Stability and chaos in real polynomial maps

Thesis presented by
Fermín Franco-Medrano, B.Sc.
As a requirement to obtain the degree of
Master of Science with specialty in Applied Mathematics
conferred by
Centro de Investigación en Matemáticas, A.C.
Thesis Advisor:
Francisco Javier Solís Lozano, Ph.D.
Guanajuato, GT, United Mexican States, June 2013.

##

Stability and chaos in real polynomial maps

Thesis presented by
Fermín Franco-Medrano, B.Sc.
As a requirement to obtain the degree of
Master of Science with specialty in Applied Mathematics
conferred by
Centro de Investigación en Matemáticas, A.C.
Approved by the Thesis Advisor:

Francisco Javier Solís Lozano, Ph.D.
Guanajuato, GT, United Mexican States, June 2013.

## Abstract

We extend and improve the existing characterization of the dynamics of general
quadratic real polynomial maps with coefficients that depend on a single
parameter $\lambda$, and generalize this characterization to cubic real
polynomial maps, in a consistent theory that is further generalized to real
$n$-th degree real polynomial maps. In essence, we give conditions for the
stability of the fixed points of any real polynomial map with real fixed
points. In order to do this, we have introduced the concept of _Canonical
Polynomial Maps_ which are topologically conjugate to any polynomial map of
the same degree with real fixed points. The stability of the fixed points of
canonical polynomial maps has been found to depend solely on a special
function termed _Product Distance Function_ for a given fixed point. The
values of this product distance determine the stability of the fixed point in
question, when it bifurcates, and even when chaos arises, as it passes through
what we have termed _stability bands_. The exact boundary values of these
stability bands are yet to be calculated for regions of type greater than one
for polynomials of degree higher than three.

## Acknowledgements

I have both the duty and pleasure of thanking the support of many people and
institutions that helped me in the process of conducting my graduate studies.
In the first place, I greatly thank the support of the National Council of
Science and Technology of the United Mexican States (CONACYT) for their
financial support as a national scholar during my studies. I thank my teachers
and the personnel of the Center for Mathematical Research (CIMAT) in
Guanajuato, for their patient and committed work from which I nurtured and
which allowed me to grow in this Mexican institution. In particular, I thank
the orientation and guidance of Professor Francisco Javier Solís Lozano, my
thesis advisor; as well as Professors Lázaro Raúl Felipe Parada, Mónica Moreno
Rocha and Daniel Olmos Liceaga, who contributed greatly to the improvement of
this work with their comments and advice; and Dr. Marcos Aurelio Capistrán
Ocampo for his support and orientation. I also thank Mr. José Guadalupe Castro
López and my friends and classmates for their support and invaluable company.
To my family and particularly to my mother, I must extend the greatest
gratitude of all.

###### Contents

  1. 1 Introduction
    1. 1.1 Elementary concepts
    2. 1.2 Hyperbolicity and criteria for stability
    3. 1.3 The Schwarzian derivative
    4. 1.4 Nonhyperbolic fixed points
    5. 1.5 Sarkovskii’s theorem
    6. 1.6 Bifurcation theory
    7. 1.7 Chaos
    8. 1.8 The period doubling route to chaos
    9. 1.9 Topological conjugacy
  2. 2 Regular-Reversal quadratic maps
    1. 2.1 Unimodal maps
    2. 2.2 Quadratic maps
    3. 2.3 Types of periodic points and chaos
    4. 2.4 Quadratic examples
  3. 3 Quadratic maps revisited
    1. 3.1 General quadratic form
    2. 3.2 Linear factors form
    3. 3.3 Canonical form
    4. 3.4 Stability and chaos in the canonical quadratic map
      1. 3.4.1 Fixed points
      2. 3.4.2 Periodic points of period two
      3. 3.4.3 Periodic points of higher period
      4. 3.4.4 Fixed points with multiplicity
      5. 3.4.5 Regular-reversality and chaos in the canonical quadratic map
    5. 3.5 Quadratic examples with the canonical map
      1. 3.5.1 Linear fixed points
      2. 3.5.2 Quadratic fixed point
      3. 3.5.3 Sinusoidal fixed point
      4. 3.5.4 When zero loses its stability
      5. 3.5.5 Asymptotic fixed point
  4. 4 Cubic maps
    1. 4.1 General Cubic Map
    2. 4.2 Linear Factors Product Form
    3. 4.3 Canonical Cubic Map
    4. 4.4 Stability and Chaos in the Canonical Cubic Map
      1. 4.4.1 Fixed Points
    5. 4.5 Higher period periodic points
    6. 4.6 Multiplicity of the fixed points
    7. 4.7 Cubic examples
  5. 5 n-th degree polynomial maps
    1. 5.1 Stability and chaos in the canonical map of degree n
    2. 5.2 Example
  6. 6 An example application of the canonical map theory
    1. 6.1 The logistic map
    2. 6.2 Harvesting strategies
    3. 6.3 Remarks on the application of the method
  7. 7 Conclusions and future work
  8. A Computational and numerical tools

##  Chapter 1 Introduction

In this chapter, we will briefly address the basic concepts of the theory of
discrete dynamical systems, so as the concept of chaos that can occur in these
systems, as much as necessary to define the relevant problem and the goals of
this thesis. Important references in this subject are [Devaney(1989),
Holmgren(1994), Elaydi(2000)], among others, and all of the proofs omitted
here can be found therein. This chapter is included with the aim of making the
work self-contained, it includes only what is now the standard basic theory of
discrete dynamical systems, and may be skipped if the reader is familiar with
it.

This chapter has the following objectives: (_i_) familiarize with the
fundamentals of the theory of discrete dynamical systems, particularly chaotic
ones; (_ii_) show the importance of the parametric dependency in families of
functions, particularly bifurcation theory; and (_iii_) present the Feigenbaum
sequence and its universality property for unimodal maps as a basis to
understand how chaos may arise through period doubling bifurcations.

