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# Monotonicity of the over-rotation intervals for bimodal maps

Sourav Bhattacharya  and  Alexander Blokh  Department of Mathematics
University of Alabama at Birmingham
Birmingham, AL 35294  Sourav Bhattacharya<EMAIL_ADDRESS>Alexander Blokh
<EMAIL_ADDRESS>
###### Abstract.

We show that the connectedness of the set of parameters for which the _over-
rotation interval_ of a bimodal interval map is constant. In other words, the
over-rotation interval is a monotone function of a bimodal interval map.

###### Key words and phrases:

Over-rotation pair, over-rotation number, pattern, periodic orbit

###### 2010 Mathematics Subject Classification:

Primary 37E05, 37E15; Secondary 37E45

##  1\. INTRODUCTION

One-dimensional combinatorial dynamics started when O. M. Sharkovsky proved
his theorem on the coexistence of periods of cycles for interval maps. To
state it, we recall the (transitive) _Sharkovsky order_ of the set natural
numbers:

$3\succ\mkern-14.0mu_{s}\;5\succ\mkern-14.0mu_{s}\;7\succ\mkern-14.0mu_{s}\;\dots\succ\mkern-14.0mu_{s}\;2\cdot
3\succ\mkern-14.0mu_{s}\;2\cdot 5\succ\mkern-14.0mu_{s}\;2\cdot
7\succ\mkern-14.0mu_{s}\;\\\ \dots\succ\mkern-14.0mu_{s}\;2^{2}\cdot
3\succ\mkern-14.0mu_{s}\;2^{2}\cdot 5\succ\mkern-14.0mu_{s}\;2^{2}\cdot
7\succ\mkern-14.0mu_{s}\;\dots\succ\mkern-14.0mu_{s}\;2^{2}\succ\mkern-14.0mu_{s}\;2\succ\mkern-14.0mu_{s}\;1.$

In what follows by the _period_ we mean the _minimal_ period. Below $I$ always
denotes a closed interval.

######  Theorem 1.1 ([11] and [12] for English translation).

If $g:I\to I$ is continuous, $m\succ\mkern-14.0mu_{s}\;n$ and $m$ is the
period of cycle of $g$ then $n$ is also the period of a cycle of $g$.

Theorem 1.1 inspired a lot of developments. One of them is the discovery of
_over-rotation numbers_ , a combinatorial tool used for classifying interval
cycles and invariant under topological conjugacy of interval maps. It can be
defined as follows.

Let $I$ be the unit interval, let $f:I\to I$ be a continuous interval map, let
$P$ be a periodic orbit of $f$ of period $q>1$, and let $m$ be the number of
points $x\in P$ such that $f(x)-x$ and $f^{2}(x)-x$ have different signs. Call
the pair $(\frac{m}{2},q)=\mathrm{orp}(P)$ the _over-rotation pair_ of $P$ and
denote it by $orp(P)$; call $\frac{m}{2q}=\rho(P)$ the _over-rotation number_
of $P$. An over-rotation pair $(p,q)$ is _coprime_ if $p$ and $q$ are coprime.
The set of _over-rotation pairs_ of all cycles of $f$ is denoted by $ORP(f)$.
Since the number $0<m\leq\frac{q}{2}$ is even, then, in an _over-rotation
pair_ $(p,q)$, $p$ and $q$ are integers and $0<\frac{p}{q}\leq\frac{1}{2}$.
The number $p$ can be interpreted as the number of times $f(x)$ goes around
$x$ as we move along the orbit of the “vector” $\overrightarrow{xf(x)}$. If
$f$ has a unique fixed point $a$, then $f(x)>x$ if $x<a$ and $f(x)<x$ if
$x>a$, and $p$ is the number of points to the right of $a$ which are mapped to
the left of $a$ (or vice versa).

######  Theorem 1.2 ([8]).

Suppose that $(p,q)$ and $(r,s)$ are _over-rotation pairs_. Moreover, suppose
that one of the following holds.

  1. (1)

$\frac{p}{q}<\frac{r}{s}$.

  2. (2)

$\frac{p}{q}=\frac{r}{s}$ so that for a coprime over-rotation pair $(k,l)$ we
have $p/k=q/l=u$ and $r/k=s/l=v$ are integers, and
$u\succ\mkern-14.0mu_{s}\;v$.

Then any interval map with a cycle of over-rotation pair $(p,q)$ has a cycle
of over-rotation pair $(r,s)$.

Definition 1.3 is based upon Theorem 1.2; to rule out trivial cases we will
from now on assume that we consider only maps with non-fixed periodic points
(otherwise orbits of all points converge to a fixed point).

######  Definition 1.3.

Given an interval map $f$, denote by $I_{f}$ the closure of the union of over-
rotation numbers of $f$-periodic points, and call $I_{f}$ the _over-rotation
interval_ of $f$.

By Theorem 1.1, any map $f$ with non-fixed periodic points has a cycle of
period 2 with over-rotation number $\frac{1}{2}$; by Theorem 1.2 if
$\frac{p}{q}=\rho(P)$ for a cycle $P$ then $[\frac{p}{q},\frac{1}{2}]\subset
I_{f}$; hence for any interval map $f$ there exists a number $r_{f},0\leq
r_{f}<\frac{1}{2},$ such that $I_{f}=[r_{f},\frac{1}{2}]$.

A continuous map $f:[0,1]\to[0,1]$ is said to have a _horseshoe_ if there
exist subintervals $I$ and $J$ of $[0,1]$ disjoint except perhaps a common
endpoint, such that $f(I)\cap f(J)\supset I\cup J$. By [8], if $f$ has a
horseshoe then $I_{f}=[0,\frac{1}{2}]$.

A map is _piecewise-monotone_ if the interval $I$ can be partitioned into
finitely many intervals (_laps_) on which $f$ is _strictly monotone_. If the
smallest number of such intervals is $2$, then $f$ is said to be _unimodal_ ;
in this case $f$ has one turning point. If the smallest number of such
intervals is $3$, then $f$ is said to be _bimodal_ ; in this case $f$ has two
turning point. In this paper we _always_ consider _only_ piecewise-monotone
maps that are increasing on their leftmost laps.

Every cycle $P$ of a map $f$ induces a cyclic permutation $\Pi$ obtained by
looking at how the map $f$ acts on the points of $P$ ordered from the left to
the right. One can introduce a relation $\sim$ on the family of all cycles
such that for two cycles $P$ and $Q$, $P\sim Q$ iff $P$ and $Q$ induce the
same permutation; $\sim$ is an equivalence relation whose equivalence classes
are called _patterns_. If an interval map $f$ has a cycle $P$ from a _pattern_
$\pi$ associated with permutation $\Pi$, say that $P$ is a _representative_ of
$\pi$ (in $f$) and $f$ _exhibits_ $\pi$ (on $P$). A permutation $\pi$ forces a
permutation $\theta$ if any continuous interval map $f$ which exhibits $\pi$
also exhibits $\theta$. By [2], _forcing_ is a partial ordering.

A useful algorithm allows one to describe all patterns forced by a pattern
$\pi$. Consider a cycle $P$ of pattern $\pi$, and denote the leftmost point of
$P$ by $a$ and the rightmost point of $P$ by $b$. Every component of $[a,b]-P$
is called a _P-basic interval_. Extend this map from $P$ to $[a,b]$ by
defining it linearly on each $P$-basic interval and call the resultant map
$f_{P}$ the $P$-_linear map_. Then, the patterns of all cycles of $f_{P}$ are
exactly the patterns forced by the pattern of $P$ (see [1] and [2]).

Over-rotation pairs and numbers for patterns are defined just like for an
interval map. Denote the _over-rotation pair_ and _over-rotation number_ of a
pattern $\pi$ by $\mathrm{orp}(\pi)$ and $\rho(\pi)$, resp. If $P$ is a
periodic orbit of pattern $\pi$, call the over-rotation interval
$I_{\pi}=[r_{\pi},\frac{1}{2}]$ of the $P$-linear map $f_{P}$ the _over-
rotation interval of $\pi$_.

######  Definition 1.4.

A _pattern_ $\pi$ is called _over-twist_ if it does not _force_ any other
_pattern_ of the same _over-rotation number_.

By Theorem 1.2, an over-twist pattern has a _coprime_ _over-rotation pair_
(i.e., an over-rotation pair $(p,q)$ where $p$ and $q$ are coprime); in
particular, over-twists of over-rotation number $\frac{1}{2}$ are of period
$2$, so from now on we consider _over-twists_ of _over-rotation number_
distinct from $\frac{1}{2}$. By [8] and by properties of forcing relation, for
any $\frac{p}{q}\in(r_{f},\frac{1}{2})$, an over-twist pattern of over-
rotation number $\frac{p}{q}$ is exhibited by a cycle of $f$; over-twists are
patterns that are _guaranteed_ to a map $f$ if the interior of $I_{f}$
contains the appropriate _over-rotation number_. This can be sharpened if $f$
is piecewise-monotone.

######  Theorem 1.5 ([5], [6], [7]).

If $f:[0,1]\to[0,1]$ is a piecewise monotone continuous map with over-rotation
interval $[r_{f},\frac{1}{2}]$, then for any
$\frac{p}{q}\in[r_{f},\frac{1}{2}]$, there exists a periodic orbit $P$ which
exhibits over-twist pattern of over-rotation number $\frac{p}{q}$.

The over-rotation interval of an over-twist pattern $\pi$ is
$[\rho(\pi),\frac{1}{2}]$ so that $r_{\pi}=\rho(\pi)$ holds. In [3] the
following version of the opposite statement is proven: a pattern $\pi$ with
_coprime over-rotation pair_ is _over-twist_ if and only if
$r_{\pi}=\rho(\pi)$.

In [9], it was proven that for a given rational number $\frac{p}{q}$ there
exists a unique unimodal over-twist pattern $\gamma_{\frac{p}{q}}$ of over-
rotation number $\frac{p}{q}$; moreover, the dynamics of
$\gamma_{\frac{p}{q}}$ was described. It was also shown that the over-rotation
interval is a monotone function of a map considered on a wide variety of one-
parameter families of unimodal maps. The goal of the present paper is to
extend this result onto the family of truncations of bimodal interval maps. It
is known (see, e.g., [10]) that any bimodal interval map can be modeled by a
truncation of a bimodal horseshoe map. So, we parameterize the family of all
truncations of a bimodal horseshoe map and then show that the set of the
parameters which correspond to a particular to a fixed over-rotation interval
is a connected subset of the parameter space. For this we will use the
explicit description of all _bimodal_ over-twist patterns obtained in a recent
paper [4]. We discuss our plans in details below.

A _bimodal horseshoe map_ is a _bimodal_ map $H_{2}:[0,1]\to[0,1]$ defined by
the following formula

$H_{2}(x)=\begin{cases}3x&\text{if $0\leq x\leq\frac{1}{3}$}\\\ 2-3x&\text{if
$\frac{1}{3}\leq x\leq\frac{2}{3}$}\\\ 3x-2&\text{if $\frac{2}{3}\leq x\leq 1$
}\\\ \end{cases}$

Evidently, $H_{2}$ has cycles that exhibit all bimodal patterns.

