (7) as in Lemma
1, we get that $|\Gamma_{\delta}|\leq M(y_{n_{\delta}}+2\delta n_{\delta})$
for some positive constant $M$. Consequently,
$\overline{\text{dim}}_{B}\Gamma\leq 1+\dim_{B}Y$. ∎

###### Proof of Theorem 3.

Consider an analytic vector field with a hyperbolic saddle, and let $t_{S}$
and $t_{U}$ be as in the statement of the theorem. Following [1, Theorem
2.15], near the hyperbolic saddle the field can be reduced to the following
(smooth) orbital normal form:

(8) $\begin{cases}\dot{x}=-x\\\ \dot{y}=\alpha y+h(x,y),\end{cases}$

where $\frac{1}{\alpha}\geq 1$ is the hyperbolicity ratio of the saddle and
$h(x,y)=O(xy^{2}),\ (x,y)\to(0,0),$ is a $C^{\infty}$ function. Since a smooth
local change of coordinates is bi-Lipschitz, it preserves the Minkowski
dimension around $0$. Therefore, it suffices to compute Minkowski dimension of
the saddle in the normal form (8).

For an arbitrary $\beta>0$, we choose a small enough neighborhood of the
saddle such that $|\alpha y+h(x,y)|\leq(1+\beta)y$. Now we choose new
transversals $t_{S}^{\prime}$ and $t_{U}^{\prime}$ that intersect the stable
and unstable manifold respectively in this neighborhood. Up to a rescaling, we
may assume that $t^{\prime}_{S}=\\{x=1\\}$ and $t^{\prime}_{U}=\\{y=1\\}$. Due
to the Flow-Box Theorem, the maximal Minkowski dimension of parts of
$\Gamma=\cup_{n\in\mathbb{N}}\Gamma_{n}$ between $t_{S}$ and $t_{S}^{\prime}$,
$t_{U}$ and $t_{U}^{\prime}$, as well as of those around $t_{U}^{\prime}$ and
$t_{S}^{\prime}$, is equal to $1+\dim_{B}Y$. Therefore,

(9) $\underline{\text{dim}}_{B}\Gamma\geq 1+\dim_{B}Y.$

To prove the theorem, it suffices to show that, for every $\beta>0$,

(10)
$\overline{\dim}_{B}\Gamma^{\prime}\leq\frac{2\beta+1+\dim_{B}Y}{1+\beta},$

where $\Gamma^{\prime}$ is the part of $\Gamma$ between $t_{S}^{\prime}$ and
$t_{U}^{\prime}$. Evidently,
$\overline{\dim}_{B}\Gamma^{\prime}=\overline{\dim}_{B}\Gamma$. Passing to
limit as $\beta\to 0$, we get:

$\overline{\dim}_{B}\Gamma\leq 1+\dim_{B}Y,$

which, along with (9), concludes the proof of the theorem.

Let us now prove (10). For $(u,v)\in\Gamma_{n}$ we have that

$-\ln u=\int_{1}^{u}-\frac{dx}{x}=\int_{y_{n}}^{v}\frac{dy}{\alpha
y+h(x,y)}\geq\int_{y_{n}}^{v}\frac{dy}{(1+\beta)y},$

that is,

$u\leq\left(\frac{y_{n}}{v}\right)^{\frac{1}{1+\beta}}.$

Similarly as in the proof of Proposition 1, the Lebesgue measure of the
$\delta$-neighborhood of trajectories $\Gamma_{n}$ arising from the nucleus is
bounded above by

$|\cup_{n\geq n_{\delta}}(\Gamma_{n})_{\delta}|\leq
y_{n_{\delta}}+\int_{y_{n_{\delta}}}^{1}\left(\frac{y_{n_{\delta}}}{y}\right)^{\frac{1}{1+\beta}}dy+D\delta.$

The integral above is of order $O(y_{n_{\delta}}^{\frac{1}{1+\beta}})$, as
$\delta\to 0$. Now we proceed similarly as in the proof of _Case 2._ in
Proposition 1. There exists $M>0$ such that, for sufficiently small
$\delta>0$,

$|\Gamma_{\delta}|\leq M(y_{n_{\delta}}^{\frac{1}{1+\beta}}+2\delta
n_{\delta}).$