###  1.1 Elementary concepts

The main objective of the theory of _discrete dynamical systems_ is to
understand the _final_ or _asymptotic_ behavior of an iterative process. If
the process is continuous, a differential equation with time being the
independent variable, then the theory intends to predict the ultimate behavior
of the solutions in the distant future ($t\rightarrow\infty$) or the distant
past ($t\rightarrow-\infty$). If the process is discrete, such as the
iteration of a real (or complex) function $f$, then the theory hopes to be
able to understand the eventual behavior of the points
$x,f(x),f^{2}(x),...,f^{n}(x)$ when $n\in\mathbb{N}$ is large, where
$f^{n}(x)$ denotes the $n$-th iteration (i.e., composition of the function
with itself). That is, the theory of dynamical systems asks the question:
where do points go and what do they do when they get there? In this work, we
aim to answer a part of this question for one of the simplest kinds of
dynamical systems: functions of a single real variable; in particular, for
real polynomial functions, having as a precedent the work of [Solís and
Jódar(2004)] for quadratic maps.

In the following, let $A\subseteq\mathbb{R}$, $x\in A$ and
$f:A\rightarrow\mathbb{R}$.

######  Definition 1.1 (Orbits).

The _forward orbit_ of $x$ is the set of points $x,\,f(x),\,f^{2}(x),\,...$
and is denoted by $O^{+}(x)$. If $f$ is a homeomorphism, we can define the
_complete orbit_ of $x$, $O(x)$, as the set of points $f^{n}(x)$ for
$n\in\mathbb{Z}$, and the _backward orbit_ of $x$, $O^{-}(x)$, as the set of
points $x,\,f^{-1}(x),\,f^{-2}(x),\,...$.

Therefore, we can restate the basic goal of discrete dynamical systems as to
know and understand all orbits of a map. Orbits can be quite complicated sets,
even for simple non-linear maps. However, there are some specially simple
orbits which play a central role in the dynamics of the whole system.

######  Definition 1.2 (Periodic Points).

The point $x$ is a fixed point of $f$ if $f(x)=x$. The point $x$ is a
_periodic point of period $n$_ if $f^{n}(x)=x$. The smallest positive $n$ for
which $f^{n}(x)=x$ is called the _prime period_ of $x$. We denote the set of
periodic points of period $n$ (not necessarily prime) by
$\mathrm{Per}_{n}(f)$, and the set of fixed points by $\mathrm{Fix}(f)$. The
set of all iterates of a periodic point forms a _periodic orbit_.

An important question is the “range of influence” of a fixed point. That is to
ask, what set of points, if any, will approach or tend to a fixed (or
periodic) point under the iteration of $f$.

######  Definition 1.3 (Stable Set or Basin of Attraction).

Let $p$ be periodic of period $n$. A point $x$ is _forward asymptotic to $p$_
if $\lim_{i\rightarrow\infty}f^{in}(x)=p$. The _stable set (or basin of
attraction)_ of $p$, denoted by $W^{s}(p)$, consists of all points forward
asymptotic to $p$.

If $p$ is not periodic, we can still define its forward asymptotic points
requiring that $|f^{i}(x)-f^{i}(p)|\rightarrow 0$ when $i\rightarrow\infty$.
Moreover, if $f$ is invertible, we can consider _backward asymptotic points_
by taking $i\rightarrow-\infty$ in the last definition. The set of backward
asymptotic points to $p$ is called the _unstable set_ of $p$ and is denoted by
$W^{u}(p)$.

Another useful concept comes from elementary calculus.

######  Definition 1.4 (Critical Points).

A point $x$ is a _critical point_ of $f$ if $f^{\prime}(x)=0$. The critical
point is _non degenerate_ if $f^{\prime\prime}(x)\neq 0$. The critical point
is _degenerate_ if $f^{\prime\prime}(x)=0$.

Critical points will prove later to be important in relation to the stable set
of a fixed point, and to determine how many attracting orbits a map may have.

Degenerate critical points can be either _maxima_ , _minima_ or _saddle
points_. Non degenerate critical points can only be maxima or minima. A
_diffeomorphism_ cannot have critical points, but its existence in non-
invertible maps is one of the main reasons why this type of maps are more
complicated.

The goal of the theory of discrete dynamical systems is then to understand the
nature of all orbits and to identify which are periodic, asymptotic, etc. In
general, this is an impossible task. For example, if $f(x)$ is a quadratic
polynomial, then explicitly finding the periodic points of period $n$ is
equivalent to solving the equation $f^{n}(x)=x$, which is a polynomial
equation of degree $2^{n}$. Computer calculations do not help in this case
either, since rounding errors tend to accumulate and make many periodic points
“invisible” to the computer. Therefore, qualitative and geometrical techniques
must be employed to understand the nature of a system.

###  1.2 Hyperbolicity and criteria for stability

A key issue once we find periodic points, is to determine the behavior of
orbits near these points; such investigation is what is called _stability
theory_. Once this is done, we can state whether a periodic point is
_attracting_ or _repelling_ (or neither). To determine this, we can part from
the following

######  Definition 1.5 (Stability).

Let $p\in A$ be a fixed point of $f$. Then,

  1. 1.

$p$ is said to be _stable_ if for any $\varepsilon>0$ there exists a
$\delta>0$ such that if $|x_{0}-p|<\delta$ then $|f^{n}(x_{0})-p|<\varepsilon$
for all integers $n\in\mathbb{Z}^{+}$. Otherwise, the fixed point $p$ is
called _unstable_.

  2. 2.

$p$ is called _attracting_ if there is a $\eta>0$ such that if
$|x_{0}-p|<\eta$ then $\lim_{n\rightarrow\infty}f^{n}(x_{0})=p$.

  3. 3.

$p$ is _asymptotically stable_ if it is both stable and attracting.

Therefore, intuitively, a fixed point is stable if nearby orbits remain near
the fixed point, and you can always find an interval around the fixed point
such that orbits starting within the interval will remain arbitrarily close to
the fixed point. The opposite is that you cannot do that; i.e. there is an
interval around the fixed point such that no matter how close to the fixed
point you start the orbit from, the orbit will go outside the interval. The
case of an asymptotically fixed point is particularly important, since in this
case all the orbits near a fixed point will approach it in the limit. The next
step is to be able to determine when a fixed point is stable or unstable. To
do this, we will need the

######  Definition 1.6 (Hyperbolic Point).

Let $p$ be a periodic point of period $n$. The point $p$ is _hyperbolic_ if
$|(f^{n})^{\prime}(p)|\neq 1$. Otherwise, it is called a _nonhyperbolic_ fixed
point. The number $(f^{n})^{\prime}(p)$ is called the _multiplier_ of the
periodic point.