For $\alpha,\beta\in[0,1]$ with $\alpha\geqslant\beta$, we call the interval
$\xi^{max}_{\alpha}=[\frac{\alpha}{3},\frac{2}{3}-\frac{\alpha}{3}]$ where
$H_{2}(x)\geqslant\alpha$ the _level_ $\alpha$ _max flat spot_ and the
interval
$\xi^{min}_{\beta}=[\frac{2}{3}-\frac{\beta}{3},\frac{2}{3}+\frac{\beta}{3}]$
where $H_{2}(x)\leqslant\beta$ the _level_ $\beta$ _min flat spot_. The map
$H_{\alpha,\beta}:[0,1]\to[0,1]$ defined by:

$H_{\alpha,\beta}(x)=\begin{cases}H(x)&\text{if
$x\notin\xi^{max}_{\alpha}\cup\xi^{min}_{\beta}$}\\\ \alpha&\text{if
$x\in\xi^{max}_{\alpha}$}\\\ \beta&\text{if $x\in\xi^{min}_{\beta}$ }\\\
\end{cases}$

is called the _bimodal truncation_ of $H$ with parameters $\alpha$ and
$\beta$. Figure 1 shows the graphs of $H_{2}$ and $H_{\alpha,\beta}$.

Figure 1. The graphs of the maps $H_{2}$ and $H_{\alpha,\beta}$ with the
latter shown in bolder lines

Recall that by [10] any bimodal map can be modeled by some $H_{\alpha,\beta}$.

Let $\mathcal{H}=\\{H_{\alpha,\beta}:\alpha,\beta\in[0,1]$ $\&$
$\alpha\geqslant\beta\\}$ be the family of all truncations of $H_{2}$. Denote
the _over-rotation interval_ of $H_{\alpha,\beta}$ by $I_{\alpha,\beta}$. It
is easy to see that if $\alpha=\beta$, $\alpha\leqslant\frac{1}{2}$, or
$\beta\geqslant\frac{1}{2}$, then all periodic points of $H_{\alpha,\beta}$
are fixed. Hence, the $\omega$-limit set $\omega(x)$ is a fixed point for
every $x$, and we are not considering such maps. Excluding these parameters,
we define the _parameter space_ as
$\mathcal{P}=\\{(\alpha,\beta)\in[0,1]\times[0,1]:\alpha\geqslant\frac{1}{2}\geqslant\beta\\}$;
$\mathcal{P}$ is a _rectangle_ with vertices
$(1,0),(1,\frac{1}{2}),(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)$. We
will study $\mathcal{P}$ using the Euclidian metric.

Let us use the following terminology to describe $\mathcal{P}$ throughout the
paper. The _points_ $(1,0)$ and $(\frac{1}{2},\frac{1}{2})$ in the _parameter
plane_ $\mathcal{P}$ are called the _focal point_ $\mathcal{F}$ and the
_vertex_ $\mathcal{V}$ respectively. Let us call the set
$\mathcal{B}=\\{(x,\frac{1}{2})|\frac{1}{2}\leq x\leq
1\\}\cup\\{(\frac{1}{2},y)|0\leq y\leq\frac{1}{2}\\}$ the _base set_.
Similarly, call the set $\mathcal{S}=\\{(x,0)|\frac{1}{2}\leq x\leq
1\\}\cup\\{(1,y)|0\leq y\leq\frac{1}{2}\\}$ the _side arm_.

Figure 2.  The Parameter space $\mathcal{P}$

Define a map $\psi:\mathcal{P}\to[0,\frac{1}{2}]$ with
$\psi(\alpha,\beta)=\rho(I_{\alpha,\beta})$. Observe that at the _focal point_
the value of $\psi$ is the _least_ namely $0$, while at any point in the
_base_ $\mathcal{B}$, the value of $\psi$ is the _greatest_ namely,
$\frac{1}{2}$. We will call the set
$\psi^{-1}(\nu)=\psi_{\nu}=\\{(\alpha,\beta)\in\mathcal{P}:\psi((\alpha,\beta)=\nu\\}$
the _Bimodal Iso-over-rotation-tract_ corresponding to $\nu$ or the
$\nu$-_Bimodal Iso-over-rotation-tract_. The main objective of our paper is to
show that for any $\nu\in[0,\frac{1}{2}]$ the $\nu$-_Bimodal Iso-over-
rotation-tract_ is a _connected set_. In other words the map
$\psi:\mathcal{P}\to[0,\frac{1}{2}]$ is a _monotone map_.

We divide our paper into three sections. Section 1 is our Introduction.
Section 2 contains some preliminary ideas. Section 3 is the main section of
the paper where we prove that _Bimodal Iso-over-rotation-tracts_ are
connected.

##  2\. PRELIMINARIES

###  2.1. Kneading Sequences

Let $f:[0,1]\to[0,1]$ be a bimodal map with turning points $0<c_{1}<c_{2}<1$.
Call $\mathcal{S}=\\{I_{0},C_{1},I_{1},C_{2},I_{2}\\}$ where
$I_{0}=[0,c_{1})$, $C_{1}=\\{c_{1}\\}$, $I_{1}=(c_{1},c_{2})$,
$C_{2}=\\{c_{2}\\}$ and $I_{2}=(c_{2},1]$ the _symbolic set_ of $f$. Order its
elements as $I_{0}<C_{1}<I_{1}<C_{2}<I_{2}$. Define a _location function_
$\mathcal{L}:[0,1]\to\mathcal{S}$ which assigns to each point $x\in[0,1]$, its
unique _location_ $\mathcal{J}(x)\in\mathcal{S}$ defined by
$x\in\mathcal{J}(x)$. Observe that if $x<y$, then
$\mathcal{J}(x)<\mathcal{J}(y)$. Let $\mathcal{S}^{\mathbb{N}}$ be the set of
all infinite sequences of symbols
$(\mathcal{J}_{0},\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J}_{3},\dots)$
where $\mathcal{J}_{i}\in\mathcal{S}$ for every $i$.

Define a function $\mathcal{I}:[0,1]\to\mathcal{S}^{\mathbb{N}}$ which
associates with each point $x\in[0,1]$, the unique sequence
$\mathcal{I}(x)=(\mathcal{J}(x),\mathcal{J}(f(x)),\mathcal{J}(f^{2}(x)),$
$\dots)\in\mathcal{S}^{\mathbb{N}}$. We call $\mathcal{J}(x)$, the _itinerary_
of the point $x$. Let
$\sigma:\mathcal{S}^{\mathbb{N}}\to\mathcal{S}^{\mathbb{N}}$ be the _shift
map_ defined by
$\sigma(\mathcal{J}_{0},\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J}_{3},\dots)=(\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J}_{3},\mathcal{J}_{4},\dots)$.
It follows that $\mathcal{I}:[0,1]\to\mathcal{S}^{\mathbb{N}}$ conjugates the
map $f:[0,1]\to[0,1]$ to the _shift map_
$\sigma:\mathcal{S}^{\mathbb{N}}\to\mathcal{S}^{\mathbb{N}}$. This allows one
to model the dynamics of the map $f$ using the properties of the
_itineraries_.

Define a _sign_ function $\Theta:\mathcal{S}\to$ $\\{-1,0,1\\}$ as follows:
for any $j$, $\Theta(I_{j})=+1$ if $f$ is _increasing_ on $I_{j}$,
$\Theta(I_{j})=-1$ if $f$ is _decreasing_ on $I_{j}$, and $\Theta(C_{j})=0$
for any $j$. Define a partial order $\succ$ on all _itineraries_ as follows.
Let
$\mathcal{J}=(\mathcal{J}_{0},\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J}_{3},\dots)$
and
$\mathcal{K}=(\mathcal{K}_{0},\mathcal{K}_{1},\mathcal{K}_{2},\mathcal{K}_{3},\dots)$
be two _itineraries_. Let
$\kappa=\min\\{i\in\mathbb{N}:\mathcal{J}_{i}\neq\mathcal{K}_{i}\\}$, and set
$\theta(\kappa-1)=\Pi_{i=1}^{\kappa-1}\Theta(\mathcal{J}_{i})=\Pi_{i=1}^{\kappa-1}\Theta(\mathcal{K}_{i})$.
Then say $\mathcal{J}$ is _stronger_ than $\mathcal{K}$ and write
$\mathcal{J}\succ\mathcal{K}$ iff $\mathcal{J}_{\kappa}>\mathcal{K}_{\kappa}$
(if $\theta(\kappa-1)=+1$) _or_ $\mathcal{K}_{\kappa}>\mathcal{J}_{\kappa}$
(if $\theta(\kappa-1)=-1$). If $\theta(\kappa-1)=0$ then $\succ$ is _not
defined_. Theorem 2.1 follows from [10].

######  Theorem 2.1.

Let $x,y\in[0,1]$ with $x>y$. Then $\mathcal{I}(x)\succeq\mathcal{I}(y)$.
Conversely, if for $x,y\in[0,1]$, we have $\mathcal{I}(x)\succ\mathcal{I}(y)$,
then $x>y$.

The _itineraries_
$\mathcal{K}_{j}=\mathcal{I}(f(c_{j}))\in\mathcal{S}^{\mathbb{N}}$ of the
_critical points_ $c_{j}$ for $j=1,2$ are called the _kneading sequences_ of
the map $f$. The _vector_
$\overrightarrow{\mathcal{K}(f)}=(\mathcal{K}_{1},\mathcal{K}_{2})$ is called
the _kneading vector_ of the map $f$.

In this paper we will use Theorem 2.1 to pinpoint the distribution of the
periodic orbits corresponding to different bimodal over-twist patterns on a
bimodal horseshoe map $H_{2}$ by comparing the itineraries of the points of
absolute maxima and minima of these orbits.

###  2.2. Some results on Degree one Circle Maps

Recall some results on circle maps that we need (see [1] for detail). Consider
the unit circle $\mathbb{S}$ and the natural projection
$\pi:\mathbb{R}\to\mathbb{S}$. If $f:\mathbb{S}\to\mathbb{S}$ is continuous,
then there is a continuous map $F:\mathbb{R}\to\mathbb{R}$ such that
$F\circ\pi=\pi\circ f$. Such a map $F$ is called a _lifting_ of $f$. It is
unique up to translation by an integer. An integer $d$ with $F(x+1)=F(x)+d$
for all $x\in\mathbb{R}$ is called the _degree_ of the map $f$ and is
independent of the choice of $F$.

Let us denoted by $\mathcal{L}_{1}$ the set of all liftings of continuous
degree one maps of $\mathbb{S}$ onto itself. Let $F\in\mathcal{L}_{1}$. We
define _upper and lower rotation numbers_ of $x\in\mathbb{R}$ for
$F\in\mathcal{L}_{1}$ as
$\overline{\rho_{F}}(x)=\displaystyle\limsup_{n\to\infty}\frac{F^{n}(x)-x}{n}$
and
$\underline{\rho_{F}}(x)=\displaystyle\liminf_{n\to\infty}\frac{F^{n}(x)-x}{n}$
respectively. If the numbers $\overline{\rho_{F}}(x)$ and
$\underline{\rho_{F}}(x)$ are equal the common value is called the _rotation
number of x_ for $F$ and is denoted by $\rho_{F}(x)$.

