\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 276, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/276\hfil Systems with spiral trajectories]
{Geometrical properties of systems with spiral trajectories in $\mathbb{R}^3$}

\author[L. Korkut, D. Vlah, V. \v{Z}upanovi\'c \hfil EJDE-2015/276\hfilneg]
{Luka  Korkut, Domagoj Vlah, Vesna \v{Z}upanovi\'c}

\address{Luka  Korkut \newline
University of Zagreb, Faculty of Electrical Engineering and Computing,
Department of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia}
\email{luka.korkut@fer.hr}

\address{Domagoj Vlah \newline
University of Zagreb, Faculty of Electrical Engineering and Computing,
Department of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia}
\email{domagoj.vlah@fer.hr}

\address{Vesna \v{Z}upanovi\'c \newline
University of Zagreb,
Faculty of Electrical Engineering and Computing,
Department of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia}
\email{vesna.zupanovic@fer.hr}

\thanks{Submitted September 28, 2015. Published October 29, 2015.}
\subjclass[2010]{37C45, 37G10, 34C15, 28A80}
\keywords{Spiral; chirp; box dimension; rectifiability; oscillatory dimension;
\hfill\break\indent  phase dimension; limit cycle}

\begin{abstract}
 We study a class of second-order nonautonomous differential equations,
 and the corresponding planar and spatial systems, from the geometrical
 point of view. The oscillatory behavior of solutions at infinity is
 measured by oscillatory and phase dimensions,
 The oscillatory dimension is defined  as the box dimension of the 
 reflected solution near the origin,  while the phase dimension is 
 defined as the box dimension of a trajectory  of the planar system 
 in the phase plane. Using the phase dimension of
 the second-order equation we compute the box dimension of a spiral
 trajectory of the spatial system. This phase dimension of the second-order
 equation is connected to the asymptotic of the associated Poincar\'e map.
 Also, the box dimension of a trajectory of the reduced normal form with
 one eigenvalue equals zero, and a pair of pure imaginary eigenvalues is
 computed when limit cycles bifurcate from the origin.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction and motivation}

We found our mathematical inspiration in the book by Tricot \cite{tricot},
 where the author introduced a new approach for studying curves.
He showed for some classes of smooth curves, nonrectifiable near the
accumulation point, that fractal dimension called box dimension, can
``measure'' the density of accumulation. Tricot computed box dimension
for class of spiral curves and chirps.
In this article by geometric properties of systems we mean type of
solution curves, which are here spirals and chirps. Furthermore we
distinguish rectifiable and nonrectifiable curves.
Whereas box dimension of rectifiable curve is trivial, we proceed to
study nonrectifiable curves using Tricot's fractal approach, and
compute the box dimension.

Since 1970, dimension theory for dynamics has evolved into an independent
field of mathematics. Together with Hausdorff dimension, box dimension was
used to characterize dynamics, in particular chaotic dynamics having
strange attractors, see \cite{fdd}.
We use the box dimension, because of countable stability of Hausdorff dimension,
its value is trivial on all smooth nonrectifiable curves, while the box
dimension is nontrivial, that is, larger than 1. The box dimension is
 suitable tool for classification of nonrectifiable curves.
Analogously, box dimension is a good tool for analysis of discrete dynamical
systems. Using box dimension we can study orbits of one-dimensional discrete
system near its fixed point. Slow convergence to stable point means higher
density of an orbit near its fixed point, which implies bigger box dimension.
Fast convergence is related to trivial box dimension.

A natural idea is that higher density of orbits reveals higher multiplicity
of the fixed point. The multiplicity of the fixed points is related to
the bifurcations which could be produced by varying parameters of a given
family of systems. Bifurcation theory provides a strategy for investigating
the bifurcations that occur within a family.

 \v{Z}ubrini\'c and \v{Z}upanovi\'c \cite{zuzu} showed
the number of limit cycles which could be produced from
weak foci and limit cycles is directly related to the box dimension
of any trajectory.
 It was discovered that the box dimension of a spiral trajectory of weak
focus signals a moment of Hopf and Hopf-Takens bifurcation. The result was
obtained using Takens normal form \cite{takens}. Using a numerical algorithm
for computation of box dimension of trajectory, it is possible to predict
change of stability of the system, through Hopf bifurcation.
Recent results Marde\v{s}i\'c,  Resman and \v{Z}upanovi\'c  \cite{MRZ},
Resman \cite{majaformal, majaabel} and Horvat Dmitrovi\'c  \cite{laho},
show efficiency of this approach to the bifurcation theory.
From asymptotic expansion of $\varepsilon$-neighborhood of an orbit,
we  read box dimension and Minkowski content from the leading term.
In the mentioned articles, it has been showed that more information
about dynamical system could be read from other terms of the asymptotic
expansion of $\varepsilon$-neighborhood.

In this article we study nonautonomous differential equation of second order,
and the corresponding systems  with spiral trajectories, in $\mathbb{R}^2$ and $\mathbb{R}^3$.
The planar system has the same type of spiral as in Takens normal form,
see \cite{takens}, which is spiral with analytic first return Poincar\'e map,
also having the same asymptotics in each direction. Here we studied graph
of solution of differential equation, as well as the corresponding trajectories
in the phase plane.  The system could produce limit cycles under perturbation,
but it is left for further research.

According to the idea of qualitative theory of differential equations,
oscillations of a class of second-order differential equations have been
considered by phase plane analysis, in Pa\v{s}i\'c, \v{Z}ubrini\'c and
\v{Z}upanovi\'c  \cite{chirp}. The novelty was a fractal approach,
connecting the box dimension of the graph of a solution and the box
dimension of a trajectory in the phase plane. Oscillatory and phase
dimensions for a class of second-order differential equations have been introduced.

On the other hand, in  \cite{zuzu,zuzulien}, the box dimension of spiral
trajectories of a system with pure imaginary eigenvalues, near singular
points and limit cycles, has been studied using normal forms, and the
Poincar\'e map. These results are based on the fact that these spiral
trajectories are  of power type in polar coordinates.
Also, it is shown that the box dimension is sensitive with respect to bifurcations,
e.g.,  it jumps from the trivial value $1$ to the value $4/3$  when the
 Hopf bifurcation occurs. Degenerate Hopf bifurcation or Hopf-Takens
bifurcation can reach even larger box dimension of a  trajectory near
a singular point. This value is related to the multiplicity of the singular
point surrounded with spiral trajectories. This phenomenon has been
discovered for  discrete systems in  Elezovi\'c, \v{Z}upanovi\'c and
\v{Z}ubrini\'c \cite{neveda} concerning saddle-node and period-doubling
bifurcations, and generalized in  \cite{laho}.
Also, in  \cite{MRZ}   there are  results about multiplicity of the
Poincar\' e map   near a weak focus, limit cycle, and saddle-loop,
obtained using the box dimension. Isochronicity of a focus has been
characterized by box dimension in Li and Wu \cite{li}. Formal normal
forms for parabolic diffeomorphisms have been characterized by fractal
invariants of the $\varepsilon-$neighborhood of a discrete orbit in
\cite{majaformal}.  All these results are related to the 16th Hilbert problem.

This work is a part of our research, which has been undertaken in order
to understand  relation between the graph of certain type of oscillatory function,
and the corresponding spiral curve in the phase plane. We believed that
chirp-like oscillations  defined by $X(\tau)=\tau^{\alpha}\sin1/\tau$ ``correspond''
to spiral oscillations  $r=\varphi ^{-\alpha}$ in the phase plane, in polar coordinates.
The relation  between these two objects has been established in \cite{cswavy},
introducing a new notion of the wavy spiral. Applications include two directions.
Roughly speaking, we consider spirals generated by chirps, and chirps generated
by spirals. If we know behavior in the phase plane, we can obtain the behavior
of the corresponding graph, and vice versa.
As examples we may consider weak foci of planar autonomous systems,
including the Li\'enard equation, because all these singularities are of
spiral power type $r=\varphi ^{-\alpha}$, $\alpha\in(0,1)$, see \cite{chirp,zuzu}.
As an application of the converse direction, from a chirp to the spiral,
we were looking for the second-order equation with an oscillatory solution
having chirp-like behavior. The Bessel equation of order $\nu$ is a nice
example of similar behavior, see \cite{bessel}. Whereas the Bessel equation
is a second-order nonautonomous equation, we interpret the equation as a
system in $\mathbb{R}^3$, using $t\to\infty$, instead of the standard
approach with a variable near the origin. The system studied in this article,
see \eqref{alfap}, coincides with the Bessel system for $p(t)=t^{-\alpha}$,
$\alpha=\nu=1/2$, and $q(t)=t$. We classify trajectories of the system
with respect to their geometrical and fractal properties.