######  Example 1.1.

Consider the diffeomorphism $f(x)=\frac{1}{2}(x^{3}+x)$. There are three fixed
points: $x=0,\,1,$ and -1. Note that $f^{\prime}(0)=1/2$ and $f^{\prime}(\pm
1)=2$. Therefore, each point is hyperbolic.

As it turns out, hyperbolic points are easy to understand in terms of whether
they are attracting or _repelling_ , as they are always one of the two.

######  Definition 1.7 (Attractor and repellor).

Let $p$ be a hyperbolic point of period $n$. If $|(f^{n})^{\prime}(p)|<1$, the
point $p$ is asymptotically stable and is called an attracting periodic point
or more shortly, an _attractor_ ; if $|(f^{n})^{\prime}(p)|>1$, $p$ is
unstable and it is called a repelling periodic point or, simply, a _repellor_.

Occasionally, the terms sink and source may also be used to refer to
attractors and repellors, respectively, stemming from the continuous dynamical
systems analog. Now, if a periodic point is attracting, it must have an
interval around it in which it is so; this is stated in the following

######  Theorem 1.1.

Let $p$ be a hyperbolic fixed point of $f$. Then if $p$ is an attractor, there
exists an open set $U\subseteq W^{s}(p)$, with $p\in U$; on the other hand, if
$p$ is a repellor, there is an open set $V\subset W^{u}(p)$, with $p\in V$.

This theorem explains the choice of the terms “attracting” an “repelling”,
since attractors and repellors have around them a basin of attraction and an
unstable set, respectively.

It is straightforward to see that $W^{s}(p)$ is an invariant set under the
action of the map. A similar result is true for periodic points of period $n$.
In this case, we have an open interval around $p$ that is mapped into itself
by $f^{n}$.

###  1.3 The Schwarzian derivative

Here, we will discuss a mathematical tool that will turn out useful in
determining some properties of maps in terms of its periodic points structure.

######  Definition 1.8 (Schwarzian Derivative).

The _Schwarzian derivative_ of a function $f$ at the point $x$ is given by

$Sf(x)=\frac{f^{\prime\prime\prime}(x)}{f^{\prime}(x)}-\frac{3}{2}\left(\frac{f^{\prime\prime}(x)}{f^{\prime}(x)}\right)^{2}.$

In particular, it was first used by [Singer(1978)] in [Singer(1978)] to
address the question of how many attracting periodic points a differentiable
map can have [Singer(1978)] (see _Singer’s Theorem_ below). It can also be
used to determine the nature of non-hyperbolic periodic points [Elaydi(2000)].
Mappings with negative Schwarzian derivative present particular dynamical
properties that will interest us in this work.

######  Theorem 1.2.

Let $P(x)$ be a real polynomial. If all roots of $P^{\prime}(x)$ are real and
distinct, then $SP(x)<0$ for all $x$.

The following theorem is useful when analyzing _topologically conjugate maps_
(a concept to be defined further below in 1.9), which in turn is useful to
determine the stability properties of one map in terms of the known such
properties of another map (the _conjugate map_).

######  Theorem 1.3.

Suppose $Sf<0$ and $Sg<0$. Then $S(f\circ g)<0$.

Also, this tells us that iteration of a map does not change the sign of its
Schwarzian derivative, i.e. the property is preserved. This immediately takes
us to

######  Corollary.

Let $Sf<0$. Then $Sf^{n}<0$ for all $n>1$.

The assumption of $Sf<0$ has several surprising implications for the dynamics
of a one-dimensional map, as we will see below. Another useful result dealing
with the geometry of a map is

######  Lemma 1.4.

If $Sf<0$, then $f^{\prime}(x)$ cannot have a positive local minimum or a
negative local maximum.

The proof of this last lemma is straightforward from the definition of the
Schwarzian derivative and elementary calculus. Note that this lemma tells us
that, if $Sf<0$, between any two consecutive critical points of $f^{\prime}$,
its graph must cross the $x$-axis; this in turn means that there must be a
critical point for $f$ between these two points, i.e. there must be a maximum
or minimum of $f$ between any pair of its inflection points.

The Schwarzian derivative can also be used to determine whether a fixed point
attracts a critical point of a map.

######  Lemma 1.5.

Let $a_{1},a_{2}$ and $a_{3}$ be fixed points of a continuously differentiable
map $g$ with $a_{1}<a_{2}<a_{3}$ and such that $Sg<0$ on the open interval
$(a_{1},\,a_{3})$. If $g^{\prime}(a_{2})\leq 1$, then $g$ has a critical point
in $(a_{1},\,a_{3})$.

The last lemma is used in the proof of the following important result due to
[Singer(1978)] ([Singer(1978)]).

######  Theorem 1.6 (Singer’s Theorem).

Suppose $Sf<0$ ($Sf(x)=-\infty$ is allowed). Suppose $f$ has $n$ critical
points. Then $f$ has at most $n+2$ attracting periodic orbits.

This last theorem allows to find directly from $f$ an upper bound for the
number of attracting periodic points. This differs from the information
provided in the analysis by _Sarkovskii’s theorem_ (see section 1.5 below) in
that the latter tells us how many periodic points there are, regardless of
whether they are attracting or not and, also, we have first to find some
periodic point and its period to be able to apply Sarkovskii’s theorem.

###  1.4 Nonhyperbolic fixed points

The stability criteria for nonhyperbolic fixed points involve the Schwarzian
derivative defined in 1.3. We will analyze the cases of the multiplier being
equal to 1 and -1 separately. The unstated proofs of the following theorems
can be found in the book by [Elaydi(2000)].

######  Theorem 1.7.

Let $p$ be a fixed point of a map $f$ such that $f^{\prime}(p)=1$. Then, if
$f^{\prime\prime\prime}(p)\neq 0$ and is continuous, the following statements
hold

  1. 1.

If $f^{\prime\prime}(p)\neq 0$, then $p$ is unstable.

  2. 2.

If $f^{\prime\prime}(p)=0$ and $f^{\prime\prime\prime}(p)>0$, then $p$ is
unstable.

  3. 3.

If $f^{\prime\prime}(p)=0$ and $f^{\prime\prime\prime}(p)<0$, then $p$ is
asymptotically stable.

Similarly, we have

######  Theorem 1.8.

Let $p$ be fixed point of a map $f$ such that $f^{\prime}(p)=-1$. If
$f^{\prime\prime\prime}(p)$ is continuous, then the following statements hold:

  1. 1.

If $Sf(p)<0$, then $p$ is an asymptotically stable.

  2. 2.