######  Theorem 2.2 ([1]).

If $F\in\mathcal{L}_{1}^{\prime}$ is a lifting of a circle map $f$ then
$\rho_{F}(x)$ exists for all $x\in\mathbb{R}$ and is independent of $x$.
Moreover, it is rational if and only if $f$ has a periodic point.

We will call this number the rotation number of $F$ and is denoted by
$\rho(F)$.

######  Lemma 2.3.

The function $\rho:\mathcal{L}_{1}^{\prime}\to\mathbb{R}$ is continuous.

For $F\in\mathcal{L}_{1}$ we define maps
$F_{l},F_{u}\in\mathcal{L}_{1}^{\prime}$ as follows:
$F_{l}(x)=\inf\\{F(y):y\geqslant x\\}$ and $F_{u}(x)=\sup\\{F(y):y\leqslant
x\\}$.

######  Theorem 2.4 ([1]).

Then the following statements are true:

  1. (1)

Let $F\in\mathcal{L}_{1}$. Then, $F_{l}(x)\leqslant F(x)\leqslant F_{u}(x)$
for every $x\in\mathbb{R}$.

  2. (2)

Let $F,G\in\mathcal{L}_{1}$ with $F\leqslant G$, then $F_{l}\leqslant G_{l}$
and $F_{u}\leqslant G_{u}$.

  3. (3)

Let $F\in\mathcal{L}_{1}^{\prime}$. Then, $F_{l}=F_{u}=F$.

  4. (4)

The maps $F\mapsto F_{l}$ and $F\mapsto F_{u}$ are Lipschitz continuous with
constant $1$ in the sup norm.

  5. (5)

The maps $F\mapsto\rho(F_{l})$ and $F\mapsto\rho(F_{l})$ are continuous.

  6. (6)

If $F_{l}(x)\neq F(x)$, then $x\in Const(F_{l})$ where $Const(F_{l})$ denotes
the union of all open intervals on which $F_{l}$ is constant.

  7. (7)

Let $F\in\mathcal{L}_{1}$ be a lifting of a circle map $f$. Then the set of
all rotation numbers of points is equal to the interval
$[\rho(F_{l}),\rho(F_{u})]$. Moreover, for each rational $a$ from this
interval there is a point $x$ such that $\pi(x)$ is periodic for $f$ and
$\rho_{F}(x)=a$. The interval $[\rho(F_{l}),\rho(F{{}_{u}})]$ is called the
_rotation interval of_ $F$. We will denote it by $Rot(F)$.

  8. (8)

The endpoints of $Rot(F)$ depend continuously on $F\in\mathcal{L}_{1}$.

###  2.3. Results on over-rotation numbers of bimodal maps [4]

The article [4] relates the _over-rotation numbers_ of bimodal interval maps
and the classical rotation numbers of degree one circle maps of the real line.
This yields a description of dynamics of _bimodal over-twist patterns_.
Namely, given an over-rotation number $\frac{p}{q}$ there are $q-2p+1$
_bimodal over-twist patterns_ $\Gamma_{r,\frac{p}{q}}$ with this over-rotation
number; each of them can be characterized by a value of
$r\in\\{0,1,2,\dots,q-2p\\}$. The patterns with $r=0$ and $r=q-2p$ are
_unimodal_ while the remaining $q-2p-1$ patterns are _strictly bimodal_.
Denote the permutation of the over-twist pattern $\Gamma_{r,\frac{p}{q}}$ by
$\Pi_{r,\frac{p}{q}}$. By [4] $\Pi_{r,\frac{p}{q}}$ is as follows.

$\Pi_{r,\frac{p}{q}}=\begin{cases}j+p&\text{if $1\leq j\leq r$}\\\
q-j+r+1&\text{if $r+1\leq j\leq r+p$}\\\ 2p-j+r+1&\text{if $r+p+1\leq j\leq
r+2p$ }\\\ j-p&\text{if $r+2p+1\leq j\leq q$}\\\ \end{cases}$ (2.1)

Let $P_{r,\frac{p}{q}}$ be a periodic orbit of a map $f$ which exhibits the
pattern $\Gamma_{r,\frac{p}{q}}$. Number its $q$ points in _spatial labelling_
as $x_{1}<x_{2}<\dots<x_{q}$ and partition them into $4$ disjoint parts.

The first $r$ points $x_{1},x_{2},\dots,x_{r}$ are called _red points_ and are
denoted by
$\mathcal{R}_{1}^{r},\mathcal{R}_{2}^{r},\mathcal{R}_{3}^{r},\dots,\mathcal{R}_{r}^{r}$.
Under the action of $f$ the _red points_ are shifted to the right by $p$
points.

The next $p$ points $x_{r+1},x_{r+2},$ $\dots,x_{r+p}$ are called _green
points_ and are denoted by
$\mathcal{G}_{1}^{r},\mathcal{G}_{2}^{r},\mathcal{G}_{3}^{r},\dots,\mathcal{G}_{p}^{r}$.
The _green points_ map onto the last $p$ points
$x_{q-p+1},\dots,x_{q-1},x_{q}$ of the orbit with a _flip_ , that is
orientation is reversed but without any expansion.

The next $p$ points $x_{r+p+1},$ $x_{r+p+2},$ $\dots,x_{r+2p}$ are called
_pink points_ and are denoted by
$\mathcal{P}_{1}^{r},\mathcal{P}_{2}^{r},\mathcal{P}_{3}^{r},$
$\dots,\mathcal{P}_{p}^{r}$. The _pink points_ map onto the first $p$ points
$x_{1},x_{2},$ $\dots,x_{p}$ of the orbit with a _flip_ but with no expansion.

The last $q-2p-r$ points are called _blue points_ and are denoted by
$\mathcal{B}_{1}^{r},\mathcal{B}_{2}^{r},\mathcal{B}_{3}^{r},\dots,\mathcal{B}_{q-2p-r}^{r}$.
They are shifted to the _left_ by $p$ points.

If $r=0$, we have no _red points_ ; if $r=q-2p$, we have no _blue points_. For
all other values of $r$, we have points of all colors.

The dynamics of $P_{3,\frac{3}{11}}$ is depicted in the Figure 3.

Figure 3.  The Bimodal over-twist pattern $\Gamma_{3,\frac{3}{11}}$

###  2.4. Some results from Algebraic Topology

We assume knowledge of basic concepts such as connectedness and path-
connectedness of topological spaces.

If $f:X\to Y$ and $g:X\to Y$ are two continuous functions then $f$ and $g$ are
said to be _homotopic_ if there exists a continuous function $H:(X,[0,1])\to
X$ such that $H(x,0)=f(x)$, $H(x,1)=g(x)$. Then we write $f\cong f^{\prime}$ ;
the map $H$ is called a _homotopy_ between $f$ and $g$. Let $(X,\tau)$ be a
topological space and $A\subset X$. A _retraction_ of $X$ onto $A$ is a
continuous map $r:X\to A$ such that $r|_{A}=id_{A}$. If such a map $r$ exists
then $A$ is called a _retract_ of $X$.

######  Definition 2.5.

Let $(X,\tau)$ be a topological space and $A\subset X$. We say that$A$ is a
_deformation retract_ of $X$ if there exists a homotopy between the identity
map on $X$ and the map that carries all of $X$ onto $A$ such that each point
of $A$ remains fixed during the homotopy.

In other words, there is a homotopy $H:X\times I\to X$ such that:

  1. (1)

$H(x,0)=id_{X}$ $\forall x\in X$.

  2. (2)

$H(x,1)\in A$

  3. (3)

$H(a,t)=a$ $\forall t\in I$ and $\forall a\in A$.

The map $r:X\to A$ defined by $r(x)=H(x,1)$ is a retraction of $X$ onto $A$
and $H$ is a homotopy between $id_{X}$ and the map $j\circ r$ where $j:A\to X$
is the inclusion map defined by $j(a)=a$ for all $a\in A$.

######  Theorem 2.6.

Let $(X,\tau)$ be a path-connected space and let $A\subset X$ be a deformation
retract of $X$. Then, $A$ is path-connected.

###### Proof.

Take $a,b\in A$. Since $(X,\tau)$ is path-connected, then there exists a
continuous map $\xi:[0,1]\to X$ with $\xi(0)=a$ and $\xi(1)=b$. Since $A$ is a
deformation retract of $X$, there exists a continuous map $H:(X,[0,1])\to X$
such that $H(x,0)=id_{X}$ and $H(x,1)|_{A}=id_{A}$. Define $r:X\to A$ such
that $r(x)=H(x,1)$. Then $r|_{A}=id_{A}$.

Define $\chi:[0,1]\to A$ by $\chi=r\circ\xi$. Then $\chi(0)=a$ and $\chi(1)=b$
and $\chi$ is continuous. Thus, $\chi$ is a path in $A$ connecting $a$ and
$b$. So, $A$ is a path-connected. ∎

##  3\. CONNECTEDNESS OF BIMODAL ISO-OVER-ROTATION-TRACTS

Let us outline our approach. Given a rational number $\frac{p}{q}$, we first
describe the set $Z_{\frac{p}{q}}$ of all points $(\alpha,\beta)$ in the
parameter space $\mathcal{P}$ such that $\psi((\alpha,\beta))=\frac{p}{q}$ and
one of the flat spots $\xi^{max}_{\alpha}$ and $\xi^{min}_{\beta}$ exhibits an
over-twist pattern of over-rotation number $\frac{p}{q}$; this set is a
continuous broken line (a _staircase_) in $\mathcal{P}$. Then we show that
$Z_{\frac{p}{q}}$ _partitions_ $\mathcal{P}$ into two _path-connected
components_ : a _component_ $\mathcal{G}_{\frac{p}{q}}$ where
$\psi((\alpha,\beta))\geqslant\frac{p}{q}$ and a _component_
$\mathcal{U}_{\frac{p}{q}}$ where $\psi((\alpha,\beta))<\frac{p}{q}$. Then we
show that the $\frac{p}{q}$-_Bimodal Iso-over-rotation-tract_
$\mathcal{I}_{\frac{p}{q}}$ is a _deformation retract_ of the component
$\mathcal{G}_{\frac{p}{q}}$. Since $\mathcal{G}_{\frac{p}{q}}$ is path-
connected, then, by Theorem 2.6, $\mathcal{I}_{\frac{p}{q}}$ is path-
connected. Finally we prove that $\psi:\mathcal{P}\to[0,\frac{1}{2}]$ is
continuous and then use it to prove connectedness of $\mu-\emph{Bimodal Iso-
over-rotation-tract}$ for an irrational number $\mu\in[0,\frac{1}{2}]$.

For two sets $A,B\subset\mathbb{R}^{2}$, $d(A,B)=\inf\\{d(x,y):x\in A$ $\&$
$y\in B\\}$.