Due to the fact that, for every $\beta>0$, $2\delta n_{\delta}<(2\delta
n_{\delta})^{\frac{1}{1+\beta}}$ for sufficiently small
$\delta<\delta_{\beta}$, we get

$|\Gamma_{\delta}|\leq 2M\max\\{y_{n_{\delta}},2\delta
n_{\delta}\\}^{\frac{1}{1+\beta}},\ \delta<\delta_{\beta}.$

Using exactly the same procedure as in the proof of Proposition 1, _Case 2._ ,
we finally get

$\limsup_{\delta\to
0}\left[2-\frac{\ln|\Gamma_{\delta}|}{\ln\delta}\right]\leq
2-\frac{1}{1+\beta}(1-\dim_{B}Y)=\frac{2\beta+1+\dim_{B}Y}{1+\beta}.$

∎

###### Proof of Theorem 4.

Due to [1, Theorem 2.19] we can use the following orbital normal form near the
semi-hyperbolic singularity

$\begin{cases}\dot{x}=-x,\\\ \dot{y}=y^{m}+h(x,y),\ m\geq 2,\end{cases}$

where $h(x,y)=O(y^{2m-1})$ is a $C^{\infty}$ function.

Now, similarly as in the proof of Theorem 3, we find a sufficiently small
neighborhood of the singularity where $|h(x,y)|\leq y^{m}$. Now, with this
bound, we proceed similarly as in the proof of Proposition 2 to show that
$\overline{\text{dim}}_{B}\Gamma\leq 1+\dim_{B}Y$. The other inequality,
$\underline{\dim}_{B}\Gamma\geq 1+\dim_{B}Y$, is deduced analogously as in
Theorem 3. ∎

######  Corollary 1 (Minkowski dimension of a monodromic polycycle).

Let $P$ be a monodromic $N$-polycycle of an analytic vector field, with
hyperbolic saddles and semi-hyperbolic singularities as vertices,
$N\in\mathbb{N}$. Let $S$ be a spiral trajectory accumulating to $P$. Let
$t_{1},t_{2},....,t_{N}$ be transversals to $($all$)$ regular sides of the
polycycle. By $Y_{k},\,k\in\\{1,2,...,N\\}$, we denote the intersections of
$S$ with transversals $t_{k}$ respectively. Then,

$\dim_{B}\,S=1+\max\left\\{\dim_{B}\,Y_{k}\colon
k\in\\{1,2,...,N\\}\right\\}.$

###### Proof.

This Corollary is a direct consequence of Theorems 3 and 4 and the finite
stability of the Minkowski dimension. ∎

##  3\. Applications to hyperbolic saddle-loops

The simplest polycycle configuration is a hyperbolic saddle-loop of an
analytic vector field. A saddle-loop is an invariant set in which the unstable
manifold of the saddle extends to its stable manifold (i.e. there exists a
homoclinic connection of the saddle).

Let $r>0$ be the hyperbolicity ratio of the saddle. It is well-known (see e.g.
[7, pg. 109]) that the first return map $P$ on any transversal to the regular
part of the loop (parametrized regularly by $x\in[0,T[$, where $x=0$
corresponds to the point on the loop) satisfies:

  1. (1)

(codimension $1$ case, $r\in\mathbb{R}^{+}\setminus\\{1\\}$)

$P(x)\,\sim Ax^{r},\ A>0,$

  2. (2)

(higher finite codimension cases, $r=1$)

$P(x)=x+\delta(x),$

where

$\delta(x)=\beta_{1}x+\alpha_{2}x^{2}(-\ln
x)+\beta_{2}x^{2}+...+\beta_{k-1}x^{k-1}+\alpha_{k}x^{k}(-\ln x)+O(x^{k}).$

The saddle loop is said [7] to be of _codimension $2k$_ if
$\delta(x)\sim\beta_{k}x^{k}$, as $x\to 0$, $\beta_{k}\neq 0$. It is said to
be of _codimension $2k+1$_ if $\delta(x)\sim\alpha_{k+1}x^{k+1}(-\ln x)$, as
$x\to 0$, with $\alpha_{k+1}\neq 0$, $k\geq 1$. We exclude here _infinite
codimension_ cases when $P\equiv\mathrm{id}$, that is, when the loop is of
_center type_. In other words, in _finite codimension_ cases there is an
accumulating spiral trajectory to the loop. In [7] it is shown that the
codimension of the saddle loop corresponds to its cyclicity in generic
unfoldings.