If $Sf(p)>0$, then $p$ is unstable.

It is worth noting that a repellor is an unstable periodic point but not all
such points are repellors since, a nonhyperbolic periodic point may also be
unstable; e.g. a “semistable” nonhyperbolic fixed point is “attracting from
the right (or left)” but “repelling from the left (or right, respectively)”.

When dealing with nonhyperbolic points, particularly when they are unstable,
it is useful to account for the concept of “semistability”, separating
stability “from the right” and “from the left”.

######  Definition 1.9 (Semistability).

A fixed point $p$ of a map $f$ is said to be semistable from the right
(respectively, from the left) if for any $\varepsilon>0$ there exists
$\delta>0$ such that if $0<x_{0}-p<\delta$ (respectively, $0<p-x_{0}<\delta$)
then $|f^{n}(x_{0})-p|<\varepsilon$ for all $n\in\mathbb{Z}^{+}$. Moreover, if
$lim_{n\rightarrow\infty}f^{n}(x_{0})=p$ whenever $0<x_{0}-p<\eta$
(respectively, $0<p-x_{0}<\eta$) for some $\eta>0$, then $p$ is said to be
semiasymptotically stable from the right (respectively, from the left).

It is then easy to prove that

######  Theorem 1.9 (Asymptotical semistability).

Let $f$ be a map, $p$ one of its fixed points and with $f^{\prime}(p)=1$ and
$f^{\prime\prime}(p)\neq 0$. Then $p$ is

  1. 1.

Semiasymptotically stable from the right if $f^{\prime\prime}(p)<0$;

  2. 2.

Semiasymptotically stable from the left if $f^{\prime\prime}(p)>0$.

###  1.5 Sarkovskii’s theorem

This theorem is of great beauty, as it constructs a new –apparently
artificial– ordering of the natural numbers in order to determine what types
of orbits may a map have. Consider then the following ordering of the natural
numbers:

$3\triangleright 5\triangleright 7\triangleright\cdots\triangleright 2\cdot
3\triangleright 2\cdot 5\triangleright 2\cdot
7\triangleright\cdots\triangleright 2^{2}\cdot 3\triangleright 2^{2}\cdot
5\triangleright\cdots$

$\triangleright 2^{3}\cdot 3\triangleright 2^{3}\cdot
5\triangleright\cdots\cdots\triangleright 2^{3}\triangleright
2^{2}\triangleright 2\triangleright 1.$

That is: first all odd numbers, then two times each odd number, $2^{2}$ times
each odd number, $2^{3}$ times each odd number, etc., and finally, the powers
of two in descending order. This is _Sarkovskii’s ordering_ of the natural
numbers and it easy to see it is exhaustive .

Having defined the above, we can state

######  Theorem 1.10 (Sarkovskii).

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ continuous. Suppose that $f$ has a
periodic point of prime period $k$. If $k\triangleright l$ in Sarkovskii’s
ordering, then $f$ also has a periodic point of period $l$.

Thus, Sarkovskii’s theorem gives a nearly complete answer to the question of
how many periodic points and of which periods can a map have, although it does
not tell us their nature in terms of their stability, nor does it tell us how
to find such points.

Some important consequences of Sarkovskii’s Theorem can be summarized as
follows:

  * •

If $f$ has periodic points with periods different from a power of 2, then $f$
has infinite periodic points of infinitely different periods.

  * •

If $f$ only has a finite number of periodic points, then necessarily all
points have periods of powers of 2.

  * •

Period 3 is the “greatest” period in Sarkovskii’s ordering, therefore, it
implies the existence of all other periods.

Sarkovskii’s theorem is without doubt a central result for the theory of
discrete dynamical systems.

###  1.6 Bifurcation theory

We will now consider families of maps instead of single maps, as this is often
useful in applications and is also the main object of study of this work. For
example, in practice, the coefficients of a polynomial map that models some
natural, physical, economic or social discrete process can only be determined
up to a certain precision, which gives rise to a family of maps as a set
bounded by the uncertainties of the coefficients.

The aim of bifurcation theory is then to study the changes that occur in a map
when its parameters are varied. This changes commonly refer qualitatively to
its periodic points structure and stability properties, but they can also
involve other changes. Consider families of functions in one real variable
which depend upon a single real parameter (uniparametric). Consider the
bivariate function $G(x,\lambda)=f_{\lambda}(x)$, where for fixed
$\lambda\in\mathbb{R}$, $f_{\lambda}(x)\in C^{\infty}$. We will assume also
that $G$ depends smoothly on $\lambda$. Some examples of such families of
functions are:

  * •

$F_{\mu}(x)=\mu x(1-x)$.

  * •

$E_{\lambda}(x)=\lambda e^{x}$.

  * •

$S_{\lambda}(x)=\lambda\sin(x)$.

  * •

$Q_{c}(x)=x^{2}+c$.

With this in mind, we will next give a rather informal definition of
bifurcation.

######  Definition 1.10 (Bifurcation).

Let $f_{\lambda}(x)$ be a uniparametric family of functions. Then, we say
there is a _bifurcation_ at $\lambda_{0}$ if there exists $\epsilon>0$ such
that if $a$ and $b$ satisfy $\lambda_{0}-\epsilon<a<\lambda_{0}$ and
$\lambda_{0}<b<\lambda_{0}+\epsilon$, then the “dynamics” of $f_{a}(x)$ is
different from the “dynamics” of $f_{b}(x)$. That is, the “dynamics” of the
function $f_{\lambda}(x)$ _changes_ when $\lambda$ passes through the value
$\lambda_{0}$.

The “changes in the dynamics” of the above definition refer to qualitative
changes in the structure of periodic points of $f_{\lambda}(x)$. To make the
above notion precise, we must consider separately the different types of
bifurcations that may occur. For real maps, the _saddle-node_ and _period
doubling_ bifurcations are the most common. We will next discuss the concept
of saddle-node bifurcation through an example and the period doubling
bifurcation is discussed in more detail below.

######  Example 1.2 (Saddle-node or Tangent Bifurcation).

Let’s consider the family of functions $E_{\lambda}(x)=\lambda e^{x}$, with
$\lambda>0$. A bifurcation occurs when $\lambda=1/e$. We have three cases:

  1. 1.

If $\lambda>1/e$ the graph of $E_{\lambda}(x)$ and $y=x$ do not intersect each
other, so that $E_{\lambda}$ has no fixed points.
$E_{\lambda}^{n}(x)\rightarrow\infty\,\forall x$.