######  Lemma 3.1.

If $(\alpha,\beta)$ and $(\gamma,\delta)$ are points in $\mathcal{P}$ co-
linear with the focal point $\mathcal{F}$ such that
$d(\mathcal{F},(\alpha,\beta))$ $\geqslant d(\mathcal{F},(\gamma,\delta))$
then $\psi((\alpha,\beta))\geqslant\psi((\gamma,\delta))$.

###### Proof.

Since, $(\alpha,\beta)$ and $(\gamma,\delta)$ are points in $\mathcal{P}$ co-
linear with the focal point $\mathcal{F}$ with $d(\mathcal{F},(\alpha,\beta))$
$\geqslant d(\mathcal{F},(\gamma,\delta))$, it follows that
$\gamma\geqslant\alpha$ and $\delta\leqslant\beta$. Thus
$\xi_{\gamma}^{max}\subseteq\xi_{\alpha}^{max}$,
$\xi_{\delta}^{min}\subseteq\xi_{\beta}^{min}$, and hence
$I_{(\alpha,\beta)}\subseteq I_{(\gamma,\delta)}$ and
$\psi((\alpha,\beta))\geqslant\psi((\gamma,\delta))$ as desired. ∎

######  Definition 3.2.

Let $\frac{p}{q}$ be a given rational number where $p$ and $q$ are _coprime_.
The _leading set_ $Z_{\frac{p}{q}}$ is the collection of points
$(\alpha,\beta)\in\mathcal{P}$ which satisfies the following:

  1. (1)

at least one of _flatspots_ $\xi_{\alpha}^{max}$ and $\xi_{\beta}^{min}$
exhibits an _over-twist pattern_ with _over-rotation number_ $\frac{p}{q}$.

  2. (2)

$\psi((\alpha,\beta))=\frac{p}{q}$.

We now describe the structure of the leading set $Z_{\frac{p}{q}}$ for a given
rational number $\frac{p}{q}$ where $p$ and $q$ are coprime. For this we first
study the distribution of the bimodal over-twist periodic orbits in a bimodal
horseshoe map $H_{2}$.

For any rational $\frac{p}{q}$ with _coprime_ $p$ and $q$, there are $q-2p$
bimodal over-twist patterns $\Gamma_{r,\frac{p}{q}}$ for
$r\in\\{0,1,2,\dots,q-2p\\}$. Out of these the patterns
$\Gamma_{0,\frac{p}{q}}$ and $\Gamma_{q-2p,\frac{p}{q}}$ are unimodal and the
remaining $q-2p-2$ patterns are strictly bimodal. Recall that in the context
of the map $H_{2}$ we have $I_{0}=[0,\frac{1}{3}],$
$I_{1}=[\frac{1}{3},\frac{2}{3}],$ $I_{2}=[\frac{2}{3},1]$.

The map $H_{2}$ has a unique cycle $P_{0,\frac{p}{q}}$ such its point of
_absolute maximum_ $M_{0}$ and its point of _absolute minimum_ $m_{0}$ lies in
$I_{1}$. Similarly, $H_{2}$ has a unique periodic orbit $P_{q-2p,\frac{p}{q}}$
such its point of _absolute maximum_ $M_{q-2p}$ and its point of _absolute
minimum_ $m_{q-2p}$ lies in $I_{1}$.

Finally, for each $r\in\\{1,2,\dots,q-2p-1\\}$, $H_{2}$ has a unique periodic
orbit $P_{r,\frac{p}{q}}$ such its point of _absolute maximum_ $M_{r}$ lies in
$I_{1}$ and its point of _absolute minimum_ $m_{r}$ lies in $I_{2}$.

Let $\mathcal{P}_{\frac{p}{q}}$ be the collection of these $q-2p+1$ cycles.
Let us study the location of these cycles on $H_{2}$. For this we introduce a
_partial ordering_ $\triangleleft$ in $\mathcal{P}_{\frac{p}{q}}$ as follows.
For $r_{1},r_{2}\in\\{0,1,2,\dots,q-2p\\}$, write $P_{r_{2},\frac{p}{q}}$
$\triangleleft$ $P_{r_{1},\frac{p}{q}}$ and say $P_{r_{2},\frac{p}{q}}$
_floats above_ $P_{r_{2},\frac{p}{q}}$ in $H_{2}$ iff
$H_{2}(M_{r_{2}})>H_{2}(M_{r_{1}})$ and $H_{2}(m_{r_{2}})>H_{2}(m_{r_{1}})$
where $M_{r_{1}}$ and $M_{r_{2}}$ are the points of _absolute maximum_ and
$m_{r_{1}}$ and $m_{r_{2}}$ be the points of _absolute minimum_ of these
orbits respectively.

Given a periodic orbit $P_{r,\frac{p}{q}}$ it is easy to see that that all its
_red_ points (if any) lies in the lapse $I_{0}$, all its _green_ and _pink_
points lies in the lapse $I_{1}$ and all its _blue_ points (if any ) lies in
the lapse $I_{2}$.

######  Lemma 3.3.

For every $1\leqslant k\leqslant q-2p-2$, $P_{k,\frac{p}{q}}$ $\triangleleft$
$P_{k+1,\frac{p}{q}}$.

###### Proof.

Since, $1\leqslant k\leqslant q-2p-2$, both $P_{k,\frac{p}{q}}$ and
$P_{k+1,\frac{p}{q}}$ contains points of all colors. Let us compare the
itineraries of $m_{k}$ and $m_{k+1}$. They are both _pink_ points in their
respective orbits. On application of $H_{2}$ both of these points will map to
the first red point $\mathcal{R}_{1}^{k}$ and $\mathcal{R}_{1}^{k+1}$ of their
respective orbits, then to a _green point_ , then to a _blue point_ and then
finally to another _pink point_ of their respective orbits. Let us call this
one _round_ of its orbit.

Both $m_{k}$ and $m_{k+1}$ will keep on performing such _rounds_ and their
itineraries will remain the same until the auspicious moment when $m_{k+1}$
hits the last _red point_ $\mathcal{R}_{k}^{k+1}$ of its orbit. Since the
number of _red points_ in $P_{k,\frac{p}{q}}$ is one less than that of
$P_{k+1,\frac{p}{q}}$, at that moment $m_{k}$ will hit the first green point
$\mathcal{G}_{1}^{k}$ of its orbit. Observe that $P_{k,\frac{p}{q}}$ attains
its absolute maximum at $\mathcal{G}_{1}^{k}$, that is,
$M_{k}=\mathcal{G}_{1}^{k}$. So, on the next step while $m_{k+1}$ will hit a
green point with corresponding symbol $I_{1}$, $m_{k}$ will hit the last
_blue_ point $\mathcal{B}^{k}_{q-2p-k}$ with symbol $I_{2}$.

Finally, observe that in each _round_ both $m_{k}$ and $m_{k+1}$ reach the
decreasing lapse $I_{2}$ of $H_{2}$ an even number of times. So, it follows
that the itinerary $H(m_{k})$ is _stronger_ than that of $H(m_{k+1})$. So, by
Theorem 2.1 it follows that $H(m_{k+1})>H(m_{k})$.

Similarly, it can be seen that the itinerary of $H_{2}(M_{k+1})$ is _stronger_
that for $H_{2}(M_{k})$ and hence by Theorem 2.1 the result follows. ∎

######  Lemma 3.4.

$\Gamma_{0,\frac{p}{q}}$ $\triangleleft$ $\Gamma_{1,\frac{p}{q}}$.

###### Proof.

$P_{0,\frac{p}{q}}$ has no red points. Comparing the itineraries of $m_{0}$
and $m_{1}$, it is easy to see that
$\mathcal{I}(H_{2}(m_{0}))=(I_{1},I_{2},\dots)$ while
$\mathcal{I}(H_{2}(m_{1}))=(I_{0},I_{1},\dots)$. Thus, clearly
$\mathcal{I}(H_{2}(m_{0}))$ is _stronger_ (in the sense of $\succ$ ) than
$\mathcal{I}(H_{2}(m_{1}))$. Thus, by Theorem 2.1,
$H_{2}(m_{0})>H_{2}(m_{1})$.

The _itineraries_ of $M_{0}$ and $M_{1}$ will remain the same till the moment
$M_{0}$ hits the first _blue point_ from the left in the spatial labelling
:$\mathcal{B}_{1}^{0}$ of its orbit $P_{0,\frac{p}{q}}$. At that point the
point $M_{1}$ maps to the last pink point $\mathcal{P}_{q}^{1}$ of its orbit
$P_{1,\frac{p}{q}}$ in the spatial labelling. Next application of $H_{2}$ maps
$M_{0}$ to some _pink_ point of its orbit while $M_{1}$ maps to the first _red
point_ $\mathcal{R}_{1}^{1}$ of its orbit. Arguing as in Lemma 3.3 it follows
that $H_{2}(M_{0})>H_{2}(M_{1})$ ∎

Similarly we can prove that:

######  Lemma 3.5.

$\Gamma_{q-2p-1,\frac{p}{q}}$ $\triangleleft$ $\Gamma_{q-2p,\frac{p}{q}}$

Using Lemma 3.3, 3.4, 3.5 we can describe the configuration of the _leading
set_ $L_{\frac{p}{q}}$ in $\mathcal{P}$. For this we need the following
definition:

######  Definition 3.6.

Let $A=[a,b]\times[a,b]$ where $a,b\in\mathbb{R}$. Let
$P_{1}=\\{a=x_{0},x_{1},\dots,x_{n+1}=b\\}$ and
$P_{2}=\\{a=y_{0},y_{1},\dots,y_{n+1}=b\\}$ be two _partitions_ of $[a,b]$.

By a $n$-_staircase_ $\mathcal{S}_{n}$ in $A$ with $2n$ steps denoted by
$S_{n}$ we mean:

  1. (1)

A set of $n$ _vertical line segments_ : $\\{x_{1}\\}\times[y_{1},y_{2}],$
$\\{x_{2}\\}\times[y_{2},y_{3}],$ $\dots,\\{x_{n}\\}\times[y_{n},y_{n+1}=b]$
(_called rises_).

  2. (2)

A set of $n$ _horizontal line segments_ :$[a=x_{0},x_{1}]\times\\{y_{1}\\},$
$[x_{1},x_{2}]\times\\{y_{2}\\},\dots,[x_{n-1},x_{n}]$ $\times\\{y_{n}\\}$
(_called treads_).

$x_{1}-x_{0}$ is called the length of the _initial tread_ while
$y_{n+1}-y_{n}$ is called the length of the _final rise_. The lengths $\Delta
x_{i}=x_{i+1}-x_{i}$ and $\Delta y_{i}=y_{i+1}-y_{i}$ are called the lengths
of the _ith tread_ and _ith rise_ respectively for $1\leqslant i\leqslant n$.

A staircase $S_{n}$ is said to be _symmetrical_ if $\Delta x_{i-1}=\Delta
y_{n+1-i}$ for every $i\in\\{1,2\dots n+1\\}$.