Why we study functions which behave like $X(\tau)=\tau^{\alpha}\sin1/\tau$,
and $r=\varphi ^{-\alpha}$ in polar coordinates?
Our starting point is Tricot's book which gives us formulas for box
dimension of $X(\tau)=\tau^{\alpha}\sin1/\tau^\beta$, $0<\alpha<\beta<1$,
and $r=\varphi ^{-\alpha}$, $0<\alpha<1$. For other parameters $\alpha$, $\beta$
these curves are rectifiable. We wanted to analyze power spirals which have
same asymptotics of the  Poincar\'e map in all directions.
Poincar\'e map which corresponds to weak focus is analytic, and limit cycles
bifurcate in the classical Hopf bifurcation. Poincar\'e  maps near general foci,
nilpotent or degenerate, as well as near polycycles are not analytic and the
logarithmic terms show up in the asymptotic expansion,
see Medvedeva \cite{medvedeva} and Roussarie \cite{16th-hilbert}.
In that case Poincar\'e  map has different asymptotics, showing characteristic
directions,  see Han, Romanovski  \cite{hanrom}.
Nilpotent focus has two different asymptotics, so we can relate that focus
with two chirps with different asymptotics.
Here, we study foci related to one chirp. Why we have chirps with $\beta=1$?
For $\alpha+1\le\beta$ we have curves which do not accumulate in the origin,
while for $\alpha+1>\beta$, if $\beta\ne1$ it is easy to see that the spiral
converges to zero in ``oscillating'' way. Wavy spirals appear in that situation,
see \cite{bessel}, \cite{cswavy}. Curves which are spirals with self intersections
like springs, could be defined by oscillatory integrals, so they appear as a
generalization of the clothoid defined by Fresnel integrals.
Asymptotics of the oscillatory integrals, which are related to singularity theory,
could be found in Arnold \cite{arnold-vol2}. Their fractal analysis is our work
in progress.
Furthermore, fixing $\beta=1$ we achieve the whole interval of nonrectifiability
both for spirals and corresponding chirps.

Also, the results about spiral trajectories in $\mathbb{R}^3$,
from \v{Z}ubrini\'c and \v{Z}upanovi\'c  \cite{cras,bilip} are extended
to the systems where some kind of Hopf bifurcation occurs.
The box dimension of a trajectory of the reduced normal form with one zero
eigenvalue, and a pair of pure imaginary eigenvalues, has been computed at
the moment of the birth of a limit cycle. Essentially, the Hopf bifurcation
studied here is a planar bifurcation, but the third equation affects the
box dimension of the corresponding trajectory in the space. We show that in
$3$-dimensional space, a limit cycle bifurcates with the box dimension of a
spiral trajectory larger than $4/3$, which is the value of the standard
planar Hopf bifurcation.

Our intention is to understand a fractal connection between oscillatority
of solutions of differential equations and oscillatority of their trajectories
in the phase space. Our work is mostly motivated by two nice formulas from
the monograph of C. Tricot \cite[p.\ 121]{tricot}. He computed the box dimension
for a class of chirps and for a class of spirals of power type in polar coordinates.
We are looking for a model to apply these formulas, and also to show that chirps
and spirals are a different manifestations of the same phenomenon.
Here we study, as a model a class of second-order nonautonomous equations,
exhibiting both chirp and spiral behavior
\begin{equation}\label{eqalfap}
\ddot x - \Big[\frac{2\,p'(t)}{p(t)} + \frac{q''(t)}{q'(t)}\Big]\dot x
+ \Big[q'^2(t) + \frac{2\,p'^2(t)}{p^2(t)} - \frac{p''(t)}{p(t)}
 + \frac{p'(t)q''(t)}{p(t)q'(t)}\Big]x = 0,
\end{equation}
for $t\in [t_0,\infty)$, $t_0>0$, where $p$ and $q$ are functions of class $C^2$.
The explicit solution is $x(t)=C_1p(t)\sin q(t) +C_2p(t)\cos q(t) $,
which is a chirp-like function. If $z=(\gamma/(t-C_3))^{\gamma}$, $\gamma>0$,
we obtain the cubic system
\begin{equation} \label{alfap}
\begin{aligned}
\dot x&= y   \\
\dot y&= -U(z) x +V(z) y  \\
\dot z&= -z^{\delta},\quad  z\in (0,z_0] ,
\end{aligned}
\end{equation}
where $\delta:=(\gamma+1)/\gamma > 1$ and
\begin{align*}
U(z) &:= q'^2(\gamma z^{-1/\gamma})
 + \frac{2\,p'^2(\gamma z^{-1/\gamma})}{p^2(\gamma z^{-1/\gamma})}
 - \frac{p''(\gamma z^{-1/\gamma})}{p(\gamma z^{-1/\gamma})}
+ \frac{p'(\gamma z^{-1/\gamma})q''(\gamma z^{-1/\gamma})}{p(\gamma z^{-1/\gamma})
 q'(\gamma z^{-1/\gamma})} , 
\\
V(z)& := \frac{2\,p'(\gamma z^{-1/\gamma})}{p(\gamma z^{-1/\gamma})}
 + \frac{q''(\gamma z^{-1/\gamma})}{q'(\gamma z^{-1/\gamma})} .
\end{align*}
It has a spiral trajectory in $\mathbb{R}^3$. In the special case $\gamma=1$,
 we obtain $z=1/(t-C_3)$ and $\delta=2$.

In this article we compute the box dimension of a spiral trajectory of the
system \eqref{alfap} exploiting the dimension of $(\alpha,1)$-chirp
$X(\tau)=\tau^{\alpha}\sin 1/\tau$, $\alpha\in(0,1)$, for $\tau>0$ small,
and also the dimension of the wavy spiral, see \cite{cswavy}.
Using a change of variable for time variable $\tau\mapsto\tau^{-1}$,
the  infinity is mapped to the origin, and such reflected  solution
of \eqref{eqalfap} with respect to time is called  the {\it reflected solution}.
We use notation $t=\tau^{-1}$.
If function $p(t)$ in \eqref{eqalfap}  is ``similar'' to $t^{-\alpha}$,
and function $q(t)$ is ``similar'' to $t$, then the reflected solution of $x(t)$
\begin{align*}
X(\tau)
&=C_1p(\frac{1}{\tau})\sin q(\tau^{-1})+ C_2p(\frac{1}{\tau})\cos q(\tau^{-1})  \\
&=\sqrt{C_1^2+C_2^2}\ p(\frac{1}{\tau})\sin(q(\tau^{-1})
+\arctan\big({\frac{C_2}{C_1}}\big), \quad \tau\in(0,\frac{1}{t_0}],
\end{align*}
is an $(\alpha,1)$-chirp-like function near the origin, see \cite{lukamaja}.
Before we obtained results connecting functions ``similar'' to $(\alpha,1)$-chirps,
and  spirals ``similar'' to $r=\varphi ^{-\alpha}$, $\alpha\in(0,1)$, in the phase plane,
see \cite{cswavy}. Applications include nonautonomous planar systems,
so here we introduce the third variable $z$ depending on the time $t$.
Furthermore, the box dimension of a  trajectory depends on $\gamma>0$.
For some values of $\gamma$ trajectory in $\mathbb{R} ^3$ is obtained
as bi-Lipschitzian image of the spiral from the phase plane, which does not
affect the box dimension. For other values, trajectory lies in the H\"olderian
surface, affecting the box dimension. The H\"olderian surface has an
infinite derivative at the origin, which is the point of accumulation of the spiral.
Spirals of the H\"olderian type have the ``tornado shape'' with  a small
bottom and wide top.