  2. 2.

When $\lambda=1/e$, the graph of $E_{\lambda}$ tangentially touches the
identity diagonal at $(1,\,1)$. $E_{\lambda}(1)=1$ is the only fixed point. If
$x<1$, $E_{\lambda}^{n}(x)\rightarrow 1$ and if $x>1$,
$E_{\lambda}^{n}(x)\rightarrow\infty$.

  3. 3.

When $0<\lambda<1/e$, the graph of $E_{\lambda}$ crosses the identity diagonal
two times, at $q$ with $E_{\lambda}^{\prime}(q)<1$ and at $p$ with
$E_{\lambda}^{\prime}(p)>1$, with which we see they are an attractor and a
repellor, respectively.

Figure 1.1 presents some of these characteristics graphically, and the
corresponding phase portraits are presented in figure 1.2. Figure 1.3 presents
a common and very useful way of depicting bifurcations of real maps, which
will be frequently used in this work: a _bifurcation diagram_ ; in it, the
horizontal axis represents the values of the parameter $\lambda$ in which the
family is varied, and the vertical axis represents the values of the periodic
points present for each $f_{\lambda}$.

Figure 1.1: Graphs of $E_{\lambda}(x)=\lambda e^{x}$ where (a) $\lambda>1/e$,
(b) $\lambda=1/e$, and (c) $0<\lambda<1/e$. Reproduced from [Devaney(1989)].
Figure 1.2: Phase portraits of $E_{\lambda}(x)=\lambda e^{x}$ for (a)
$\lambda>1/e$, (b) $\lambda=1/e$, and (c) $0<\lambda<1/e$. Reproduced from
[Devaney(1989)]. Figure 1.3: Bifurcation diagram of $E_{\lambda}(x)=\lambda
e^{x}$. $x$ is plotted against $\lambda$. Reproduced from [Devaney(1989)].

The above example suggests that bifurcations may occur when non-hyperbolic
fixed points are present. Indeed, this the only case when bifurcations of
fixed points occur (in the sense of one fixed point giving birth to two
others), as the next theorem demonstrates.

######  Theorem 1.11.

Let $f_{\lambda}$ be a one-parameter family of functions and suppose that
$f_{\lambda_{0}}(x_{0})=x_{0}$ and $f^{\prime}_{\lambda_{0}}(x_{0})\neq 1$.
Then there are intervals $I$ about $x_{0}$ and $N$ about $\lambda_{0}$ and a
smooth function $p:N\rightarrow I$ such that $p(\lambda_{0})=x_{0}$ and
$f_{\lambda}(p(\lambda))=p(\lambda)$. Moreover, $f_{\lambda}$ has no other
fixed points in $I$.

###### Proof.

Consider the function defined by $G(\lambda,x)=f_{\lambda}(x)-x$. By
hypothesis, $G(\lambda_{0},x_{0})=0$ and

$\frac{\partial G}{\partial
x}(\lambda_{0},x_{0})=f^{\prime}_{\lambda_{0}}(x_{0})-1\neq 0,$ (1.1)

using the other hypothesis. Then, by the Implicit Function Theorem
[Marsden(1974)], there are intervals $I$ about $x_{0}$ and $N$ about
$\lambda_{0}$, and a smooth function $p:N\rightarrow I$ such that
$p(\lambda_{0})=x_{0}$ and $G(\lambda,p(\lambda))=0$ for all $\lambda\in N$;
that is, $p(\lambda)$ is fixed by $f_{\lambda}$ for $\lambda\in N$. Moreover,
$G(\lambda,x)\neq 0$ unless $x=p(\lambda)$. ∎

The above theorem obviously holds for periodic points by replacing
$f_{\lambda}$ with $f^{n}_{\lambda}$.

###  1.7 Chaos

In this section we will precise what we understand by the mathematical concept
of _chaos_. It must be said though, that there is not a single universally
accepted definition for this concept, as variations occur; however, they
preserve the same general “spirit”. We will here adopt one of the most popular
definitions, due to [Devaney(1989)].

Before we can properly define the concept of chaos, we need some other
auxiliary concepts.

######  Definition 1.11 (Topological transitivity).

Let $(X,\,d)$ be a metric space and $f:\,X\rightarrow X$ a map. Then $f$ is
said to be _topologically transitive_ if for any pair of nonempty open sets
$U,\,V\subset X$ there exists $k>0$ such that $f^{k}(U)\cap V\neq\emptyset$.

In plain words, what this definition states is that a map $f$ is topologically
transitive if any arbitrarily small set of the domain of $f$ contains points
that are mapped by $f$ to any other arbitrarily small set of the codomain in a
finite number of iterations. An immediate consequence of topological
transitivity is that the domain cannot be divided into two nonempty disjoint
sets which are invariant under $f$. It is easy to see that if $f$ has a dense
orbit in $X$ then it is topologically transitive (the converse is also true
for compact subsets of $\mathbb{R}$ or $S^{1}$).

Another necessary concept for the definition of chaos is the following.

######  Definition 1.12 (Sensitive Dependence to Initial Conditions).

Let $(X,\,d)$ be a metric space and $f:\,X\rightarrow X$ a map. $f$ is said to
posses _sensitive dependence to initial conditions_ if there exists
$\varepsilon>0$ such that for any $x\in X$ and any neighborhood $N$ of $x$,
there exists $y\in N$ and $n\geq 0$ such that
$|f^{n}(x)-f^{n}(y)|>\varepsilon$.

Intuitively, sensitive dependence to initial conditions means that no matter
how close two points are to each other, they will eventually be separated by
at least some distance $\varepsilon$ under a finite number of iterations of
$f$.

With these preliminary definitions we can now state the main definition of
this section.

######  Definition 1.13.

Let $(X,\,d)$ be a metric space and $f:\,X\rightarrow X$ a map. Then $f$ is
said to be _chaotic_ on $X$ if

  1. 1.

$f$ is topologically transitive,

  2. 2.

the set of periodic points of $f$ is dense in $X$, and

  3. 3.

$f$ has sensitive dependence on initial conditions.