######  Theorem 3.7.

$Z_{\frac{p}{q}}$ is a symmetrical staircase with $2(q-2p+1)$ steps in
$\mathcal{P}$ with initial tread and final rise lying in the _side arm_
$\mathcal{S}$.

###### Proof.

The location of the points $m_{r}$ and $M_{r}$ of the over-twist periodic
orbits $P_{r,\frac{p}{q}}$ in $H_{2}$ is elucidated by Lemma
LABEL:criticalpoints:overtwist:location. To describe $Z_{\frac{p}{q}}$ we take
an arbitrary point $(\alpha,\beta)$ in $\mathcal{P}$ and waggle it in
$\mathcal{P}$ and look for moments when at least one of $\xi_{\alpha}^{max}$
and $\xi_{\beta}^{min}$ is part of $P_{r,\frac{p}{q}}$ for some $r$. First we
take $\alpha=1$ and $\beta=\frac{1}{2}$. At this point the map
$H_{\alpha,\beta}$ has no _non-fixed periodic_ point and hence its _dynamics_
is not interesting. No,we slowly _decrease_ the value of $\beta$ keeping
$\alpha$ fixed at $1$. There will be some unique value $\beta_{0}$ of $\beta$
when $\xi_{\beta}^{min}$ will hit $H_{2}(m_{0})$. At this moment
$\xi_{\beta}^{min}$ is part of the orbit $P_{0,\frac{p}{q}}$ while
$\xi_{\alpha}^{max}$ is not. Now, we fix $\beta=\beta_{0}$ and slowly decrease
$\alpha$. For some time only $\xi_{\beta}^{min}$ will be part of
$P_{0,\frac{p}{q}}$, but then we will encounter some unique value
$\alpha=\alpha_{0}$ of $\alpha$ when $\xi_{\alpha}^{max}$ hits $H_{2}(M_{0})$.
At the point $(\alpha_{0},\beta_{0})$ both $\xi_{\alpha}^{max}$ and
$\xi_{\beta}^{min}$ will be part of $P_{0,\frac{p}{q}}$. Now, we keep
$\alpha=\alpha_{0}$ and slowly decrease $\beta$. As $\beta$ moves away from
$\beta_{0}$, $\xi_{\beta}^{min}$ will no longer be part of $P_{0,\frac{p}{q}}$
and only $\xi_{\alpha}^{max}$ will remain part of $P_{0,\frac{p}{q}}$. This
will continue for some time. Then for some $\beta=\beta_{1}$,
$\xi_{\beta}^{min}$ will hit $H_{2}(m_{0})$. At that moment
$\xi_{\beta}^{min}$ and $\xi_{\alpha}^{max}$ will both be periodic but parts
of different orbits. While $\xi_{\beta}^{min}$ is now part of
$P_{1,\frac{p}{q}}$, $\xi_{\alpha}^{max}$ will remain part of
$P_{0,\frac{p}{q}}$. Now, keeping keeping $\beta$ fixed at $\beta_{1}$ we
decrease $\alpha$ from $\alpha_{0}$. Now, the orbit $P_{0,\frac{p}{q}}$ is
completely cut off from our map $H_{\alpha,\beta}$ and for some time only
$\xi_{\beta}^{min}$ will remain part of $P_{1,\frac{p}{q}}$ until for some
value of $\alpha_{1}$ of $\alpha$, $\xi_{\alpha}^{max}$ hits $H_{2}(M_{1})$
and both _flat spots_ become part of $P_{1,\frac{p}{q}}$. Repeating the
process $q-2p+1$ times( for each of the $q-2p+1$ _over-twist periodic orbits_
$P_{r,\frac{p}{q}}$ ), it is easy to see that $Z_{\frac{p}{q}}$ is a
_staircase_ with $2(q-2p+1)$ steps. The _symmetry_ of the _staircase_ follows
from the _symmetry_ of $H_{2}$. ∎

If $(\alpha,\beta)\in Z_{\frac{p}{q}}$, then by definition
$\psi((\alpha,\beta))=\frac{p}{q}$. The next illustrates what happens if
$\psi((\alpha,\beta))=\frac{p}{q}$ but $(\alpha,\beta)\notin Z_{\frac{p}{q}}$:

######  Lemma 3.8.

If $\psi((\alpha,\beta))=\frac{p}{q}$ but $(\alpha,\beta)\notin
Z_{\frac{p}{q}}$, then there exists $\kappa_{1},\kappa_{2}\geq 0$ such that
$(\alpha-\kappa_{1},\beta+\kappa_{2})\in Z_{\frac{p}{q}}$.

###### Proof.

Since, $\psi(\alpha,\beta)=\frac{p}{q}$, so $H_{\alpha,\beta}$ has a periodic
orbit $P$ which exhibits an over-twist pattern corresponding to over-rotation
number $\frac{p}{q}$. Also, since, $(\alpha,\beta)\notin Z_{\frac{p}{q}}$, so
$P$ lies in the interior of $[0,1]-(\xi_{max}^{\alpha}\cup\xi_{min}^{\beta})$.
Thus, there exits $\kappa_{1},\kappa_{2}\geq 0$ such that one of the flat
spots $\xi_{max}^{\alpha-\kappa_{1}}$ and $\xi_{min}^{\beta-\kappa_{2}}$ hits
a point $P$ while the other flat spot stays outside all points of $P$. Thus,
$(\alpha-\kappa_{1},\beta+\kappa_{2})\in Z_{\frac{p}{q}}$. ∎

######  Lemma 3.9.

If $\frac{p}{q},\frac{r}{s}\in\mathbb{Q}$, $g.c.d(p,q)=g.c.d(r,s)=1$ and
$\frac{p}{q}<\frac{r}{s}$, then
$d(\mathcal{F},Z_{\frac{p}{q}})<d(\mathcal{F},Z_{\frac{r}{s}})$.

###### Proof.

We first note that $\frac{p}{q}\neq\frac{r}{s}$, so $Z_{\frac{p}{q}}\cap
Z_{\frac{r}{s}}\neq 0$, since otherwise $\psi$ would become multi-valued at
the point of intersection. So, either
$d(\mathcal{F},Z_{\frac{p}{q}})<d(\mathcal{F},Z_{\frac{r}{s}})$ or
$d(\mathcal{F},Z_{\frac{p}{q}})>d(\mathcal{F},Z_{\frac{r}{s}})$. Suppose if
possible $d(\mathcal{F},Z_{\frac{p}{q}})>d(\mathcal{F},Z_{\frac{r}{s}})$.
Choose a point $(\alpha,\beta)\in Z_{\frac{p}{q}}$ and $(\gamma,\delta)\in
Z_{\frac{r}{s}}$ such that the points $(\alpha,\beta)$, $(\gamma,\delta)$ and
$\mathcal{F}$ are _co-linear_. Then ,
$d(\mathcal{F},(\alpha,\beta)>d(\mathcal{F},(\gamma,\delta))$. By Lemma 3.1
$H_{\alpha,\beta}$ _forces_ $H_{\gamma,\delta}$. Hence,
$\psi((\alpha,\beta))=\frac{p}{q}\geqslant\psi((\gamma,\delta))=\frac{r}{s}$
which is a contradiction. ∎

To show that the $\frac{p}{q}$-_Iso-over-rotation-tract_
$\mathcal{I}_{\frac{p}{q}}$ is connected, we define two more sets:

######  Definition 3.10.

Let $\mathcal{G}_{\frac{p}{q}}$ be the set of all points $(\alpha,\beta)$ in
the parameter plane $\mathcal{P}$ such that
$\psi((\alpha,\beta))\geqslant\frac{p}{q}$.

######  Definition 3.11.

Let $\mathcal{U}_{\frac{p}{q}}$ be the set of all points $(\alpha,\beta)$ in
the parameter plane $\mathcal{P}$ such that
$\psi((\alpha,\beta))<\frac{p}{q}$.

######  Theorem 3.12.

$\mathcal{U}_{\frac{p}{q}}$ is precisely the component of
$\mathcal{P}-Z_{\frac{p}{q}}$ containing the focal point $\mathcal{F}$.

###### Proof.

Let $(x,y)$ be any point in the component of $\mathcal{P}-Z_{\frac{p}{q}}$
containing the focal point $\mathcal{F}$. Then clearly,
$d((x,y),\mathcal{F})<d(Z_{\frac{p}{q}},\mathcal{F})$. So, for every $(s,t)\in
Z_{\frac{p}{q}}$, we have $d((s,t),\mathcal{F})>d((x,y),\mathcal{F})$. Hence,
$\psi((x,y))\leqslant\frac{p}{q}$. If $\psi((x,y))=\frac{p}{q}$ and
$(x,y)\notin Z_{\frac{p}{q}}$, then by Lemma 3.8, there exists
$\kappa_{1},\kappa_{2}\geq 0$ such that $(x-\kappa_{1},y+\kappa_{2})\in
Z_{\frac{p}{q}}$. Now, clearly,
$d((x-\kappa_{1},y+\kappa_{2}),\mathcal{F})<d((x,y),\mathcal{F})$ which is a
contradiction. Thus, $\psi((x,y))<\frac{p}{q}$.

Now, suppose that for some $(x,y)\in\mathcal{P}$ we have
$\psi((x,y))<\frac{p}{q}$. We claim that
$d((x,y),\mathcal{F})<d(Z_{\frac{p}{q}}$. By way of contradiction, suppose
$d((x,y),\mathcal{F})<d(Z_{\frac{p}{q}},\mathcal{F})$. Then, there exists
$(w,z)\in Z_{\frac{p}{q}}$ such that
$d((x,y),\mathcal{F})>d((w,z),\mathcal{F})$. But,
$\psi((x,y))<\frac{p}{q}=\psi((w,z))$ which contradicts Lemma 3.1. This proves
the theorem. ∎

######  Corollary 3.13.

$\mathcal{U}_{\frac{p}{q}}$ is path-connected.

###### Proof.

From Theorem 3.12 it follows that $\mathcal{U}_{\frac{p}{q}}$ is precisely the
component of $\mathcal{P}-Z_{\frac{p}{q}}$ containing the _focal point_
$\mathcal{F}$. Since, $\mathcal{P}$ is path-connected and from Theorem 3.7 it
follows that $Z_{\frac{p}{q}}$ is path-connected, so
$\mathcal{U}_{\frac{p}{q}}$ is path-connected. ∎

######  Lemma 3.14.

$\mathcal{G}_{\frac{p}{q}}$ is path-connected.

###### Proof.

Clearly, $\mathcal{G}_{\frac{p}{q}}$ = $\mathcal{P}-\mathcal{U}_{\frac{p}{q}}$
and from Corollary 3.13 $\mathcal{G}_{\frac{p}{q}}$ is path-connected. Hence,
the result follows. ∎

From Theorem 2.6, it follows that to show that the _Iso-over-rotation-tract_
$\mathcal{I}_{\frac{p}{q}}$ is connected, it is sufficient to show that
$\mathcal{I}_{\frac{p}{q}}$ is a deformation retract of
$\mathcal{G}_{\frac{p}{q}}$. This is exactly what we will show next.