In the following Theorem 5, we apply our results from Section 2 to give a
correspondence between the codimension of the loop and the Minkowski content
of its (any) spiral trajectory. The correspondence between the codimension of
the saddle-loop and the Minkowski dimension of spiral trajectories is $2$–$1$.

######  Theorem 5 (Minkowski dimension of a saddle-loop).

The Minkowski dimension of a spiral trajectory $S$ in an analytic vector field
that has a finite-codimension saddle-loop as its $\alpha/\omega$-limit set
depends only on the codimension of the saddle-loop. More precisely, if $k\geq
1$ is the codimension of the saddle loop, then:

$\dim_{B}S=\begin{cases}2-\frac{2}{k},&k\text{ even,}\\\
2-\frac{2}{k+1},&k\text{ odd}.\end{cases}$

###### Proof.

In codimension $1$ and $2$ cases, the first return map $P$ is hyperbolic, so
the Minkowski dimension of the set of points of intersection of any spiral
with a regular transversal to the saddle-loop (i.e. of an orbit of $P$) is $0$
(see [2, Lemma 1]). Now, due to the Flow-Box Theorem, Theorem 3 and the finite
stability of the Minkowski dimension, the dimension of the spiral is _trivial_
, $\dim_{B}S=1$.

For even codimensions $k>2$, the Minkowski dimension of the set of points of
intersection of any spiral with a transversal to the saddle-loop (i.e. of an
orbit of $P$) is $1-\frac{2}{k}$ due to [2, Theorem 1]. Again, using the Flow-
Box Theorem and Theorem 3 we conclude that $\dim_{B}S=2-\frac{2}{k}$.

For odd codimensions $k>2$, the Minkowski dimension of the set of points of
intersection of any spiral with a transversal to the saddle-loop is
$1-\frac{2}{k+1}$ (see [5, Theorem 2]). Therefore,
$\dim_{B}S=2-\frac{2}{k+1}.$ ∎

##  4\. Applications to hyperbolic $2$-cycles

In this section we focus on analytic vector fields with a hyperbolic
$2$-saddle polycycle $\Gamma_{2}$ with hyperbolicity ratios $r_{1}$ and
$r_{2}$ (see Figure 2). We apply our fractal methods to the classical results
from [6] (and references therein) about the cyclicity of the $2$-cycle in the
_non-degenerate_ and the _degenerate_ case.

Figure 2. A non-trivial hyperbolic $2$-cycle

###  4.1. _Non-degenerate_ $2$-cycles

######  Theorem 6 (Cyclicity of non-degenerate $2$-cycles, [6]).

If the conditions

(11) $r_{1}\neq 1,\quad r_{2}\neq 1,\quad r_{1}r_{2}\neq 1$

hold, then the polycycle $\Gamma_{2}$ is of cyclicity less than or equal to
$2$ in any $C^{\infty}$-unfolding. If, moreover,

$(r_{1}-1)(r_{2}-1)<0,$

there exists a two-parameter $C^{\infty}$-versal unfolding $(X_{\lambda})$ in
which $\Gamma_{2}$ is of cyclicity $2$. Otherwise, there exists a two-
parameter $C^{\infty}$-versal unfolding in which $\Gamma_{2}$ is of cyclicity
$1$.

We now state our fractal result for non-degenerate $2$-cycles from Theorem 6.
Note that, since $r_{1}r_{2}\neq 1$, the $2$-cycle of Theorem 7 is not of
_center-type_ ($P\neq\mathrm{id}$), but has accumulating spiral trajectories
(_focus-type_).

######  Theorem 7 (Minkowski dimension of a non-degenerate $2$-polycycle).

Let $\Gamma_{2}$ be a monodromic $2$-polycycle of an analytic vector field,
non-degenerate in the sense of (11). The Minkowski dimension of any spiral
trajectory accumulating on $\Gamma_{2}$ is _trivial_ :

$\dim_{B}S=1.$

###### Proof.

Let $t_{1}$ and $t_{2}$ be any regularly parametrized transversals to
heteroclinic connections of $\Gamma_{2}$ not intersecting the saddles. The
first return maps $P_{i}:t_{i}\to t_{i},\ i\in\\{1,2\\}$, as compositions of
regular diffeomorphisms and corner maps of the saddles, satisfy