It is interesting to notice that our results  about the box dimension of
planar trajectories of a system with pure imaginary eigenvalues,
show that the box dimension of any trajectory depends on the exponents
of the system. In $\mathbb{R} ^3$ we have already found an example,
see \cite{cras,bilip}, where dimension depends on the coefficients
of the systems, which will be the case in \eqref{alfap}.
See \cite{bilip} for the computation of the box dimension of  the system
\begin{equation}\label{spp}
\dot r=a_1rz,\quad
\dot \varphi=1,\quad
\dot z=b_2z^2,
\end{equation}
in cylindrical coordinates. If $a_1/b_2\in(0,1]$ then any spiral
trajectory $\Gamma$ of \eqref{spp} has the box dimension
$\dim_B\Gamma=\frac{2}{1+a_1/b_2}$ near the origin.

\section{Definitions}

Let us introduce some definitions and notation.
For $A\subset\mathbb{R}^N$  bounded we define
 {\it{  $\varepsilon$-neighborhood}} of  $A$ as:
$A_\varepsilon:=\{y\in\mathbb{R}^N\:d(y,A)<\varepsilon\}$.
By {\it lower $s$-dimensional  Minkowski content} of $A$, $s\ge0$ we mean
$$
\mathcal{M}_*^s(A):=\liminf_{\varepsilon\to0}\frac{|A_\varepsilon|}{\varepsilon^{N-s}},
$$
and analogously for the {\it upper $s$-dimensional Minkowski content} $\mathcal{M}^{*s}(A)$.
The lower and upper box dimensions of $A$ are
$$
\underline\dim_BA:=\inf\{s\ge0\:\mathcal{M}_*^s(A)=0\}
$$
 and analogously
$\overline\dim_BA:=\inf\{s\ge0\:\mathcal{M}^{*s}(A)=0\}$.
If these two values coincide, we call it simply the box dimension of $A$,
and denote by $\dim_BA$. It will be our situation.
If $0<\mathcal{M}_*^d(A)\le\mathcal{M}^{*d}(A)<\infty$ for some $d$, then we say
 that $A$ is {\it Minkowski nondegenerate}. In this case obviously $d=\dim_BA$.
In the case when lower or upper $d$-dimensional Minkowski contents of $A$
are $0$ or $\infty$, where $d=\dim_BA$,  we say that $A$ is {\it degenerate}.
For more details on these definitions see e.g.\ Falconer \cite{falc},
and \cite{zuzu}.

Let $x:[t_0,\infty)\to\mathbb{R}$, $t_0>0$, be a continuous function.
We say that $x$ is an {\it oscillatory function}  near $t=\infty$
if there exists a sequence $t_k\searrow \infty$
such that $x(t_k)=0$, and functions $x|_{(t_k,t_{k+1})}$ intermittently
change sign for $k\in \mathbb{N}$.

Let  $u:(0,t_0]\to\mathbb{R}$, $t_0>0$, be a continuous function.
We say that $u$ is an {\it oscillatory function} near the origin if
there exists a sequence $s_k$ such that $s_k\searrow 0$ as
$k\to \infty $, $u(s_k)=0$ and restrictions $u|_{(s_{k+1},s_k)}$
intermittently change sign, $k\in \mathbb{N}$.

Let us define $X:(0,1/t_0]\to\mathbb{R}$ by $X(\tau)=x(1/\tau)$.
 We say that $X(\tau)$ is oscillatory near the origin if $x=x(t)$
is oscillatory near $t=\infty$.
We measure the rate of oscillatority of $x(t)$ near $t=\infty$ by the rate
of oscillatority of $X(\tau)$ near $\tau=0$.
More precisely, the {\it oscillatory dimension} $\dim_{osc}(x)$ (near $t=\infty$)
is defined as the box dimension of the graph of $X(\tau)$ near $\tau=0$.
In Radunovi\'c, \v{Z}ubrini\'c and \v{Z}upanovi\'c \cite{RaZuZu} box
dimension of unbounded sets has been studied.

Assume now that $x$ is of class $C^1$. We say that $x$ is a
{\it phase oscillatory} function if the following stronger condition holds: the set
$\Gamma=\{(x(t),\dot x(t)):t\in[t_0,\infty)\}$ in the plane is a spiral
converging to the origin.

By a spiral here we mean the graph of a function $r=f(\varphi)$,
$\varphi\geq\varphi_1>0$, in polar coordinates, where
\begin{equation} \label{def_spiral}
\parbox{10cm}{
$f:[\varphi_1,\infty)\to(0,\infty)$  is such that
$f(\varphi)\to 0$  as $\varphi\to\infty$,
$f$ is radially decreasing (i.e., for any fixed $\varphi\geq\varphi_1$
the function  $\mathbb{N}\ni k\mapsto f(\varphi+2k\pi)$  is decreasing),
}
\end{equation}
which is the  definition from \cite{zuzu}. By a spiral we also mean a mirror
image of the spiral \eqref{def_spiral}, with respect to the $x$-axis.

The phase dimension $\dim_{\rm ph}(x)$ of the function $x(t)$
is defined as the box dimension of the corresponding spiral
$\Gamma=\{(x(t),\dot x(t)):t\in[t_0,\infty)\}$.

We use a result for box dimension of  graph $G(X)$ of standard
 $(\alpha,\beta)$-\emph{chirps} defined by
\begin{equation}\label{standard_chirp}
X_{\alpha,\beta}(\tau)=\tau^{\alpha}\sin(\tau^{-\beta}) .
\end{equation}
For $0<\alpha<\beta$ we have
\begin{equation}\label{dim}
\dim_B G(X_{\alpha,\beta})=2-(\alpha+1)/(\beta+1) ,
\end{equation}
 and the same for $X_{\alpha,\beta}(\tau)=\tau^{\alpha}\cos(\tau^{-\beta})$,
see Tricot  \cite[p.\ 121]{tricot}.
Also we use a result for box dimension of spiral $\Gamma$
defined by $r=\varphi^{-\alpha}$, $\varphi\ge\varphi_0>0$,  $\dim_B\Gamma=2/(1+\alpha)$ when
 $0<\alpha\le1$, see Tricot \cite[p.\ 121]{tricot} and some generalizations
from \cite{zuzu}.
Oscillatory and phase dimensions are fractal dimensions, which are well known
tool in the study of dynamics, see survey article \cite{fdd}.

For two real functions $f(t)$ and $g(t)$ of real variable
we write $f(t)\simeq g(t)$, and say that functions are {\it comparable}
as $t\to0$ (as $t\to\infty$),
if there exist positive constants $C$ and $D$ such that
$C\,f(t)\le g(t)\le D\,f(t)$ for all $t$ sufficiently close to $t=0$
(for all $t$ sufficiently large). For example, for a function $F:U\to V$
with $U,V\subset\mathbb{R}^2$, $V=F(U)$, the condition $|F(t_1)-F(t_2)|\simeq
|t_1-t_2|$ means that $f$ is a bi-Lipschitz mapping, i.e.,\ both $F$ and
$F^{-1}$ are Lipschitzian.

We say that {\it function $f$ is comparable of class $k$ to  power
$t^{-\alpha}$} if $f$ is class $C^k$ function, and
$f^{(j)}(t)\simeq t^{-\alpha-j}$ as $t\to\infty$, $j=0,1,2,\dots, k$.

Also, we write $f(t)\sim g(t)$ if $f(t)/g(t)\to1$ as $t\to\infty$, and say
that {\it function $f$ is comparable of class $k$ to  power
$t^{-\alpha}$ in the limit sense}  if $f$ is class $C^k$ function,
$f(t)\sim t^{-\alpha}$ and
$f^{(j)}(t)\sim (-1)^j\alpha(\alpha+1)(\alpha+j-1) t^{-\alpha-j}$ as
$t\to\infty$, $j=1,2,\dots, k$.

We write $f(t)=O(g(t))$ as $t\to0$ (as $t\to\infty$) if there exists positive
constant $C$ such that $|f(t)|\leq C|g(t)|$. We write $f(t)=o(g(t))$ as
$t\to\infty$ if for every positive constant $\varepsilon$ it holds
$|f(t)|\leq \varepsilon|g(t)|$ for all $t$ sufficiently large.