So a chaotic map “mixes up” its domain set by means of its topological
transitivity but everywhere you look there are periodic points arbitrarily
close and, finally, it is practically impossible to predict orbits
numerically; these are the three elements that make up a chaotic map, under
this definition. Actually, it was later shown [Banks et al.(1992)Banks,
Brooks, Cairns, Davis, and Stacey] that topological transitivity and denseness
of periodic points imply sensitive dependence to initial conditions, but no
other pair of conditions imply the other. However, for continuous maps on
intervals of $\mathbb{R}$, topological transitivity _implies_ that the set of
periodic points is dense [Vellekoop and Berglund(1994)], so that in this case,
by the above, topological transitivity alone actually implies chaos.

###  1.8 The period doubling route to chaos

Perhaps one of the most important types of bifurcations is the _period
doubling bifurcation_. This type of bifurcation will play a central role in
the main results of this work. The most commonly know map which presents
period doubling bifurcation is the famous logistic map. The issue is that the
_logistic quadratic map_ $f_{\lambda}(x)=\lambda x(1-x)$ has simple dynamics
for $0\leq\lambda\leq 3$ but it becomes chaotic when $\lambda\geq 4$ (we will
precise the notion of chaos in the next section). The natural question here
is: how exactly is it that $f_{\lambda}$ becomes chaotic as $\lambda$
increases? Where do the infinitely many periodic points come from when
$\lambda$ is large? In this section we will see that “period doubling”
bifurcation is central to this.

_Sarkovskii’s theorem_ provides a partial answer to the question of how many
periodic points emerge when the parameter is varied. Before $f_{\lambda}$ can
have infinite periodic points of distinct periods, it must first have periodic
points of all the periods of the form $2^{j}$ with $j\in\mathbb{N}$. Local
bifurcation theory provides two typical ways in which these points may emerge:
by means of saddle-node bifurcations or through period doublings. The question
is then which type of bifurcations occur as $f_{\lambda}$ becomes chaotic.

The typical way in which $f_{\lambda}$ becomes chaotic is that $f_{\lambda}$
undergoes a series of period doubling bifurcations. This is not always the
case, but it is a typical “route to chaos”.

######  Theorem 1.12 (Period Doubling Bifurcation).

Suppose that

  1. 1.

$f_{\lambda}(x_{0})=x_{0}$ for all $\lambda$ in an interval around
$\lambda_{0}$.

  2. 2.

$f_{\lambda}^{\prime}(x_{0})=-1$.

  3. 3.

$\frac{\partial(f_{\lambda}^{2})^{\prime}}{\partial\lambda}|_{\lambda=\lambda_{0}}(x_{0})\neq
0.$

Then there exists an interval $I$ around $x_{0}$ and a function
$p:I\rightarrow\mathbb{R}$ such that

$f_{p(x)}(x)\neq x$

but

$f_{p(x)}^{2}(x)=x.$

###### Proof.

Let $G(\lambda,\,x)=f^{2}_{\lambda}(x)-x$. Then, from hypotheses (1) and (2),

$\frac{\partial G}{\partial\lambda}(\lambda_{0},\,x_{0})=0$

so that we cannot apply the implicit function theorem directly. Thus, to
rectify the situation, we set

$B(\lambda,\,x)=\begin{cases}G(\lambda,\,x)/(x-x_{0})&\mathrm{if}\,x\neq
x_{0},\\\ \frac{\partial G}{\partial
x}(\lambda_{0},\,x_{0})&\mathrm{if}\,x=x_{0}.\end{cases}$ (1.2)

Straightforward calculation of the limits of the derivatives allow us to
verify that $B$ is smooth and satisfies

$\displaystyle\frac{\partial B}{\partial x}(\lambda_{0},\,x_{0})$
$\displaystyle=\frac{1}{2}\frac{\partial^{2}G}{\partial
x^{2}}(\lambda_{0},x_{0})$ (1.3) $\displaystyle\frac{\partial^{2}B}{\partial
x^{2}}(\lambda_{0},\,x_{0})$
$\displaystyle=\frac{1}{3}\frac{\partial^{3}G}{\partial
x^{3}}(\lambda_{0},x_{0}).$

We now observe that, once again using hypotheses (1) and (2),

$\displaystyle B(\lambda_{0},\,x_{0})$ $\displaystyle=\frac{\partial
G}{\partial x}(\lambda_{0},\,x_{0})$ (1.4)
$\displaystyle=\left.\frac{\partial}{\partial
x}f^{2}_{\lambda_{0}}(x)\right|_{x=x_{0}}-1$
$\displaystyle=\left.\frac{\partial}{\partial
x}f_{\lambda_{0}}\left(f_{\lambda_{0}}(x)\right)\right|_{x=x_{0}}-1$
$\displaystyle=\left[\left.\frac{\partial}{\partial
y}f_{\lambda_{0}}(y)\right|_{y=f_{\lambda_{0}}(x)}\cdot\frac{\partial}{\partial
x}f_{\lambda_{0}}(x)\right]_{x=x_{0}}-1$ $\displaystyle=\frac{\partial
f_{\lambda_{0}}}{\partial x}(f_{\lambda_{0}}(x_{0}))\frac{\partial
f_{\lambda_{0}}}{\partial x}(x_{0})-1$ $\displaystyle=\left[\frac{\partial
f_{\lambda_{0}}}{\partial x}(x_{0})\right]^{2}-1$
$\displaystyle=\left[f_{\lambda_{0}}^{\prime}(x_{0})\right]^{2}-1$
$\displaystyle=(-1)^{2}-1$ $\displaystyle=0.$

Moreover, now using hypothesis (3),

$\displaystyle\frac{\partial B}{\partial\lambda}(\lambda_{0},\,x_{0})$
$\displaystyle=\frac{\partial}{\partial\lambda}\left(\frac{\partial
G}{\partial x}(\lambda_{0},\,x_{0})\right)$ (1.5)
$\displaystyle=\frac{\partial}{\partial\lambda}\left(\frac{\partial}{\partial
x}f^{2}_{\lambda_{0}}(x_{0})-1\right)$
$\displaystyle=\frac{\partial^{2}}{\partial\lambda\partial
x}f^{2}_{\lambda_{0}}(x_{0})$
$\displaystyle=\frac{\partial}{\partial\lambda}(f^{2}_{\lambda_{0}})^{\prime}(x_{0})$
$\displaystyle\neq 0.$

Therefore, by the implicit function theorem [Marsden(1974)], there exists a
$C^{1}$ map $p(x)$ defined on an interval around $x_{0}$ such that
$p(x_{0})=\lambda_{0}$ and $B(p(x),x)=0$. So that, for $x\neq x_{0}$,

$\frac{1}{x-x_{0}}G(p(x),x)=0.$

Consequently, $G(p(x),x)=0$ and then $f^{2}_{p(x)}(x)=x$ and thus $x$ is of
period 2 for $\lambda=p(x)$ and for $x$ in an interval around $x_{0}$. Notice
that $x$ is not fixed by $f_{p(x)}$ because we can apply theorem 1.11 in this
case and, then, the curve of fixed points is unique.