$P_{i}(x)\simeq x^{r_{1}r_{2}},\,x\to 0.$

Due to [2, Lemma 1], the Minkowski dimension of its orbit (i.e. of the
intersection of the spiral with $t_{1}$ and $t_{2}$) is $0$. By Corollary 1,
the Minkowski dimension of the entire spiral is $1$. ∎

###  4.2. _Degenerate_ $2$-cycles

In [6], Mourtada distinguishes between three families of _degenerate_
$2$-cycles (when some of non-degeneracy conditions $r_{1}\neq 1,\,r_{2}\neq 1$
or $r_{1}r_{2}\neq 1$ do not hold):

  * •

$\mathcal{C}_{1}=\\{\Gamma_{2}\colon r_{1}r_{2}\neq 1\\}$,

  * •

$\mathcal{C}_{2}=\\{\Gamma_{2}\colon
r_{1}r_{2}=1,\,r_{1}\not\in\mathbb{Q}\\}$,

  * •

$\mathcal{C}_{3}=\\{\Gamma_{2}\colon r_{1}r_{2}=1,\,r_{1}\in\mathbb{Q}\\}$.

In the remainder of this paper we focus only on fractal analysis on families
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ and comparison with their known
cyclicities. The cyclicity of polycycles in $\mathcal{C}_{3}$ is more
complicated due to the presence of independent _Ecalle-Roussarie compensators_
from two resonant saddles (see [7]).

####  4.2.1. Family $\mathcal{C}_{1}$

######  Theorem 8 (Cyclicity of $\mathcal{C}_{1}$, Theorem 1 in [6]).

Let $\Gamma_{2}$ be a $2$-cycle belonging to $\mathcal{C}_{1}$ and tangent to
a planar vector field $X_{0}$. Then $\Gamma_{2}$ is of cyclicity less than or
equal to $2$ in every $C^{\infty}$ family $(X_{\lambda})$ unfolding $X_{0}$.
Furthermore, there exists a three-parameter $C^{\infty}$-versal unfolding
$(X_{\lambda})$ in which $\Gamma_{2}$ is of cyclicity $2$.

Again, same as in the case of non-degenerate $2$-cycles in Theorem 7, it is
clear here (by form of $P\neq\mathrm{id}$) that $\Gamma_{2}\in\mathcal{C}_{1}$
is not of center-type, but has spiral trajectories.

######  Theorem 9 (Minkowski dimension of $\mathcal{C}_{1}$).

Let a degenerate $2$-cycle $\Gamma_{2}$ of an analytic vector field belong to
the family $\mathcal{C}_{1}$. The Minkowski dimension of any spiral trajectory
$S$ accumulating on $\Gamma_{2}$ is _trivial_ , $\dim_{B}S=1$.

###### Proof.

The proof is analogous to the proof of Theorem 7. ∎

####  4.2.2. Family $\mathcal{C}_{2}$

First note that here, since $r_{1}r_{2}=1$, an extra assumption of _non-
trivial_ polycycles is requested in Theorems 10 and 11 to exclude identity
first return maps, that is, $2$-cycles that are of center type. Therefore, we
consider only monodromic cases of polycycles with spiraling trajectories.

Let $\Gamma_{2}\in\mathcal{C}_{2}$ be a $2$-cycle of an analytic vector field.
The following analysis of the first return maps on transversals to both
heteroclinic connections is due to Mourtada [6]. Since the ratios of
hyperbolicity $r_{1}$ and $r_{2}$ of the saddles are irrational, the saddles
are analytically linearizable, and there are $C^{\infty}$ transversals
$\sigma_{i},\tau_{i}$ near the saddles such that the associated corner Dulac
maps $D_{i}:\sigma_{i}\to\tau_{i}$ are given by

$y_{i}=D_{i}(x_{i})=x_{i}^{r_{i}},\,i=1,2.$

For more details see [6, pg. 78]. On the other hand, regular transition maps
$R_{1}:\tau_{2}\to\sigma_{1}$ and $R_{2}:\tau_{1}\to\sigma_{2}$ are given by