In the sequel we shall consider the functions of the form
$y=p(\tau)\sin(q(\tau))$ or $y=p(\tau)\cos(q(\tau))$.
If $p(\tau)\simeq \tau^\alpha$, $q(\tau)\simeq \tau^{-\beta}$,
$q'(\tau)\simeq \tau^{-\beta-1}$ as $\tau\to 0$ then we say that $y$ is
an $(\alpha,\beta)$-{\it chirp-like function}.


\section{Spiral trajectories in $\mathbb{R}^3$}

In this section we describe solutions of equation \eqref{eqalfap} and
trajectories of system \eqref{alfap}, with respect to box dimension,
specifying a class of functions $p$ and $q$. Let $p(t)$ be comparable
to power $t^{-\alpha}$, $\alpha>0$, in the limit sense, and let $q(t)$
be comparable to $K t$, $K>0$, in the limit sense.
Depending on ${\alpha}$, we have rectifiable spirals with trivial box
dimension equal to $1$, or nonrectifiable spirals with nontrivial box
dimension greater than $1$.
The box dimension will not exceed $2$ even in $\mathbb{R}^3$, because
these spirals lie on a surface. Mapping spiral from the plane to
the Lipschitzian surface does not affect the box dimension,
see \cite{cras,bilip}, while mapping to H\"olderian surface affects
the box dimension.


To explain fractal behavior of the system \eqref{alfap} we need a lemma
dealing with a bi-Lipschitz map. The idea is to use a generalization
of the result about box dimension of a class of planar spirals
from \cite[Theorem 4]{cswavy}, see Theorem \ref{gen_wavy} below.
It is well known result from \cite{falc}, that box dimension is preserved
by bi-Lipschitz map. Putting together these two results we will obtain
desired results about \eqref{alfap}.
For the sake of simplicity, we deal with trajectory $\Gamma$ of the solution
of the system \eqref{alfap}  defined by
\begin{equation}\label{spiral}
\begin{aligned}
 x(t)&=p(t) \sin q(t)    \\
 y(t)&= p'(t)\sin q(t) +p(t)q'(t)\cos q(t)   \\
 z(t)&=1/t^{\gamma}\,.
\end{aligned}
\end{equation}
We can assume, without the loss of generality, that $q(t)$ is comparable
to $t$, in the limit sense, by contracting time variable $t$ by factor
$K$ and also contracting $x$ by factor $K^{\alpha}$, $y$ by factor $K^{\alpha+1}$,
and $z$ by factor $K^{\gamma}$. Notice that rescaling of spatial variables
by a constant factor is a bi-Lipschitz map, so the box dimension of
trajectory $\Gamma$ is preserved.

Trajectory  $\Gamma$  has projection $\Gamma_{xy}$ to $(x,y)$-plane which
is a planar spiral satisfying conditions of Theorem \ref{gen_wavy} below.
In the following lemma we will prove that the mapping
between planar spiral $\Gamma_{xy}$ and spacial spiral $\Gamma$ is
bi-Lipschitzian near the origin. We prove lemma using definition of
bi-Lipschitz mapping. Interesting phenomenon appeared in spiral $\Gamma_{xy}$,
defined in polar coordinates and generated by a chirp.
The radius $r(\varphi)$  is not decreasing function, there are some regions
where $r(\varphi)$  increases causing some waves on the spiral.
We introduced notion of wavy spiral in \cite{cswavy}. Also, the waves are
found in the spiral generated by Bessel functions, and  by generalized
Bessel functions, depending on the parameters in the equation, see \cite{bessel}.
 Furthermore, the surface containing the space spiral $\Gamma$ contains
 points with infinite derivative, showing  some vertical  regions.

\begin{lemma}\label{Domagoj} 
Let the map $B:\mathbb{R}^2\times\{0\}\to \mathbb{R}^3$ be defined as
$B(x(t),y(t),0)=(x(t),y(t),z(t))$, where coordinate functions are given by
\eqref{spiral}. Let $p(t)\in C^2$ is comparable of class $1$  to $t^{-\alpha}$,
$\alpha\in(0,1)$, in the limit sense, and $p''(t)\in o(t^{-\alpha})$, as
$t\to\infty$. Let $q(t)\in C^2$ is comparable of class $1$  to $K t$, $K>0$,
in the limit sense, and $q''(t)\in o(t^{-2})$, as $t\to\infty$.
Let $\Gamma$ is defined by parametrization $(x(t),y(t),z(t))$ from
\eqref{spiral} and $\Gamma_{xy}$ is the projection of $\Gamma$ to $(x,y)$-plane.
If $\gamma\geq\alpha$ then map $B|_{\Gamma_{xy}}$ is bi-Lipschitzian near the origin.
\end{lemma}

\begin{proof}
Without the loss of generality we assume $K=1$.
It is clear that $B(\Gamma_{xy})=\Gamma$. We have to prove that there exist
two positive constants $K_1,K_2$ such that
\begin{equation}\label{bil}
\begin{aligned}
&K_1 d( (x(t_1),y(t_1),0), (x(t_2),y(t_2),0))\\
&\leq d( B(x(t_1),y(t_1),0), B(x(t_2),y(t_2),0))\\
&\leq K_2 d( (x(t_1),y(t_1),0), (x(t_2),y(t_2),0))  ,
\end{aligned}
\end{equation}
where $d$ is Euclidian metrics and $t_1,t_2>t_0$, for $t_0$ sufficiently large.
Notice that, without loss of generality, $t_1\leq t_2$.
It is obvious that by $K_1=1$ the left hand side inequality is satisfied.
To prove right hand side inequality, first  we prove
\[
(z(t_1)-z(t_2))^2\le C\left((x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2\right),
\]
and then the right inequality will be satisfied. From the proof of the planar
case, Theorem \ref{gen_wavy}, we know that
\begin{gather}\label{fi}
\varphi (t) =t+\frac{\pi}2 + O(t^{-1})
=\frac{\pi}2+t\left(1+O\left(t^{-2}\right)\right), \quad t\to \infty, \\
\label{fialfa}
f(\varphi)\simeq \varphi ^{-\alpha}, \quad \varphi\to \infty .
\end{gather}
From the generalization of \cite[Lemma 3]{cswavy} used in the proof of
Theorem \ref{gen_wavy}, using assumptions on $p$ and $q$, it follows
that there exists $C_1\in (0,1)$, such that for every $\Delta\varphi$,
$\frac{\pi}{3}\le \Delta\varphi\le 2\pi+\frac{\pi}{3}$, holds
$$
f(\varphi)-f(\varphi + \Delta\varphi )\ge  \Delta\varphi \alpha C_1\varphi ^{-\alpha-1},
$$
for $\varphi$ sufficiently large. Let
\begin{equation}\label{fi12}
\begin{gathered}
\varphi_1=\varphi(t_1)=\pi/2 +t_1\left(1+O\left(t_1^{-2}\right)\right),\\
\varphi_2=\varphi(t_2)=\pi/2 +t_2\left(1+O\left(t_2^{-2}\right)\right).
\end{gathered}
\end{equation}