On the other hand, from the chain rule we have

$\frac{d}{dx}B(p(x),x)=\frac{\partial B}{\partial x}+\frac{\partial
B}{\partial\lambda}p^{\prime}(x)=0.$

From which,

$p^{\prime}(x)=-\frac{\frac{\partial B}{\partial x}(p(x),x)}{\frac{\partial
B}{\partial\lambda}(p(x),x}.$ (1.6)

We know that, for $x=x_{0}$, the denominator in (1.6) is different from zero
and, on the other hand, the numerator is equal to
$\frac{1}{2}\frac{\partial^{2}G}{\partial x^{2}}$ and

$\displaystyle\frac{\partial^{2}G}{\partial x^{2}}(\lambda_{0},x_{0})$
$\displaystyle=\frac{1}{2}\left.\left\\{f^{\prime\prime}_{\lambda}(f_{\lambda}(x))[f^{\prime}_{\lambda}(x)]^{2}+f^{\prime}_{\lambda}(f_{\lambda}(x))f^{\prime\prime}_{\lambda}(x)\right\\}\right|_{x=x_{0},\,\lambda=\lambda_{0}}$
(1.7)
$\displaystyle=\frac{1}{2}\left\\{f^{\prime\prime}_{\lambda_{0}}(x_{0})[-1]^{2}-f^{\prime\prime}_{\lambda_{0}}(x_{0})\right\\}$
$\displaystyle=0.$

Where we have once again used hypothesis (2). Ergo, $p^{\prime}(x_{0})=0$. ∎

Also, differentiating equation (1.6) again and evaluating in $x=x_{0}$, it is
easy to see that

$p^{\prime\prime}(x_{0})=-\frac{\frac{\partial^{2}B}{\partial
x^{2}}(\lambda_{0},x_{0})}{\frac{\partial
B}{\partial\lambda}(\lambda_{0},x_{0})}.$ (1.8)

We already know that the denominator in (1.8) above is not null and, also,

$\displaystyle\frac{\partial^{2}B}{\partial x^{2}}(\lambda_{0},x_{0})$
$\displaystyle=\frac{1}{3}\frac{\partial^{3}G}{\partial x^{3}}$ (1.9)
$\displaystyle=\frac{1}{3}\left\\{f^{\prime\prime\prime}_{\lambda}(f_{\lambda}(x))\left[f^{\prime}_{\lambda}(x)\right]^{3}+2f^{\prime\prime}_{\lambda}(f_{\lambda}(x))f^{\prime}_{\lambda}(x)+f^{\prime\prime}_{\lambda}(f_{\lambda}(x))f^{\prime\prime}_{\lambda}(x)f^{\prime}_{\lambda}(x)\right.$
$\displaystyle\quad\left.\left.+f^{\prime}_{\lambda}(f_{\lambda}(x))f^{\prime\prime\prime}_{\lambda}(x)\right\\}\right|_{x=x_{0},\,\lambda=\lambda_{0}}$
$\displaystyle=\frac{1}{3}\left\\{-2f^{\prime\prime\prime}_{\lambda_{0}}(x_{0})-3[f^{\prime\prime}_{\lambda_{0}}(x_{0})]^{2}\right\\}$
$\displaystyle=\frac{2}{3}\left\\{\frac{f^{\prime\prime\prime}_{\lambda_{0}}(x_{0})}{f^{\prime}_{\lambda_{0}}(x_{0})}-\frac{3}{2}\left[\frac{f^{\prime\prime}_{\lambda_{0}}(x_{0})}{f^{\prime}_{\lambda_{0}}(x_{0})}\right]^{2}\right\\}$
$\displaystyle=\frac{2}{3}\,Sf_{\lambda_{0}}(x_{0}),$

where $Sf_{\lambda}$ is the Schwarzian derivative of $f_{\lambda}$ and, yet
once again, we used hypothesis (2) from the theorem above. Therefore,
substituting in equation (1.8) we have

$p^{\prime\prime}(x_{0})=-\frac{-\frac{2}{3}\,Sf_{\lambda_{0}}(x_{0})}{\frac{\partial}{\partial\lambda}(f^{2}_{\lambda_{0}})^{\prime}(x_{0})}.$
(1.10)

So that, if $Sf_{\lambda}\neq 0$ then $p^{\prime\prime}(x_{0})\neq 0$ and the
curve $p(x)$ is either concave left or right. Along with $p(x_{0})=0$ this
gives us an idea of the geometry of the period 2 curve $p$ near the
bifurcation point $x_{0}$. Notice that $x_{0}$ does not bifurcate in the sense
of giving birth to another fixed point (as was the case in the pitchfork
bifurcation) but, rather, $x_{0}$ continues to be the only fixed point in a
interval around $\lambda_{0}$ (see theorem 1.11).

In the case of the logistic map, the period doubling bifurcation repeats over
and over again for the fixed point $x^{*}=\frac{\lambda-1}{\lambda}$ as the
value of the parameter $\lambda$ is increased, starting from $\lambda=3$:

  * •

Between $\lambda_{0}=1<\lambda<3=\lambda_{1}$, $f_{\lambda}$ has a single
stable fixed point, $x^{*}$ (of period $2^{0}=1$).

  * •

For $\lambda_{1}=3<\lambda<1+\sqrt{6}=\lambda_{2}$, $x^{*}$ loses its
stability and ’gives rise’ to an attracting cycle of period two ($2^{1}$),
whose orbital points separate from $x^{*}$ as $\lambda$ increases.

  * •

For $\lambda_{2}=1+\sqrt{6}<\lambda<3.54409=\lambda_{3}$ [Elaydi(2000)], the
attracting cycle of period two loses in turn its stability and gives rise to
an attracting cycle of period $2^{2}$.

  * •

For $\lambda>3.54409=\lambda_{3}$, the period four cycle also loses its
stability and bifurcates into an asymptotically stable cycle of period
$2^{3}$.