######  Theorem 3.15.

For a given rational number $\frac{p}{q}$ where $p$ and $q$ are coprime,
$\mathcal{I}_{\frac{p}{q}}$ is a deformation retract of
$\mathcal{G}_{\frac{p}{q}}$.

###### Proof.

Take any point $(x,y)\in\mathcal{G}_{\frac{p}{q}}$. Then
$\psi((x,y)\geqslant\frac{p}{q}$. We consider the _path_
$\Phi_{(x,y)}:[0,1]\to\mathcal{P}$ defined by
$\Phi_{(x,y)}(t)=(x(1-t)+t,y(1-t))$ (which is the _directed line segment_ in
$\mathcal{P}$ joining $(x,y)$ to the _focal point_ $\mathcal{F}$). We now
start to move the point $(x,y)$ along the path $\Phi_{(x,y)}(t)$ towards the
focal point $\mathcal{F}$ a study happens with the value of $\psi$ as we move
along. By Lemma 3.1 it follows that the value of $\psi$ decreases
monotonically as we move along the path $\Phi_{(x,y)}$ towards the focal point
$\mathcal{F}$. So, the set
$\mathcal{T}_{\frac{p}{q}}=\\{t\in[0,1]|\psi(\Phi_{(x,y)}(t))=\frac{p}{q}\\}$
is non-empty and bounded below. Let
$\tau_{\frac{p}{q}}=inf\mathcal{T}_{\frac{p}{q}}$.

Consider the function
$\mathcal{D}_{\frac{p}{q}}:\mathcal{G}_{\frac{p}{q}}\times[0,1]\to\mathcal{G}_{\frac{p}{q}}$
defined by:

$\mathcal{D}_{\frac{p}{q}}((x,y),t)=\begin{cases}\Phi_{(x,y)}(t)&\text{if
$t<\tau_{\frac{p}{q}}$}\par\\\ \Phi_{(x,y)}(\tau_{\frac{p}{q}})&\text{if
$t\geqslant\tau_{\frac{p}{q}}$}\\\ \end{cases}$

$\mathcal{D}_{\frac{p}{q}}$ is clearly continuous and is thus a deformation
retract of $\mathcal{G}_{\frac{p}{q}}$ onto $\mathcal{I}_{\frac{p}{q}}$. ∎

######  Corollary 3.16.

For a given rational number $\frac{p}{q}$ where $p$ and $q$ are coprime, the
Iso-over-rotation-tract $\mathcal{I}_{\frac{p}{q}}$ is connected.

###### Proof.

$\mathcal{I}_{\frac{p}{q}}$ is a deformation retract of
$\mathcal{G}_{\frac{p}{q}}$ From Theorem 3.15, for a given rational number
$\frac{p}{q}$ where $p$ and $q$ are coprime, $\mathcal{I}_{\frac{p}{q}}$ is a
deformation retract of $\mathcal{G}_{\frac{p}{q}}$. Also, from Lemma 3.14
$\mathcal{G}_{\frac{p}{q}}$ is path-connected. Thus, from Theorem 2.6 the
result follows. ∎

Consider the map $\psi:\mathcal{P}\to[0,\frac{1}{2}]$. Combining results
Theorem 2.4 and [4] we have:

######  Theorem 3.17.

The map $\psi:\mathcal{P}\to[0,\frac{1}{2}]$ is continuous.

From Corollary 3.16 for each rational number $\frac{p}{q}\in[0,\frac{1}{2}]$,
$\psi^{-1}(\frac{p}{q})$ is connected.

Now, take an irrational number $\mu\in[0,\frac{1}{2}]$. We now show that
$\psi^{-1}(\mu)$ is also connected.

######  Theorem 3.18.

For each irrational number $\mu\in[0,\frac{1}{2}]$ the Iso-over-rotation-tract
$\mathcal{I}_{\mu}=\psi^{-1}(\mu)$ is connected.

###### Proof.

Consider the sets
$\mathcal{G}_{\mu}=\displaystyle\bigcup_{\frac{p}{q}>\mu}\mathcal{G}_{\frac{p}{q}}$
and
$\mathcal{U}_{\mu}=\displaystyle\bigcup_{\frac{p}{q}<\mu}\mathcal{U}_{\frac{p}{q}}$.
It follows from Theorem 3.13 and Corollary 3.14 that the sets
$\mathcal{G}_{\mu}$ and $\mathcal{U}_{\mu}$ are connected. Hence,
$\mathcal{A}=\mathcal{P}-(\mathcal{G}_{\mu}\cup\mathcal{U}_{\mu})$ is
connected. From Theorem 3.17 it readily follows that $\mathcal{A}$ is
precisely the set of points $(x,y)$ of $\mathcal{P}$ such that
$\psi((x,y))=\mu$ and hence the result follows. ∎

From Corollary 3.16 and Theorem 3.18 we finally have the main result of this
paper:

######  Corollary 3.19.

The map $\psi:\mathcal{P}\to[0,\frac{1}{2}]$ is monotone.

## References

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# Thermal Analysis of Malignant Brain Tumors by Employing a Morphological
Differentiation-Based Method in Conjunction with Artificial Neural Network

Hamed Hani
Department of Integrated Systems Engineering
The Ohio State University
Columbus, OH 43210
<EMAIL_ADDRESS>
Afsaneh Mojra
Department of Mechanical Engineering
K.N. Toosi University of Technology
Tehran, Iran
<EMAIL_ADDRESS>
Spine Research Institute: spine.osu.edu

###### Abstract

In this study, a morphological differentiation-based method has been
introduced which employs temperature distribution on the tissue surface to
detect brain tumor’s malignancy. According to the common tumor CT scans, two
different scenarios have been implemented to describe irregular shape of the
malignant tumor. In the first scenario, tumor has been considered as a polygon
base prism and in the second one, it has been considered as a star-shaped base
prism. By increasing the number of sides of the polygon or wings of the star,
degree of the malignancy has been increased. Constant heat generation has been
considered for the tumor and finite element analysis has been conducted by the
ABAQUS software linked with a PYTHON script on both tumor models to study
temperature variations on the top tissue surface. This temperature
distribution has been characterized by 10 parameters. In each scenario, 98
sets of these parameters has been used as inputs of a radial basis function
neural network (RBFNN) and number of sides or wings has been selected to be
the output. The RBFNN has been trained to identify malignancy of tumor based
on its morphology. According to the RBFNN results, the proposed method has
been capable of differentiating between benign and malignant tumors and
estimating the degree of malignancy with high accuracy.

_Keywords_ Brain tumor $\cdot$ Tumor differentiation $\cdot$ Morphological
analysis $\cdot$ Artificial neural network $\cdot$ Finite element method

##  1 Introduction

Brain tumors are categorized into benign and malignant. Despite considerable
advances in the diagnosis and the treatment of the brain tumors, the mortality
from malignant brain tumors is still high Kateb et al. (2009). Malignant
tumors are cancerous and are made up of cells that grow out of control Baish
and Jain (2000). It has been found that cells in the peripheral areas of the
malignant tumor have a high invasive rhythm to the surrounding tissue Guarino
et al. (2007). Therefore, border of a malignant tumor has sharp edges that
result in a polygonal shape or star-shaped morphology of the tumor Golston et
al. (1992). Malignant tumors are deeply fixed in the surrounding tissue by the
sharp edges. The sharpness increases rapidly since the tumor needs more
invasions to the nearby tissue for the growth Condeelis and Pollard (2006). On
the contrary, benign tumors are often smooth and round and easy to remove
since they are not attached to the surrounding tissue Shah et al. (1995).
Surgery is usually the first step in the treatment of the brain tumors. The
goal is to remove as much of the tumor as possible while maintaining
neurological function. Surgical treatments for malignant brain tumors are not
easy to handle because they do not have clear borders Argani et al. (2001). In
order to improve the surgeon’s ability in defining the tumor’s border and
avoid injury to vital brain areas in the operating room, image-guided surgery
is performed. Intraoperative imaging is a revolutionary tool in the modern
neurosurgery Illingworth (1995). Intraoperative imaging techniques especially
intraoperative MRI (iMRI), help neurosurgeons achieve the goal of maximum
tumor resection with least morbidity Schulder and Carmel (2003). The main
drawbacks of using iMRI is the patient positioning during the surgery for a
proper imaging and also limitation of using the surgical instruments because
of the presence of a strong magnetic field.
In order to avoid limitations and high expenses of intraoperative imaging,
many researches have focused on improving the performance of the preoperative
tumor imaging techniques. These techniques mainly include magnetic resonance
imaging (MRI) and computed tomography scan (CT scan). In the MRI, an injected
contrast agent is used which makes the cancerous tumor brighter than the
surrounding normal tissue. During the scan, there is a rapid increase in the
signal intensity of a malignant tumor immediately 1 to 2 minutes after the
injection. The intensity decreases in the following minutesKobayashi and
Brechbiel (2005). For a benign mass, the rise in the intensity is much slower.
Inaccuracy in the time and the intensity measurements results in a probability
that benign and malignant tumors have overlap in their morphological
appearance Barentsz et al. (1996). It was proved that the specificity of MRI
to correctly predict a benign tumor is limited. Moreover, Specificity of MRI
would be decreased by reducing the tumor size. Therefore, MRI is usually
recommended after a malignancy is detected by other methods in order to have
more information about the extent of the cancer.
CAT or CT scanning is an accurate medical test that combines x-ray with
computerized technology to detect malignancy. The main drawback of this method
is using high doses of radiation, leading to the possibility of lung cancer or
breast cancer as a consequence Lee et al. (2004). X-rays also damage DNA
itself Spotheim-Maurizot and Davidkova (2011). CT scanning provides images in
shades of grey; occasionally the shades are similar, making it difficult to
distinguish between the normal and abnormal tissues. To overcome such
deficiency a contrast agent may be injected into the bloodstream. Main
problems of the injection include pathological side-effects such as nausea and
vomiting, hypotension and extravasation of the contrast which can be severe
enough to require skin grafting Rull and Tidy (2015).
During recent years, many researches have focused on improving the procedure
of defining tumor morphology in the preoperative imaging techniques. Wu et al.
(2012) used the level set method to segment ultrasound breast tumors
automatically and used a genetic algorithm to detect indicative features for
the support vector machine (SVM) to detect tumor malignancy. The proposed
system could discriminate benign from malignant breast tumors with high
accuracy and short feature extraction time. Huang et al. (2013) evaluated the
value of using 3D breast MRI morphological features to differentiate malignant
and benign breast tumor. Malignancy of a tumor was investigated by using a
number of extracted morphological features in the breast MRI. Jen and Yu
(2015) introduced a method for abnormality detection in the mammograms based
on an abnormality detection classifier (ADC) which extracts a couple of
distinctive features including first-order statistical intensities and
gradients. In this method, image preprocessing techniques were used to obtain
more accurate breast tissue segmentation. Han et al. (2015) provided an
improved segmentation algorithm which is a combination of the fuzzy clustering
segmentation and the fuzzy edge enhancement. Results showed that the fuzzy
clustering segmentation is highly efficient for complex brain tissues, and the
images after the fuzzy clustering segmentation provide a solid foundation for
3D processing and help to acquire better 3D visualization of the brain tumors.
Ramya and Sasirekha (2015) developed a robust segmentation algorithm in order
to diagnose tumors in the MR images. In this method, a 4th-order partial
differential equation was employed to denoise images to improve segmentation
accuracy. Zhang et al. (2016) proposed a wavelet energy-based method to
classify the MR images. This approach had a three-stage system which detects
characteristics indicative of the abnormal brain tissues. Shirazi and Rashedi
(2016) used a combination of support vector machine (SVM) and mixed
gravitational search algorithm (MGSA) to improve the classification accuracy
in the mammography images. Xia et al. (2016) proposed a novel voting ranking
random forests (VRRF) method for the image classification and developed a
center-proliferation segmentation (CPS) method. This method showed good
performance in the image classification with strong robustness.
In the present study, a palpation-based method was used for scanning of the
brain tissue in order to detect and follow the malignancies. The method avoids
the main aforementioned drawbacks of the imaging techniques since it is only
based on the tissue palpation. It is called “tactile thermography” and maps
the thermal parameters of the tissue mainly the temperature and the heat flux
on the tissue surface. Sadeghi-Goughari and Mojra (2015) estimated thermal
parameters of the brain tumors by introducing the tactile thermography method
as a new noninvasive thermal imaging method. In this method, the brain tissue
was mechanically and thermally loaded and the temperature and the heat flux
variations on the tissue surfaces were recorded. A number of thermal
parameters were extracted and optimized by an artificial neural network to
verify tumor existence and its depth. The main objective of this study is to
evaluate the capability of the proposed method in detecting sharp morphology
of the tumor indicative of the tumor malignancy and also to verify its
sensitivity to the sharpness increase that can be used in the follow-up
procedure for evaluating the tumor growth.
To this end, a malignant tumor is simulated in the brain tissue by two
scenarios that represent the invading sharp edges of the tumor. Moreover, the
tumor is considered as a heat source in the thermal analysis since the cell
number and the overall cell metabolism are considerably increased in the tumor
relative to the normal tissue. By conducting a thermal analysis, the
temperature distribution on the tissue surface is obtained for tumors with
varying penetration into the surrounding tissue. By the use of the artificial
neural network, a number of variables are extracted from the temperature map
and will be used as the malignancy indicative features.