We first consider several cases where $\alpha\leq\gamma<1$.
First, let $|\varphi_2-\varphi_1|\le \frac{\pi}3$.
From \cite[Proposition 1]{bessel}, and \eqref{fialfa}, \eqref{fi12}, we have
\begin{align*}
&\sqrt{(x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2}\\
&\ge \frac{2}{\pi}(\varphi_2-\varphi_1)\min\{f(\varphi_1),f(\varphi_2)\}
\ge   C_2(t_2-t_1)t_2^{-\alpha}.
\end{align*}
Hence,
\begin{equation}\label{b}
\begin{aligned}
(z(t_1)-z(t_2))^2
&=\Big(\frac1{t_1^{\gamma}}-\frac1{t_2^{\gamma}}\Big)^2
 =\frac{(t_2^{\gamma}-t_1^{\gamma})^2}{t_1^{2\gamma}t_2^{2\gamma}}\\
&\leq \frac{(t_2-t_1)^2(t_2^{\gamma-1}+t_1^{\gamma-1})^2}{t_1^{2\gamma}t_2^{2\gamma}} \\
&\leq c_2\frac{(t_2-t_1)^2 t_1^{2(\gamma-1)}}{t_1^{2\gamma}t_2^{2\gamma}}
 = c_2\frac{(t_2-t_1)^2}{t_1^2t_2^{2\gamma}}
 \frac{C_2^2t_2^{-2\alpha}}{C_2^2t_2^{-2\alpha}}\\
&\leq  c_2\frac{(x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2}
 { C_2^2 t_1^2t_2^{2(\gamma-\alpha)}}\\
&\leq C\left((x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2\right).
\end{aligned}
\end{equation}
For the case $2\pi + \frac{\pi}3\ge |\varphi_2-\varphi_1|\ge \frac{\pi}3$, we have
\begin{align*}
&\sqrt{ (x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2 }\\
&\ge f(\varphi_1)-f(\varphi_2)\\
&=  f(\varphi_1)-f(\varphi_1+(\varphi_2-\varphi_1))\ge C_1 ( \varphi_2-\varphi_1)\alpha{\varphi_1}^{-\alpha -1}\\
&\ge  C_3 t_1^{-\alpha -1}(t_2-t_1).
\end{align*}
Then again
\begin{equation}\label{c}
\begin{aligned}
(z(t_1)-z(t_2))^2
&=c_3\frac{(t_2-t_1)^2}{t_1^2t_2^{2\gamma}}
 \le c_3\frac{ (x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2 }{C_3^2 t_1^{2(\gamma-\alpha)}}\\
&\leq C \left( (x(t_1)-x(t_2))^2+((y(t_1)-y(t_2))^2 \right) .
\end{aligned}
\end{equation}
For the case $|\varphi_2-\varphi_1|\ge 2\pi + \frac{\pi}3$, we define
 $n:=[\frac{\varphi_2-\varphi_1- \frac{\pi}3}{2\pi}]$. Then we have
\begin{align*}
&\sqrt{(x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2 }\ge f(\varphi_1)-f(\varphi_2)\\
&= \sum_{i=0}^{n-1}\left(f(\varphi_1+2i\pi)- f(\varphi_1+(i+1)2\pi)\right)
 + f(\varphi_1+2n\pi)-f(\varphi_2)\\
&\ge  \sum_{i=0}^{n-1}2\pi\alpha C_1(\varphi_1+2i\pi)^{-\alpha-1}
 +\frac{\pi}3\alpha C_1(\varphi_1+2n\pi)^{-\alpha-1}\\
&\ge \frac{\pi}3\alpha C_1\sum_{i=0}^{n}(\varphi_1+2i\pi)^{-\alpha-1}
 = \frac{\pi}3\alpha C_1\sum_{i=0}^{n}{(2\pi)}^{ -\alpha-1}
 (\frac{\varphi_1}{2\pi}+i)^{ -\alpha-1} \\
&\ge  \frac{\pi}3\alpha C_1{2\pi}^{ -\alpha-1}
 \int_{\frac{\varphi_1}{2\pi}}^{\frac{\varphi_1}{2\pi}+n-1}x^{ -\alpha-1}dx\\
&\ge C_4\varphi_1^{-\alpha-1}(\varphi_2-\varphi_1)
 \ge  C_5 t_1^{-\alpha-1}(t_2-t_1).
\end{align*}
Furthermore
\begin{equation}\label{d}
\begin{aligned}
(z(t_1)-z(t_2))^2&=c_5\frac{(t_2-t_1)^2}{t_1^2t_2^{2\gamma}}
 \le c_5\frac{ (x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2 }{C_5^2 t_1^{2(\gamma-\alpha)}}\\
& \le  C\left((x(t_1)-x(t_2))^2+(y(t_1)-y(t_2))^2\right) .
\end{aligned}
\end{equation}
From \eqref{b}, \eqref{c}, \eqref{d}, the right hand side of inequality
\eqref{bil} follows with $K_2=\sqrt{1+C}$, where $C$ is (in all three cases)
sufficiently small if $t_0$ is large enough.

For $t_0$ sufficiently large, it is easy to see that
$$
(z(t_1)-z(t_2))^2=\Big(\frac{1}{t_1^{\gamma}}-\frac{1}{t_2^{\gamma}}\Big)^2
\leq \Big(\frac{1}{t_1^2}-\frac{1}{t_2^2}\Big)^2 ,
$$
if $\gamma>2$. On the other hand, for $1\leq\gamma\leq 2$, considering
$$
(z(t_1)-z(t_2))^2
\leq \frac{(t_2-t_1)^2(t_2^{\gamma-1}+t_1^{\gamma-1})^2}{t_1^{2\gamma}t_2^{2\gamma}}
\leq c_6\frac{(t_2-t_1)^2 t_2^{2(\gamma-1)}}{t_1^{2\gamma}t_2^{2\gamma}}
= c_6\frac{(t_2-t_1)^2}{t_1^{2\gamma}t_2^2} ,
$$
the rest of the proof is analogous as for the case $\alpha\leq\gamma<1$.
\end{proof}


\begin{theorem}[Trajectory in $\mathbb{R}^3$] \label{phase}
 Let $p(t)\in C^3$ be a function  comparable of class $2$ to  power $t^{-\alpha}$,
$\alpha>0$, in the limit sense, and $p^{(3)}(t)\in O(t^{-\alpha-3})$, as
$t\to\infty$. Let $q(t)\in C^3$ be a function comparable of class $1$ to
$K t$, $K>0$ in the limit sense, $q''(t)\in o(t^{-3})$, as $t\to\infty$, and
$q^{(3)}(t)\in o(t^{-2})$, as $t\to\infty$.
\begin{itemize}

\item[(i)] The phase portrait
$\Gamma_{xy}=\{(x(t),\dot x(t))\in\mathbb{R}^2: t\in[t_0,\infty)\}$
of any solution is a spiral near the origin. Phase dimension of any solution
of the equation \eqref{eqalfap} is equal to $\dim_{\rm ph}(x)=\frac 2{1+\alpha}$,
for $\alpha\in(0,1)$.

\item[(ii)]  The trajectory $\Gamma $ of the system \eqref{alfap}
 has box dimension $\dim_B\Gamma =\frac 2{1+\alpha}$ for $\alpha\in(0,1)$ and
$\gamma\geq\alpha$.

\item[(iii)]  Trajectory $\Gamma $ of the system \eqref{alfap} has box dimension
$\dim_B\Gamma =2-\frac{\alpha +\gamma}{1+\gamma}$ for $\alpha\in(0,1)$ and
$0<\gamma<\alpha$.

\item[(iv)] The trajectory  $\Gamma $ of the system \eqref{alfap}
 for $\alpha >1$ is rectifiable and  $\dim_B\Gamma =1$.
\end{itemize}
\end{theorem}

The graphs of trajectories \eqref{spiral} for different values of parameter
$\alpha$ can be seen in Figures \ref{fig1}--\ref{fig3}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1} % tri1_4.pdf
\end{center}
\caption{System \eqref{alfap} for $p(t)=t^{-\frac14}$, $q(t)=t$ and
$\gamma=1$, Lipschitz case.} \label{fig1}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig2} % tri1_1.pdf
\end{center}
\caption{System \eqref{alfap} for $p(t)=t^{-1}$, $q(t)=t$ and
$\gamma=1$, Lipschitz case.} \label{fig2}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig3} % tri3_1.pdf
\end{center}
\caption{System \eqref{alfap} for $p(t)=t^{-3}$, $q(t)=t$
and $\gamma=1$, H\" older case.} \label{fig3}
\end{figure}

\begin{remark} \label{rmk1} \rm
In Theorem \ref{phase} the box dimension of the spiral  $\Gamma$
 has been computed,  and all values satisfy
$\frac{2}{1+\alpha} \leq \dim_B\Gamma< 2-\alpha$ for $\alpha\in (0,1)$.
\end{remark}