$x_{1}=R_{1}(y_{2})=\beta_{2,1}y_{2}\left[1+\alpha_{1}y_{2}^{k_{1}-1}+o(y_{2}^{k_{1}-1})\right]$

and

$x_{2}=R_{2}(y_{1})=\beta_{1,2}y_{1}\left[1+\alpha_{2}y_{1}^{k_{2}-1}+o(y_{1}^{k_{2}-1})\right]$

where $\beta_{1,2},\beta_{2,1}>0$ and $2\leq
k_{i}\in\mathbb{N}\cup\\{\infty\\}$, and where $k_{i}=\infty$ implies
$\alpha_{i}=0$. The first return maps $P_{1}=R_{1}\circ D_{2}\circ R_{2}\circ
D_{1}:\sigma_{1}\to\sigma_{1}$ and $P_{2}=R_{2}\circ D_{1}\circ R_{1}\circ
D_{2}:\sigma_{2}\to\sigma_{2}$ are then given by:

$\displaystyle P_{1}(x_{1})=\beta_{2,1}\beta_{1,2}^{r_{2}}x_{1}\Big{[}1+$
$\displaystyle\left(\alpha_{1}\beta_{1,2}^{r_{2}(k_{1}-1)}x_{1}^{k_{1}-1}+o(x_{1}^{k_{1}-1})\right)+$
(12)
$\displaystyle\qquad\qquad+\left(r_{2}\alpha_{2}x_{1}^{r_{1}(k_{2}-1)}+o(x_{1}^{r_{1}(k_{2}-1)})\right)\Big{]},$

and

$\displaystyle P_{2}(x_{2})=\beta_{1,2}\beta_{2,1}^{r_{1}}x_{2}\Big{[}1+$
$\displaystyle\left(\alpha_{2}\beta_{2,1}^{r_{1}(k_{2}-1)}x_{2}^{k_{2}-1}+o(x_{2}^{k_{2}-1})\right)+$
(13)
$\displaystyle\qquad\qquad+\left(r_{1}\alpha_{1}x_{2}^{r_{2}(k_{1}-1)}+o(x_{2}^{r_{2}(k_{1}-1)})\right)\Big{]}.$

Notice that $P_{1}$ is hyperbolic/tangent to identity if and only if $P_{2}$
is hyperbolic/tangent to identity. Indeed,

$\beta_{1,2}\beta_{2,1}^{r_{1}}=\beta_{1,2}^{r_{1}r_{2}}\beta_{2,1}^{r_{1}}=\left(\beta_{2,1}\beta_{1,2}^{r_{2}}\right)^{r_{1}}.$

Moreover, exactly one of the inequalities $k_{1}-1<r_{1}(k_{2}-1)$ or
$k_{2}-1<r_{2}(k_{1}-1)$ holds. On the contrary, if we assume that both hold,
we get

$k_{1}-1<r_{1}(k_{2}-1)<r_{1}r_{2}(k_{1}-1)=k_{1}-1,$

which is obviously a contradiction. On the other hand, if we assume that
neither one holds, we have

$k_{1}-1\geq r_{1}(k_{2}-1)\geq r_{1}r_{2}(k_{1}-1)=k_{1}-1.$

which would imply $k_{1}-1=r_{1}(k_{2}-1)$ and $k_{2}-1=r_{2}(k_{1}-1)$. As a
consequence, $r_{1},\,r_{2}\in\mathbb{Q}$, which is a contradiction.

Moreover, in the case $\beta_{1,2}\beta_{2,1}^{r_{1}}=1$,
$|\alpha_{1}|+|\alpha_{2}|\neq 0$. Otherwise $P_{1}=P_{2}=\mathrm{id}$, which
is the trivial case that is not considered here. This is a consequence of the
_quasi-analyticity_ of the first return maps around hyperbolic saddle
polycycles in analytic planar vector fields [4], that states that the Taylor
map that associates to a Dulac germ its Dulac asymptotic expansion is
injective (i.e., trivial expansion implies the trivial germ).

######  Theorem 10 (Cyclicity in $\mathcal{C}_{2}$, [6], p. 83).

Let $X_{0}$ be an analytic vector field with a non-trivial $($in the sense
that the first return map is not equal to the identity$)$ $2$-cycle
$\Gamma_{2}\in\mathcal{C}_{2}$ tangent to $X_{0}$. In the notation as above,

  1. (1)

If $\beta_{1,2}\beta_{2,1}^{r_{1}}\neq 1$, then the cyclicity of $\Gamma_{2}$
$($in any $C^{\infty}$ unfolding $(X_{\lambda}))$ is not greater than $3$.