##  2 Materials and Methods

Figure 1 and Figure 2 are CT scan images of two malignant brain tumors. It can
be inferred that while a benign tumor has an almost smooth round shape,
malignancy can be identified by the existence of invading edges. Two major
morphologies of the malignant brain tumors are identified by solid lines in
Figure 1 and Figure 2. The first morphology resembles a polygonal shape while
the second one resembles a star. In the present numerical analysis, the tumor
malignancy was simulated in two different scenarios:

  1. 1.

Tumor was considered as an n-sided polygonal based prism. Area of the
polygonal base and consequently the volume of the tumor and the distance of
uppermost vertex (D) were kept unchanged for all number of sides (n) (Figure
3).

  2. 2.

Tumor was considered as a star polygonal based prism. This star was formed by
an inscribed circle with constant radius of R=10mm and different corner
vertices. Tumor penetration inside the nearby tissue was increased by
increasing the number of corner vertices. Each group of two intersecting edges
was called a wing. The number of wings was increased by keeping the star
polygonal base area constant (and the tumor volume) (Figure 4).

Table 1: Dimensions of simulated tissue containing a malignant tumor, all dimensions are in millimeters. X | Y | Z | D | R
---|---|---|---|---
120 | 60 | 25 | 12 | 10
Figure 1: CT scan image of a malignant tumor resembles a polygon presented as
scenario 1. Figure 2: CT scan image of a malignant tumor resembles a star
presented as scenario 2.

(a)

(b)

Figure 3: Mid cross section of the brain tissue model (rectangular cuboid) in
the ABAQUS environment including (a) a pentagonal based prismatic tumor; (b) a
decagonal based prismatic tumor.

(a)

(b)

Figure 4: Mid cross section of the brain tissue model (rectangular cuboid) in
the ABAQUS environment including (a) a 4 wing star polygonal based prismatic
tumor; (b) a 10 wing star polygonal based prismatic tumor.

Geometrical dimensions of the sample brain tissue containing a malignant tumor
is presented in Table 1. In order to describe mechanical behavior of the brain
tissue, the elasticity parameters of the brain tissue obtained by Soza et al.
(2005)) were used(Table 2). The Young’s modulus of tumor was considered to be
10 times of the brain tissue while the Poisson’s ratio was the same Shiddiqi
et al. (2010).

Table 2: Elastic properties of the brain tissue. Young’s modulus (Pa) | Poisson’s ratio
---|---
9210.87 | 0.458344

Study of energy transport in the biological systems involves various
mechanisms including conduction, advection and metabolism Gore and Surawicz
(2003). In this study these phenomena were considered in a steady-state
thermal analysis; a constant thermal conductivity equal to 0.6 W/m2K was
considered for the brain tissue Elwassif et al. (2006), blood perfusion and
metabolic activities were considered by assuming a constant heat generation
equal to 100000 W/m3 for the tumor. A convective heat transfer between the top
tissue surface and the surrounding environment was considered with a
convective heat transfer coefficient equals 20 W/m2K. The bottom surface of
the brain tissue sample was assumed to have a constant temperature of 33.1°C
equal to the temperature of the blood vessel which was assumed to be in
contact with the tissue. Side surfaces were insulated.
ABAQUS software (version 6.14) was employed to perform finite element analysis
(FEA) under 3D axisymmetric conditions. A compressive strain was applied to
the top surface of the tissue and the temperature variation was recorded on
it. In order to measure temperature variations while the tissue was loaded
mechanically, the step named “COUPLED TEMP-DISPLACEMENT” was selected. A “time
increment” and a “time period” should be assigned to the step for that the
default values were assumed. For both rectangular cube and prism, a 4-noded
tetrahedral with thermal coupling (C3D4T) element type was used.
Rather than thermal boundary conditions, mechanical boundary conditions were
also defined. The bottom surface of the tissue was totally fixed while the top
surface was loaded by a compressive strain equals to 6% which corresponds to a
4 mm compression. Adhesion of the malignant tumor to its surrounding tissue
was satisfied by applying ‘TIE’ constraint between the brain tissue and the
tumor.
Mesh independency was examined for the tissue sample including a decagonal
based prismatic tumor. Three stages of mesh refinement were performed and
temperature values were measured. For three models with 21796, 29725 and 44029
elements, the computational time was 19.27, 23.44 and 29.82 seconds,
respectively. The maximum relative difference between the temperature values
in the models with 21796 elements and 29725 elements was less than 1%, so the
computational grid with 21796 elements was selected (Table 3).

Table 3: Mesh independency evaluation for the sample brain tissue including a malignant tumor. decagonal based prismatic tumor | 10 wing star polygonal based prismatic tumor
---|---
Number of elements |  Element type |  Maximum error relative to selected model (%) |  Computational time (sec) |  Number of elements |  Element type |  Maximum error relative to selected model (%) |  Computational time (sec)
21796 |  C3D4T |  0 |  19.27 |  22362 |  C3D4T |  0 |  21.49
29725 |  C3D4T |  0.0751 |  23.44 |  29869 |  C3D4T |  0.0665 |  23.56
44029 |  C3D4T |  0.0382 |  29.82 |  43366 |  C3D4T |  0.0269 |  30.81

Similarly, mesh independency was examined for the tissue sample including the
star polygonal based prismatic tumor with 10 wings. For three models with
22362, 29869 and 43366 elements, the computational time was 21.49, 23.56 and
30.81 seconds, respectively. The maximum relative error between the
temperature values in the models with 22362 elements and 29869 elements was
less than 1%, so the computational mesh with 22362 elements was selected
(Table 3).
The sample model of the brain tissue was numerically analyzed while a tumor
with irregular borders was included as a prism with a polygonal base or a star
polygonal base. The number of sides of the polygon and wings of the star were
changed to the study the effects of the irregularity increase on the thermal
parameters. Table 4 lists the number of sides and wings of the tumor model. A
total number of 98 polygonal based prisms and 98 star polygonal based prisms
were modeled and analyzed by the ABAQUS software linked with a PYTHON script
to automatically change the number of sides and wings. MATLAB software
(version 8.6) was used to plot the thermal outputs and extract thermal
variables that would be employed as the inputs of an artificial neural
network.

Table 4: Number of sides and wings of both tumor models. Initial value | Increment | Final value
---|---|---
3 | 1 | 100

###  2.1 Artificial neural network

Artificial neural networks (ANNS) try to model the processing capabilities of
actual human nervous systems. They are consist of numerous simple computing
components, called neurons that generates an input layer, one or more hidden
layers and an output layer Steuber and Jaeger (2013). ANNs are mostly used for
functions approximation which may be single variable or multivariable. In
order to recognize inputs patterns, suitable weights for the connections must
be derived. Therefore, the network should be trained to obtain proper desired
results.
In this study, radial base function neural network (RBF) was used that employs
the radial basis functions as the activation functions Park and Sandberg
(1991). The output of this network is a linear combination of neuron
parameters and radial basis function of the inputs. Properties of the employed
network are listed in Table 5.

Table 5: Neural network’s properties Performance function |  Network initialization function |  Network derivative function |  Number of inputs |  Number of outputs |  Transfer function |  Type of neurons in 1st layer |  Type of neurons in 2nd layer |  Number of epochs
---|---|---|---|---|---|---|---|---
MSE |  initlay |  defaultderiv |  10 |  1 |  Tansig |  Radbas |  Purelin |  100

Performance of this network was evaluated by mean squared error (MSE) that is
the average of the squares of the errors and error is the difference between
the desired value of a variable and the estimated value by the network
(equation 3). The ideal performance of a neural network occurs when values of
the MSE are zero. The error is defined as equation 1:

$e_{i}=\hat{x}_{i}-x_{i}$ (1)

Where $\hat{x}_{i}$ is the estimated value and $x_{i}$ is the desired value
that is obtained from the ABAQUS runs in this study. Mean of the errors for
all data (98 data for each tumor model) is then calculated by equation 2:

$\mu=\frac{1}{n}\sum_{i=1}^{n}e_{i}$ (2)
$MSE=\frac{1}{n}\sum_{i=1}^{n}e_{i}^{2}$ (3)

Where $n$ is the number of the network inputs. The variance is the average of
the squared differences from the mean (equation 4):

$\sigma^{2}=\frac{1}{n}\sum_{i=1}^{n}(e_{i}-\mu)^{2}$ (4)

In order to have a better criterion for evaluating the network performance,
it’s better to use the root mean square error (RMSE) which has the same unit
as the variable being estimated (equation 5). The square root of the variance
is known as the standard deviation (equation 6).

$RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}e_{i}^{2}}$ (5)

$\sigma=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(e_{i}-\mu)^{2}}$ (6)

##  3 Results and Discussion

Figure 5 is a display of the temperature contours in the mid cross section of
the model obtained from the ABAQUS runs. Tumor existence in the tissue
increases the temperature in the vicinity of the tumor. At further distance
from the tumor margin, the temperature reduces and the rate of reduction is
not proportional to the distance from the tumor center. Temperature gradients
are considerable in the tumor vicinity. It can be also inferred from Figure 7
that the area of the tumor-affected region depends on the tumor shape to a
great extent. While the tumor effect spreads smoothly over the whole tissue
area for a polygonal based prismatic tumor, it has a localized distribution
for a star polygon. Consequently, temperature distribution pattern can be
correlated with the shape of the malignant tumor.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5: Temperature contours in the mid cross section of the brain tissue
model including: a pentagonal, heptagonal, decagonal and 15 sided polygonal
based prismatic tumor (top from left to right) and a 5 wing, 7 wing, 10 wing,
and 30 wing star polygonal based prismatic tumor (bottom from left to right).

Temperature variation was studied on the paths defined in Figure 3(a) and
Figure 4a. and by varying the number of sides and wings of the tumor (Figure
4(a). The diagrams show that the location of the maximum temperature on the
tissue surface corresponds to the location of the tumor center inside the
tissue where the tumor distance is the minimum relative to the tissue surface.
Moreover, while the tumor volume was kept unchanged, increasing the number of
sides and wings results in the elevation of the maximum surface temperature.
These achievements offer two opportunities:

  1. 1.

The temperature variation is indicative of the tumor existence. Therefore,
temperature map can be used for the tumor detection task.

  2. 2.

For a specific malignant tumor, the temperature map can be recorded in the
successive examinations. Comparison between the maps can be indicative of the
malignancy progression in a period of time.

(a)

(b)

Figure 6: Temperature distribution on a path which passes from the center of
the top tissue surface for brain tissue sample including a) polygonal based
prismatic tumors; b) star polygonal based prismatic tumors.

In Figure 7 variation of the maximum temperature on the tissue surface is
investigated by increasing the number of sides and wings. For the polygonal
based tumor model, increase of the number of sides from 3 to 100 results in
the elevation of the maximum surface temperature from 29.7°C to 30.5°C. For
the star polygonal based tumor, by increasing the number of the wings from 3
to 100, maximum surface temperature increases from 30.3°C to 30.8°C. However,
maximum temperature variations are less than 0.02°C and 0.01°C by increasing
the number of sides and wings more than 20, respectively. Therefore,
sensitivity of the surface temperature to the variation of the sides and wings
of the tumor reduces when the sharpness of the tumor morphology increases.

Figure 7: Correlation between the maximum surface temperature and number of
sides of the polygonal based prismatic tumor and number of wings of the star
polygonal based prismatic tumor.

In order to find a quantitative criterion for the tumor detection and the
malignancy progression, the temperature curve on the tissue surface was
interpolated by a 4th-order Fourier series (equation 7). Fitting error was
less than 1% for all tumor models. By fitting this curve to each set of data
achieved from the ABAQUS runs, 10 coefficients were extracted for each tumor
which are $a_{i}^{\prime}$ s,$b_{i}^{\prime}$ s and w in equation 7.

$T(x)=a_{0}+\sum_{i=1}^{4}(a_{i}coswx+b_{i}sinwx)$ (7)

These coefficients were used as the inputs of a radial basis function (RBF)
artificial neural network (ANN). This network (RBFNN) provided the link
between the inputs that were the surface temperature interpolating function’s
coefficients and outputs that were the corresponding number of sides or wings
of the malignant tumor.

###  3.1 Polygonal based prismatic tumor

The RBFNN was trained to estimate number of sides by using 98 samples of the
brain tissue including a polygonal based prismatic tumor with different number
of sides of the polygonal base. 68 samples from the whole datasets were used
randomly for training the network and 30 remaining samples were used for the
testing procedure.

For a better comprehension of the extent and the distribution of the
coefficients of equation 7, the mean $\bar{x}$ the minimum $x_{min}$, and
maximum $x_{max}$ of these coefficients for the polygonal base tumor model are
listed in Table 6. The employed neural network transfer function was
"Tansig"that takes values only between $-1$ and $1$. Therefore, the
coefficients were normalized in the range of $[-11]$. Normalization of the
dataset would prevent misinterpretation in defining the contribution of each
input in the neural network.

Table 6: Variation of the Fourier series coefficients for the polygonal based prismatic tumor. Coefficient | $x_{min}$ | $\bar{x}$ | $x_{max}$
---|---|---|---
$a_{0}$ | 28.8860 | 29.1522 | 29.1666
$a_{1}$ | 0.6511 | 0.9437 | 0.9583
$a_{2}$ | 0.1349 | 0.2394 | 0.2462
$a_{3}$ | 0.0326 | 0.0701 | 0.0737
$a_{4}$ | 0.0101 | 0.0203 | 0.0223
$b_{1}$ | 0.0017 | 0.0042 | 0.0062
$b_{2}$ | 0.000983 | 0.0034 | 0.0059
$b_{3}$ | -0.000574 | 0.0018 | 0.0043
$b_{4}$ | -0.000914 | 0.0010 | 0.0031
$w$ | 51.7375 | 52.6726 | 53.0506

The box plot of the normalized coefficients is plotted in Figure 8 that is a
standardized way of displaying the distribution of data based on five
characteristics: minimum, maximum, median, first quartile, and third quartile.
In a box plot, the rectangle expands from the first quartile to the third
quartile and the inner line represents the median. A segment inside the
rectangle shows the median and whiskers above and below the box show the
minimum and the maximum of the data. It can be inferred from Figure10 that
$a_{0}$,$a_{1}$,$a_{2}$ and $a_{3}$ have the most compact distributions which
means that these coefficients are not too sensitive to the variation of the
number of sides and consequently do not have major contribution in the
proposed network training. Contribution of these coefficients in the network
is providing a general similarity between the patterns of temperature
variations. On the contrary, $b_{1}$,$b_{3}$ and $b_{4}$ have the widest
distributions. Moreover, based on the position of the median, a coefficient
may have normal or abnormal distribution. According to the box plot,
$a_{2}$,$a_{4}$, $b_{3}$, and $b_{4}$ have the most normal distributions that
mean that by varying the number of sides, distribution of these coefficients
is not concentrated below or above the medians. Normal distribution
facilitates the network training and reduces the normal deviations.
Distribution of $a_{0}$, $a_{1}$ and $a_{3}$ are far from being normal.

Figure 8: Box plot of the normalized coefficients for the polygon base
prismatic tumor. Figure 9: Performance plots of data training by the proposed
RBFNN for the polygonal based prismatic tumor.

Performance of the proposed RBFNN in data training was evaluated by four
different plots in Figure9. The top left figure is a plot of the network
output that has overlap with the corresponding real values. The top right
figure is a plot the linear regression of the real or desired values versus
the network outputs. Slope of the fitted line by the RBFNN was close to 1 that
means that the estimated values are very close to the real values. RMSE was
equal to $4.142×10^{-13}$. Table 7 provides the value of the evaluating
parameters for both training and testing datasets.

Table 7: Evaluating parameters of the proposed RBFNN for the polygonal based prismatic tumor. Training dataset | Testing dataset
---|---
RMSE | $\mu$ | $\sigma$ | RMSE | $\mu$ | $\sigma$
$4.142×10^{-13}$ | $2.099×10^{-14}$ | $4.167×10^{-13}$ | $3.310×10^{-12}$ | $-1.937×10^{-12}$ | $2.727×10^{-12}$

###  3.2 Star polygonal based prismatic tumor

For the second tumor model, the RBFNN was trained to estimate the number of
tumor wings by using 98 samples of the tissue including star polygonal based
prismatic tumors. Number of the training and testing datasets were 68 and 30,
respectively. Mean $\bar{x}$ , minimum $x_{mean}$ , and maximum $x_{max}$ of
the extracted coefficients are listed in Table 8.

Table 8: Variation of the Fourier series coefficients for the star polygonal based prismatic tumor. Coefficient | $x_{min}$ | $\bar{x}$ | $x_{max}$
---|---|---|---
$a_{1}$ | 0.8480 | 1.0025 | 1.0119
$a_{2}$ | 0.1652 | 0.2044 | 0.2118
$a_{3}$ | 0.0370 | 0.0441 | 0.0470
$a_{4}$ | 0.0087 | 0.0124 | 0.0155
$b_{1}$ | 0.0015 | 0.0039 | 0.0060
$b_{2}$ | 0.00019 | 0.0030 | 0.0055
$b_{3}$ | -0.0015 | 0.0012 | 0.0043
$b_{4}$ | -0.0019 | 0.0007 | 0.0044
$w$ | 51.2349 | 51.6295 | 52.1607

The box plot of the normalized coefficients is plotted in Figure 10. Similar
to the polygonal based prismatic tumor, the box plot shows that
$a_{0}$,$a_{1}$,and $a_{2}$ have the most compact distributions and $a_{4}$,
$b_{1}$ and $b_{2}$ have the most normal distributions.

Figure 10: Box plot of the normalized coefficients for the star-shaped base
prismatic tumor.

Performance of the RBFNN was evaluated by the prescribed parameters for both
training and testing datasets (Table9). For both malignant tumor models,
R-value was close to 1 and insignificant values of RMSE are indicative of
perfect fit of the designed network to the numerical data. RMSE, errors mean
($\mu$) and standard deviation ($\sigma$) are almost zero.

Table 9: Evaluating parameters of the proposed RBFNN for the star polygonal based prismatic tumor. Training dataset | Testing dataset
---|---
RMSE | $\mu$ | $\sigma$ | RMSE | $\mu$ | $\sigma$
$7.028×10^{-13}$ | $-2.099×10^{-13}$ | $6.742×10^{-13}$ | $2.846×10^{-12}$ | $1.987×10^{-12}$ | $2.072×10^{-12}$

##  4 Conclusion

In the present study, a morphologically malignant tumor was simulated in a
cuboid sample of the brain tissue and thermal effect of the tumor on the
surrounding tissue was investigated. Considering CT-scan images, morphology of
the malignant tumor was defined by two major scenarios; polygonal based
prismatic tumor and star polygonal based prismatic tumor. Main characteristic
of both morphologies is having corner vertices and multiple edges. The tumor
was considered as a biological heat source in the tissue and the temperature
map on the tissue surface was obtained. An interpolating function was fitted
to the temperature map and ten distinct variables were extracted. The tumor
growth and malignancy progression was linked to the increase of the corner
vertices of the tumor. Subsequently, 98 polygonal based prismatic tumors with
different number of sides and 98 star polygonal based prismatic tumors with
different number of wings were modeled and thermally analyzed. The
aforementioned variables were extracted for all tumor models. Numerical
results showed that the temperature of the normal tissue is affected by the
tumor existence and the pattern of temperature variation has agreement with
the tumor morphology. Moreover, the extracted thermal variables for all tumor
models were used as the inputs of a radial base function neural network
(RBFNN) and the number of sides and wings were estimated. The RBFNN analysis
offered that the proposed method has the potential to be employed as a
quantitative tool for measuring the malignancy progression over time.

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