\begin{remark} \label{rmk2} \rm
Regarding rectifiability of trajectory  $\Gamma $ of system \eqref{alfap}
 from Theorem \ref{phase}, the assumptions on functions $p$ and $q$ could
be weakened. For instance, for
\begin{align*}
p_1(t)&=t^{-\alpha}\log^k(t)  \\
p_2(t)&=t^{-\alpha}\log(\log(\ldots\log(t))),\quad k \text{ times}  \\
q_1(t)&=t\log^l(t)  \\
q_2(t)&=t\log(\log(\ldots\log(t))),\quad l \text{ times}
\end{align*}
where $\alpha>1$ and $k,l\in\mathbb{N}$ if we take $x(t)=p_i(t)\sin(q_j(t))$,
$i,j=1,2$, it is easy to see that curve $\Gamma$ is also rectifiable.
If $\alpha\le1$ we expect nonrectifiability and the same box dimension as
in the case with no logarithmic terms. This comes from  \cite[Remark 9]{zuzu},
saying that  spirals $r=\varphi^{-\alpha}(\log\varphi)^{\beta}$, $\varphi\ge\varphi_1$, where $\beta\ne0$
and $\alpha\in(0,1)$ have box dimension equal to $d:=2/(1+\alpha)$ (the same as
for the spiral $r=\varphi^{-\alpha}$), but their $d$-dimensional Minkowski content is
degenerate. See that degeneracy at Figures \ref{fig4}--\ref{fig6}.

We did not prove that all our statements are valid for $p_i, q_i$, $i=1,2$
 with logarithmic terms, because in order to do it, we would have to extend
theorems from \cite{zuzu} for that cases,  making this article too long.
On the other hand, from the dynamical point of view, spirals
$r=\varphi^{-\alpha}(\log\varphi)^{\beta}$ are not trajectories of vector fields.
The Poincar\'e maps or first return maps of foci, which are not weak,
have logarithmic terms in the asymptotic expansion. Asymptotics is different
in the characteristic directions. These directions could be seen after blowing
up when polycycle appears from the focus. The directions with different
asymptotic pass through singularities of the polycycle. The logarithmic terms
are produced by singularities of the polycycle.
Spiral  $r=\varphi^{-\alpha}(\log\varphi)^{\beta}$ has the same asymptotics in all directions,
 which is a different situation.
\end{remark}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig4} %  power-spiral.pdf
\end{center}
\caption{Spiral $r=\varphi^{-1/2}$, in polar coordinates}. \label{fig4}
\end{figure}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig5} %  power-log-spiral.pdf
\end{center}
\caption{Spiral $r=\varphi^{-1/2}\log\varphi$, in polar coordinates}.
\label{fig5}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\textwidth]{fig6} %  power-log2-spiral.pdf
\end{center}
\caption{Spiral $r=\varphi^{-1/2}\log^2 \varphi$, in polar coordinates}.
\label{fig6}
\end{figure}

\begin{remark} \label{rmk3}  \rm
In the Introduction, it was briefly explained why we did not
set $q(t)\sim t^{\beta}$ for $\beta\neq 1$. Here we  show figures
concerning those cases.
If $X(\tau)=\tau^{\alpha}\sin1/\tau^\beta$,  for $\alpha+1\le\beta$ using the
described procedure, we have a planar curve which does not accumulate near
the  origin, see Figure \ref{fig7}.

If $\alpha+1>\beta$ and $\beta\ne1$, we have spiral converging to zero 
in ``oscillating'' way, see Figure \ref{fig8}.
Figure \ref{fig9} shows focus with different asymptotic in the direction 
of $x$-axes.
\end{remark}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig7} %  deg-spiral1.pdf
\end{center}
\caption{Part of unbounded curve $\Gamma_1=\{(x(t),\dot x(t)): t\in[t_0,\infty)\}$,
 for $x(1/\tau)=X(\tau)=\tau^{1/2}\sin(1/\tau)^{7/4}$, rotated by $\pi/2$ clockwise.}
\label{fig7}
\end{figure}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig8} %  deg-spiral2.pdf
\end{center}
\caption{Spiral $\Gamma_2=\{(x(t),\dot x(t)): t\in[t_0,\infty)\}$, 
for $x(1/\tau)=X(\tau)=\tau^{1/2}\sin(1/\tau)^{3/4}$.}
\label{fig8}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig9} %  nilpot.pdf
\end{center}
\caption{Nilpotent focus with characteristic direction along $x$-axes.}
\label{fig9}
\end{figure}


\begin{remark} \label{rmk4} \rm
System \eqref{alfap} coincides with the Bessel system of order $\nu$ 
for $p(t)=t^{-\alpha}$, $\alpha=\nu=1/2$, and $q(t)=t$. 
The Bessel equation of order $\nu$ has phase dimension equal to $4/3$, 
for the proof see \cite[Corollary 1]{bessel}. This is a consequence of 
a fact that the Bessel functions in some sense ``behave'' like chirps
 $x(t)=t^{-1/2}\sin\left(t+\theta_0\right)$, $\theta_0\in\mathbb{R}$, 
as $t\to\infty$. Although, this background connection is pretty intuitive, 
the proof is long, complex and technically exhausting.
\end{remark}

The following theorem gives sufficient conditions for rectifiability of 
a spiral lying into the  H\"olderian  surface $z=g(r)$, 
$g(r)\simeq r^\beta$, $\beta>0$. Spiral is called H\"older-focus spiral if 
it lies in the H\"olderian  surface, and tend to the origin.

\newpage 

\begin{theorem}[Rectifiability in $\mathbb{R}^3$] \label{rectif}
 Let $f[\varphi_1,\infty)\to(0,\infty)$,
$\varphi_1>0$, $f(\varphi)\simeq \varphi^{-\alpha}$,   $|f'(\varphi)|\le C \varphi^{-\alpha -1}$,  
$\alpha>1$,  $r=f(\varphi) $  define a rectifiable spiral. 
Assume that $g\:(0,f(\varphi_1))\to(0,\infty)$
is a function of class $C^1$ such that
\[
g(r)\simeq r^\beta,\quad |g'(r)|\le D r^{\beta -1}, \quad \beta>0.
\]
 Let $\Gamma$ be a H\"older-focus spiral defined by $r=f(\varphi)$, 
$\varphi\in[\varphi_1,\infty)$, $z=g(r)$, then $\Gamma$ is rectifiable spiral.
\end{theorem}

\begin{proof}
The corresponding parametrization of  spiral $\Gamma$ in Cartesian space 
coordinates is
\begin{equation}\label{system1}
x=f(\varphi)\cos \varphi,\quad
y=f(\varphi)\sin \varphi\quad
z=g(f(\varphi)).
\end{equation}

For the length $l(\Gamma$) of this spiral we have
\begin{align*}
l(\Gamma)
&=\int_{\varphi_1}^{\infty}\sqrt{\dot x^2(\varphi)+\dot y^2(\varphi)+\dot z^2(\varphi)}d\varphi\\
&= \int_{\varphi_1}^{\infty}\sqrt{f^2(\varphi)+f'^2(\varphi)+g'^2(f(\varphi))f'^2(\varphi)}d\varphi\\
&\le C\int_{\varphi_1}^{\infty} \sqrt{\varphi^{-2\alpha}+\varphi^{-2\alpha-2}
 +\varphi^{-\alpha(2\beta-2)}\varphi^{-2\alpha-2}}d\varphi\\
&=C\int_{\varphi_1}^{\infty} \sqrt{\varphi^{-2\alpha}+\varphi^{-2\alpha-2}+\varphi^{-2\alpha\beta-2}}d\varphi.
\end{align*}
If $-2\alpha\beta-2\le -2\alpha$, then
$$ 
l(\Gamma)\le C\int_{\varphi_1}^{\infty}\varphi^{-\alpha}d\varphi\ < \infty,
$$
and if $-2\alpha\beta-2> -2\alpha$, then
$$ 
l(\Gamma)\le C\int_{\varphi_1}^{\infty}\varphi^{-\alpha\beta-1}d\varphi < \infty.
$$
\end{proof}


\begin{proof}[Proof of Theorem \ref{phase}]

(i) Without the loss of generality we take solution  $x(t)=p(t)\sin q(t)$ 
of the equation \eqref{eqalfap}. Spiral trajectory $\Gamma$ of system 
\eqref{alfap} is defined by \eqref{spiral}. Then $\Gamma_{xy}$ is 
the projection of $\Gamma$  in $(x,y)-$plane. Using Theorem \ref{gen_wavy}
below, we obtain that $\Gamma_{xy}$ is a spiral near the origin 
and $\dim_B\Gamma_{xy} = \dim_{\rm ph}(x)=\frac 2{1+\alpha}$.