  2. (2)

If $\beta_{1,2}\beta_{2,1}^{r_{1}}=1$ and $|\alpha_{1}|+|\alpha_{2}|\neq 0$,
then the cyclicity of $\Gamma_{2}$ $($in any $C^{\infty}$ unfolding
$(X_{\lambda}))$ is not greater than $\epsilon$, where:

$\epsilon:=\begin{cases}2+k_{1}+\lfloor\frac{k_{1}-1}{r_{1}}\rfloor&\text{if
}k_{1}-1<r_{1}(k_{2}-1),\\\
2+k_{2}+\lfloor\frac{k_{2}-1}{r_{2}}\rfloor&\text{if
}k_{2}-1<r_{2}(k_{1}-1).\end{cases}$

Note the surprising fact that the cyclicity, unlike in all the previous cases,
cannot be read only from a single first return map.

We now state our ’fractal’ version of Theorem 10. The goal is to read the
Mourtada’s upper bound on the cyclicity of the $2$-cycle
$\Gamma_{2}\in\mathcal{C}_{2}$ from the Minkowski dimension of only one
trajectory accumulating to $\Gamma_{2}$. However, as can be expected in the
light of the comment before Theorem 10, the dimension of the spiral trajectory
will not suffice. In order to read Mourtada’s upper bound, we need additional
fractal data, see Corollary 2 below.

######  Theorem 11 (Fractal version of Theorem 10).

Let $X_{0}$ be an analytic vector field with a non-trivial $2$-cycle
$\Gamma_{2}\in\mathcal{C}_{2}$. Let $r:=\min\\{r_{1},\,r_{2}\\}$ be the
minimal hyperbolicity ratio. Any spiral trajectory accumulating to the
polycycle has the same Minkowski dimension $d\in[1,2)$. Moreover, the
cyclicity of $\Gamma_{2}$ in $C^{\infty}$ unfoldings of $X_{0}$ is at most

(14) $\left\lfloor 3+(1+r)\frac{d-1}{2-d}\right\rfloor.$

###### Proof.

By Theorem 10, if $\beta_{1,2}\beta_{2,1}^{r_{1}}\neq 1$ then the cyclicity of
the polycycle is at most $3$. On the other hand, by (4.2.2) and (4.2.2), the
first return maps $P_{1}$ and $P_{2}$ are hyperbolic and, therefore, the
intersections of any spiral with a transversal to the polycycle has Minkowski
dimension $0$ (see [2, Lemma 1]). By Corollary 1 we conclude that $d=1$.

Consider now the case when $\beta_{1,2}\beta_{2,1}^{r_{1}}=1$. By (4.2.2) and
(4.2.2), it follows that $|\alpha_{1}|+|\alpha_{2}|\neq 0$. Indeed, if
$|\alpha_{1}|+|\alpha_{2}|=0$, the first return maps are equal to the identity
and the polycycle is trivial (of center type), which is a contradiction with
the assumption. Therefore, in (4.2.2) and (4.2.2), at least one of $k_{1}$ and
$k_{2}$ is finite. Without loss of generality (see the discussion at the
beginning of the section) we assume that

(15) $k_{1}-1<r_{1}(k_{2}-1).$

The Minkowski dimension of an orbit of $P_{1}$ is $1-\frac{1}{k_{1}}$ and the
Minkowski dimension of an orbit of $P_{2}$ is $1-\frac{1}{r_{2}(k_{1}-1)+1}$
(see [2, Theorem 1]). Now we distinguish two cases:

  1. (1)

$r_{1}>1$

Since $r_{1}r_{2}=1$, it follows that $r_{2}<1$, so $r_{2}(k_{1}-1)<k_{1}-1$.
By Corollary 1 we conclude that $d=2-\frac{1}{k_{1}}\in\mathbb{Q}$.