(ii) The map  $B\:(x(t),y(t),0)\to(x(t),y(t),z(t))$ is a bi-Lipschitz 
map near the origin for $\gamma\geq\alpha$, see Lemma \ref{Domagoj}.
It is clear that $\Gamma=B(\Gamma_{xy})$ and it is easy to see that 
subset $S\subseteq\Gamma$, for which $B$ is not a bi-Lipschitz map, 
is rectifiable and therefore $\dim_{B}S=1$. 
The box dimension of set $\Gamma$ is preserved under bi-Lipschitzian 
mappings and under removing $S\subseteq\Gamma$ such that $\dim_{B}S=1$, 
see \cite[p.\ 44]{falc}, so it follows form \rm{(i)} that 
$\dim_B\Gamma=\frac{2}{1+\alpha}$.

(iii) Without the loss of generality we take $K=1$. The rest of the proof 
is similar as in (ii), but using \cite[Theorem 9]{bilip} instead 
of Lemma \ref{Domagoj}.

(iv) Without the loss of generality we take $K=1$, because rectifiability 
is also unaffected by rescaling of spatial variables. 
Let $r=f(\varphi)$ define curve $\Gamma_{xy}$ in polar coordinates. 
Notice that $f(\varphi)\simeq\varphi^{-\alpha}$ and 
$|f'(\varphi)|\leq C\varphi^{-\alpha-1}$, see the proof of Theorem \ref{gen_wavy} 
from the Appendix. Respecting $r(t)=\sqrt{x(t)^2+\dot{x}(t)^2}$, 
we take $g(r)$ such that $g(r)\simeq z(t)$ and $g'(r)\in O(z'(t))$, 
using $z(t)$ from \eqref{spiral}. As $r(t)\simeq t^{-\alpha}$ and 
$|r'(t)|\leq D_1 t^{-\alpha-1}$, we obtain $g(r)\simeq r^{\gamma/\alpha}$
and $|g'(r)|\leq D r^{\gamma/\alpha-1}$, so using Theorem \ref{rectif} 
and the fact that rectifiability is invariant to bi-Lipschitz mapping, 
as $g(r)\simeq z(t)$, we prove the claim.
\end{proof}

\begin{remark} \label{rmk5} \rm
Notice that in the proof of Theorem \ref{phase} (iii), the H\"older case, 
 we used \cite[Theorem 9]{bilip}, but in the proof of Theorem \ref{phase} (ii), 
the Lipschitz case; we could not use the analogue to \cite[Theorem 7]{bilip} 
and we had to devise Lemma \ref{Domagoj}. 
The reason is  the fact that assumptions in \cite[Theorem 9]{bilip} 
about the spiral $r=f(\varphi)$, in polar coordinates, regarding 
function $f$ being decreasing and $|f'(\varphi)|\simeq\varphi^{-\alpha-1}$, 
as $t\to\infty$, can be replaced by weaker assumptions. 
By carefully examining the proof, we see that function $f$ does not have 
to be decreasing and we can take $|f'(\varphi)|\in O(\varphi^{-\alpha-1})$, 
as $t\to\infty$. Regardlessly, these assumptions are necessary 
in \cite[Theorem 7]{bilip}.
\end{remark}

It is interesting to study the Poincar\'e or the first return map 
associated to a spiral trajectory. The following result is about asymptotics 
of the Poincar\'e map near focus of the planar spiral from Theorem \ref{phase} (i).


\begin{proposition}[Poincar\'e map] \label{prop1}
 Assume $\Gamma$ is the planar spiral from Theorem \ref{phase}~(i). 
Let $P:(0,\varepsilon)\cap\Gamma\to(0,\varepsilon)\cap\Gamma$ 
be the Poincar\'e map with respect to any axis that passes through the origin.
Then the map $P$ has the form $P(r)=r+d(r)$, where 
$-d(r)\simeq r^{\frac{1}{\alpha}+1}$ as $r\to 0$.
\end{proposition}

\begin{proof}
Let $\Gamma$ be defined by $r=f(\varphi)$. Analogously as in the proof of 
Theorem \ref{gen_wavy}, see \cite[Theorem 4]{cswavy}, it is easy to see 
that $-d(r)=f(\varphi)-f(\varphi+2\pi)\simeq \varphi^{-\alpha-1}$  as 
$\varphi\to\infty$ and $r\simeq\varphi^{-\alpha}$ as $\varphi\to\infty$.
 From this follows $-d(r)\simeq r^{\frac{1}{\alpha}+1}$ as $r\to 0$.
\end{proof}

The projection of a solution of system \eqref{alfap} is a spiral in 
$(x,y)$-plane. For other two coordinate planes we have the following theorem.

\begin{theorem}[Projections] \label{projection}
 Let $p(t)\in C^2$ be a function  comparable of class $1$ to  power 
$t^{-\alpha}$, $\alpha>0$, and $p''(t)\in O(t^{-\alpha})$, as $t\to\infty$. 
Let $q(t)\in C^2$ be a function comparable of class $1$ to $K t$, $K>0$, 
and $q''(t)\in O(t^{-1})$, as $t\to\infty$.

If $\alpha\in(0,1)$ then projections $G_{xz}$ and $G_{yz}$ of a 
trajectory \eqref{spiral}, $\gamma>0$ of the system 
\eqref{alfap} to $(x,z)-$plane and $(y,z)$-plane, respectively, are 
$(\alpha/\gamma,1/\gamma)$-chirp-like functions, and  
$\dim_B G_{xz}=\dim_B G_{yz}=2-\frac{\alpha+\gamma}{1+\gamma}$.
\end{theorem}

\begin{proof}
Without the loss of generality let $K=1$. Projection $G_{yz}$ is
\begin{align*}
Y(z)
&=y\big(z^{-1/\gamma}\big)\\
&=p'\big(z^{-1/\gamma}\big)\sin q\big(z^{-1/\gamma}\big)
+ p\big(z^{-1/\gamma}\big)q'\big(z^{-1/\gamma}\big)
\cos q\big(z^{-1/\gamma}\big) \\
&=\sqrt{p'^2\big(z^{-1/\gamma}\big)+p^2\big(z^{-1/\gamma}\big)q'^2
 \big(z^{-1/\gamma}\big)}  \\
&\quad\times \sin\Big(z^{-1/\gamma} + \arctan\frac{p\big(z^{-1/\gamma}\big)
  q'\big(z^{-1/\gamma}\big)}{p'\big(z^{-1/\gamma}\big)}\Big).
\end{align*}
For functions  
\[
P(z)=\sqrt{p'^2\big(z^{-1/\gamma}\big)
+ p^2\big(z^{-1/\gamma}\big)q'^2\big(z^{-1/\gamma}\big)}\]
 and 
\[
Q(z)=z^{-1/\gamma} + \arctan\frac{p\big(z^{-1/\gamma}\big) 
q'\big(z^{-1/\gamma}\big)}{p'\big(z^{-1/\gamma}\big)}
\]
we have $P(z)\simeq z^{\frac{\alpha}{\gamma}}$,  
$P'(z)\simeq z^{\frac{\alpha}{\gamma}-1}$, 
$Q(z)\simeq z^{\frac{-1}{\gamma}}$, $Q'(z)\simeq z^{-\frac{1}{\gamma}-1}$ 
as $z\to 0$. So $Y(z)$ is $(\alpha/\gamma,1/\gamma)$-chirp-like function.
To calculate the box dimension of  $G_{yz}$  we apply  \cite[Theorem 5]{cswavy}.

The proof for projection $G_{xz}$ is analogous.
\end{proof}

\begin{remark} \label{rmk6} \rm
In other words an oscillatory dimension of the solution of 
\eqref{eqalfap}, under  assumptions of previous theorem concerning 
$p$ and $q$, is equal to $\dim_{osc}x=\frac{3-\alpha}2$, if  $\alpha\in (0,1)$.
\end{remark}


\section{Limit cycles}

Limit cycles are interesting object appearing in differential equations. 
 In particular, we consider a system having its linear part in Cartesian 
coordinates with a conjugate pair $\pm\omega i$ of pure imaginary eigenvalues
 with $\omega>0$, and the third eigenvalue is equal to zero. 
The corresponding normal form in cylindrical coordinates is
\begin{equation}\label{space}
\begin{aligned}
\dot r &= a_1 rz+a_2 r^3+a_3rz^2+O(|r,z|)^4 \\
\dot \varphi &= \omega+O(|r,z|)^2\\
\dot z &= b_1r^2+b_2z^2+b_3r^2z+b_4z^3+O(|r,z|)^4,
\end{aligned}
\end{equation}
where $a_i$ and $b_i\in\mathbb{R}$ are coefficients of the system.
Such systems and their bifurcations are treated in 
Guckenheimer-Holmes \cite[Section 7.4]{gh}. The fold-Hopf bifurcation 
and  cusp-Hopf bifurcation have been studied in Harlim and 
Langford \cite{harlim} and the references therein, showing that 
system \eqref{space} can exhibit much richer dynamics then singular 
points and periodic solutions. Notice that in system \eqref{alfap} 
there are no limit cycles for any acceptable function $p(t)$. 
We hypothesize that the limit cycle could be induced by introducing 
perturbation in the last equation, $\dot{z}=-z^2$.

Here we make a note about box dimension of a spiral trajectory of the 
simplified system \eqref{space} at the moment of the birth of limit 
cycles in $(x,y)$-coordinate plane. In \cite{zuzu}, \cite{zuzulien}
 we studied planar system consisting of first two equations 
from \eqref{system}, and  made  fractal analysis of the Hopf bifurcation 
of the system. We proved that box dimension of a spiral trajectory  
becomes nontrivial at the moment of bifurcation. The Hopf bifurcation occurs 
with box dimension equal to $4/3$, furthermore degenerate Hopf bifurcation 
or Hopf-Takens bifurcation occurs with the box dimension greater than $4/3$. 
The more limit cycles have been related to larger box dimension. 
Analogous results have been showed for discrete systems in \cite{laho}, 
and applied to continuous systems via Poincar\' e map.
On the other hand in \cite{cras} and \cite{bilip} $3$-dimensional spirals
 have been studied. Here we consider reduced system
\begin{equation}\label{system}
\begin{aligned}
\dot r &=r(r^{2l}+\sum_{i=0}^{l-1}a_ir^{2i})\\
\dot\varphi &=1\\
\dot z &= b_2z^2+\dots+b_nz^n.
\end{aligned}
\end{equation}
First two equations are standard normal form of codimension $l$, 
where  the Hopf-Takens bifurcation occurs, see \cite{takens}.  
The third equation gives us the case where spiral trajectories 
lie on Lipschitzian or H\" olderian surface, depending on the first exponent. 
The H\" olderian surface has infinite derivative in the origin, geometrically 
it is a cusp.

We are interested in the change of the box dimension with respect to 
the third equation at the moment of birth of limit cycles.
We proved for the standard planar model that the Hopf bifurcation occurs 
with box dimension equal to $4/3$ and the Hopf-Takens occurs with larger 
dimensions. Here we prove that on the H\" olderian surface a limit cycle 
occurs with the box dimension greater than $4/3$.

\begin{theorem}[Limit cycle] \label{hopf}
Let $l=1$ in the system \eqref{system} and $b_p<0$ be the first nonzero 
coefficient in the third equation and $a_0=0$. 
Then then  a trajectory $\Gamma$ 
near the origin has the following properties:
\begin{itemize}
\item[(i)] if $2\le p\le 3$ then 
\[
\dim_B\Gamma=\frac{4}{3},
\]

\item[(ii)]if $p\ge 4$ then 
\begin{equation}\label{holddim}
\dim_B\Gamma=\frac32-\frac{1}{2p}.
\end{equation}
\end{itemize}
\end{theorem}

\begin{proof}
Using  \cite[Theorem 9]{zuzu}, the solution of the first two equations 
of \eqref{system} satisfy $r\simeq \varphi ^{-1/2}$, having 
$\dim_B\Gamma_{xy}=4/3$, where $\Gamma_{xy}$ is orthogonal projection of 
space trajectory $\Gamma$ to $(x,y)$ plane.
From the third equation we obtain $z\simeq r^{\frac2{p-1}}$, so for
$2\le p\le 3$ we obtain the Lipschitzian surface, while for $p\ge 4$
surface is H\" olderian. Applying
 \cite[Theorem 7 (a)]{bilip} we obtain $\dim_B\Gamma=4/3$ for
 $2\le p\le 3$, because the box dimension is invariant for the Lipschitzian case. 
For the H\" olderian case we apply  \cite[Theorem 9 (a)]{bilip}, 
where $\alpha=1/2$ and $\beta=2/(p-1)$. 
So, we obtain $\dim_B\Gamma=\frac32-\frac{1}{2p}$.
\end{proof}


\begin{remark} \label{rmk7} \rm
The box dimension of a trajectory at the moment of planar Hopf bifurcation 
is equal to $4/3$, also for $3$-dimensional case with spiral trajectory 
lying in the  Lipschitzian surface. Situation is different for spiral 
trajectory contained in the H\" olderian surface, the box dimension of a 
space spiral trajectory tends to $3/2$. Only one limit cycle could be produced, 
but dimension increases caused by the H\" olderian behavior near the origin.
Notice that if we apply  formula \eqref{holddim} obtained for the H\" olderian case, 
to  the Lipschitzian case $p=3$ we will get correct result $4/3$.
For $l>1$ degenerate Hopf bifurcation or  Hopf-Takens  bifurcation appears, 
where $l$ limit cycles could be born, and  the box dimension of the space 
spiral trajectory is equal to $\dim_B\Gamma=\frac{(4l-1)p-2l+1}{2lp}$ using 
the same arguments.
\end{remark}

\section{Appendix: Auxiliary results}

First  we have a Generalization of \cite[Theorem 4]{cswavy}.

\begin{theorem} \label{gen_wavy}
Let $p(t)\in C^3$ be a function comparable of class $2$ to  
power $t^{-\alpha}$, $\alpha>0$, in the limit sense, and 
$p^{(3)}(t)\in O(t^{-\alpha-3})$, as $t\to\infty$. Let $q(t)\in C^3$ 
be a function comparable of class $1$ to $K t$, $K>0$ in the limit sense, 
$q''(t)\in o(t^{-3})$, as $t\to\infty$, and $q^{(3)}(t)\in o(t^{-2})$, as $t\to\infty$.
Define $x(t)=p(t) \sin q(t)$ and continuous function 
$\varphi (t)$ by $\tan \varphi(t)=\frac{\dot{x}(t)}{x(t)}$.
\begin{itemize}
\item[(i)] If $\alpha\in(0,1)$ then the planar curve 
$\Gamma:=\{(x(t),\dot x(t))\in\mathbb{R}^2: t\in[t_0,\infty)\}$ 
is a spiral $r=f(\varphi)$, $\varphi\in(-\infty,-\phi_0]$, near the origin, and
    $$
    \dim_{\rm ph}(x):=\dim_B\Gamma=\frac{2}{1+\alpha} .
    $$

\item[(ii)] 
If $\alpha>1$ then the planar curve $\Gamma$ is a rectifiable spiral 
near the origin.
\end{itemize}
\end{theorem}

\begin{proof}
After substitution of time variable by $u=t/K$ and respective rescaling 
of the $x$ and $y$ axes, we continue assuming $K=1$. 
The rest of the proof is analogous to the proof of \cite[Theorem 4]{cswavy}, 
but carefully taking care about the more general conditions on $q$. 
Rescaling of the $x$ and $y$ axes in the plane is a bi-Lipschitz map, 
so the box dimension remains preserved.
\end{proof}

\subsection*{Acknowledgments}
This work was supported in part by Croatian Science Foundation 
under project IP-2014-09-2285.

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\end{document}