By Theorem 10, cyclicity in case (15) is at most

$2+k_{1}+\left\lfloor\frac{k_{1}-1}{r_{1}}\right\rfloor=\left\lfloor
2+k_{1}+\frac{k_{1}-1}{r_{1}}\right\rfloor=\left\lfloor
3+\frac{d-1}{2-d}+r_{2}\frac{d-1}{2-d}\right\rfloor.$

  2. (2)

$r_{1}<1$

It follows that $r_{2}>1$, so $r_{2}(k_{1}-1)>k_{1}-1.$ Similarly as in the
previous case we get that $d=2-\frac{1}{1+r_{2}(k_{1}-1)}\not\in\mathbb{Q}$.
The cyclicity of the polycycle is at most

$2+k_{1}+\left\lfloor\frac{k_{1}-1}{r_{1}}\right\rfloor=\left\lfloor
2+k_{1}+\frac{k_{1}-1}{r_{1}}\right\rfloor=\left\lfloor
3+r_{1}\frac{d-1}{2-d}+\frac{d-1}{2-d}\right\rfloor.$

In the symmetrical case $k_{2}-1<r_{2}(k_{1}-1)$ in (15) similar conclusions
hold, and the statement of the theorem follows. ∎

The following corollary unites previous results in non-degenerate and
degenerate $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ cases.

######  Corollary 2.

Let $\Gamma_{2}$ be a non-trivial $2$-saddle polycycle of an analytic vector
field such that $\Gamma_{2}\notin\mathcal{C}_{3}$. Let $S$ be its one
accumulating spiral trajectory.

  1. (1)

If $\dim_{B}S=1$, then the cyclicity is at most $3$.

  2. (2)

If $d:=\dim_{B}S\in(1,2)$, then the upper bound on cyclicity is given by
formula (14) of Theorem 11, and

$r=\min\left\\{\frac{d_{1}-1}{d_{2}-1},\frac{d_{2}-1}{d_{1}-1}\right\\},$

$d_{1}\in(0,1)$ and $d_{2}\in(0,1)$ Minkowski dimensions of sequences obtained
as intersections of spiral $S$ with transversals to the two heteroclinic
connections. Note also that $d=1+\max\\{d_{1},d_{2}\\}$.

###### Proof.

By Theorems 7, 9 and 11, $\dim_{B}S=1$ for non-degenerate cycles, for family
$\mathcal{C}_{1}$ and in the case $r_{1}r_{2}\neq 1$ in $\mathcal{C}_{2}$. In
all those cases the first return maps are strongly hyperbolic or hyperbolic.
In all these families, by Theorems 6, 8 and 10 of Mourtada, the upper bound on
cyclicity is $3$. On the other hand, if $r_{1}\cdot r_{2}=1$ and
$r_{1},\,r_{2}\notin\mathbb{Q}$, the first return maps on both transversals
are tangent to the identity, with multiplicities $\gamma_{1}$ and $\gamma_{2}$
striclty greater than $1$. It is easy to check by e.g. [2] that
$r_{1}=\frac{\gamma_{1}}{\gamma_{2}}$ and
$r_{2}=\frac{\gamma_{2}}{\gamma_{1}}$. By [2], $d_{1}=1-\frac{1}{\gamma_{1}}$
and $d_{2}=1-\frac{1}{\gamma_{2}}$, and the above formula for
$r:=\min\\{r_{1},r_{2}\\}$ follows. ∎

######  Remark 1.

Note that a trivial saddle polycycle is of center type (no spiral
trajectories). The first return map on transversals to heteroclinic
connections is equal to the identity. The Minkowski dimension of just one
periodic trajectory close to the polycycle is $1$ (moreover, of finite
length). On the other hand, the continuum of periodic trajectories
accumulating on the polycycle is an open set of non-zero area and its
Minkowski dimension is equal to $2$. None of the two makes much sense to
consider. Therefore, we exclude this case from our fractal considerations.

In all non-trivial cases, by quasi-analyticity of first return maps around
hyperbolic polycycles of planar analytic vector fields, the first return map
on transversals is never the identity, but either tangent to the identity or
(strongly) hyperbolic. Therefore its orbits on transversals to heteroclinic
connections have Minkowski dimension belonging to $[0,1)$, by e.g. [2]. By
Corollary 1, the Minkowski dimension of a spiral trajectory $S$ around the
non-trivial hyperbolic saddle polycycle then satisfies $\dim_{B}S\in[1,2)$.

## Acknowledgements

This research was supported by Croatian Science Foundation (HRZZ) Grant
PZS-2019-02-3055 from Research Cooperability program funded by the European
Social Fund. The first two authors are supported by the Special Research Fund
(BOF number: BOF21BL01) of Hasselt University. The third author is supported
by Croatian Science Foundation (HRZZ) Grant UIP-2017-05-1020. The first and
the third author are also supported by the bilateral Hubert-Curien Cogito
grant 2023-24.

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