The above mentioned process repeats itself indefinitely and produces a
sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}$. The _Feigenbaum sequence_
$\\{\lambda_{k}\\}_{k=0}^{\infty}$ describes the values of the parameter
$\lambda$ for which _period doubling bifurcations_ arise in a unimodal map
[Feigenbaum(1978)]. The sequence converges exponentially. The important point
to remark here is that the rate of convergence of these sequences (whose
values depend on the specific unimodal map), is constant for every unimodal
function and is given by $\delta\approx 4.669201609\cdots$, which gives rise
to the so-called property of _universality_. In fact, its importance deserved
naming it _Feigenbaum’s constant_ , after [Feigenbaum(1978)] discovered it in
[Feigenbaum(1978)] (though amazingly ignored for over a decade).

###  1.9 Topological conjugacy

Here we will define the important concept of _topological conjugacy_ which
relates the dynamics of two different maps by means of a third one. The
importance of conjugacy is that it preserves the stability and chaotic
properties of a map, so that one map can be analyzed by means of analyzing
another one, presumably simpler, so that the analysis becomes easier.

######  Definition 1.14 (Topological Conjugacy).

Let $f:A\rightarrow A$ and $g:B\rightarrow B$ be two maps. Then $f$ and $g$
are said to be _topologically conjugate_ , denoted $f\approx g$, if there
exists a homeomorphism $h:A\rightarrow B$ such that $h\circ g=g\circ h$. The
homeomorphism is called a _topological conjugacy_. We also say that $f$ is
$h$-conjugate to $g$ to emphasize the importance of the homeomorphism $h$.

It is easy to prove that conjugacy is an equivalence relation. Topologically
conjugate maps have equivalent dynamics. For example, if $f\approx g$ and $x$
is a fixed point of $f$ then $h(x)$ is a fixed point of $g$; and, also,
$f^{k}\approx g^{k}$ for all $k\in\mathbb{N}$. Similarly, periodic orbits of
$f$ are mapped to analogue periodic orbits of $g$.

######  Theorem 1.13.

Let $f:A\rightarrow A$ be $h$-conjugate to $g:B\rightarrow B$. If $f$ is
chaotic on $A$, then $g$ is chaotic on $B$.

Theorem 1.13 will play a key role in the importance of the study of “canonical
polynomial maps” in chapters 3, 4 and 5. The proof of this theorem can be
found in [Elaydi(2000)], where it can be seen that, in fact, the topological
conjugacy $h$ does not need to be a homeomorphism for theorem 1.13 to hold, it
only needs to be onto, continuous and open, which leads to the concept of

######  Definition 1.15 (Semiconjugacy).

The maps $f:A\rightarrow A$ and $g:B\rightarrow B$ are said to be
_semiconjugate_ if there exists a map $h:A\rightarrow B$ onto, continuous and
open, such that $h\circ f=g\circ h$.

That been stated, it can be shown that theorem 1.13 holds with “conjugate”
replaced by “semiconjugate”.

##  Chapter 2 Regular-Reversal quadratic maps

In this chapter the work of [Solís and Jódar(2004)] will be discussed, as the
main precedent to the present work, where an application of bifurcation theory
and chaos in discrete dynamical systems is analyzed to determine the
conditions in which chaos arises for a family of unimodal maps, namely
quadratic maps. If the asymptotic behavior and parametric dependency of
quadratic maps is understood, the results can be generalized to more complex
maps, starting to set a precedent for cubic maps and polynomial maps of higher
order. The only original contribution in this chapter is given by some of the
examples in the corresponding section below.

The main objective of this chapter is to analyze, for a given unimodal
quadratic map, under what circumstances does the property of period doubling
hold and when it disappears, which means that chaos is no longer a
possibility, so as to be able to control the appearance (or not) of chaotic
behavior by controlling the value of the parameter.

###  2.1 Unimodal maps

Consider a one-parameter family of real discrete dynamical systems of the form
$x_{n+1}=g(x_{n},\lambda)$ with _iteration function_
$g(x_{n},\lambda)=g_{\lambda}(x_{n})$, $g:J\times I\rightarrow\mathbb{R}$,
$I$, $J$ intervals, and $g$ smooth on $x$ and $\lambda$.

######  Definition 2.1 (Unimodal Map).

An _iteration function (map)_ $f:I\subset\mathbb{R}\rightarrow I$, with $I$ an
interval, is called _unimodal_ if it is smooth111This requirement follows from
the computability of the Schwarzian derivative. (at least $C^{3}(I)$) and with
a unique maximal point. By extension, we call a family of maps unimodal if
every member of the family is unimodal.

In the following, let $g(x_{n},\lambda)$ be unimodal and with a _fixed point_
$x=x_{p}(\lambda)$.

######  Definition 2.2 (Eigenvalue function).

The _eigenvalue function_ $\phi:\mathbb{R}\rightarrow\mathbb{R}$ corresponding
to the fixed point $x=x_{p}(\lambda)$ of the one-parameter family of maps
$g(x_{n},\lambda)$ is

$\phi(\lambda)=\frac{\partial g}{\partial x}(x_{p}(\lambda)),$ (2.1)

defined for the values of $\lambda$ for which the fixed point exists.

That is, the eigenvalue function gives us the multiplier of a fixed point as a
function of the parameter of the family of maps. The eigenvalue function for
fixed $\lambda$ is simply the derivative of the map at the fixed point, i.e.
the “traditional” multiplier. Notice that $\phi$ is continuous since $g$ is
smooth. The fixed point $x_{p}(\lambda)$ is _asymptotically stable_ for the
values of $\lambda$ for which $-1<\phi(\lambda)<1$, agreeing with our notion
of a hyperbolic attracting fixed point.

######  Definition 2.3 (Region of Type 1).

The _region of type 1_ of $x_{p}(\lambda)$ is the set
$\\{(\lambda,\,\phi(\lambda))|-1<\phi(\lambda)<1\\}\subset\mathbb{R}^{2}$.

The region of type 1 of $x_{p}$ is the set of values of the parameter
$\lambda$ for which $x_{p}$ is a stable fixed point, along with its
corresponding values of the eigenvalue function $\phi$ (multipliers).
Therefore, the region of type 1 of a fixed point is its _stability region_ as
a function of the parameter $\lambda$. Since $g(x,\lambda)$ is smooth on both
$\lambda$ and $x$, regions of type 1 consist of the countable union of
connected subsets of $\mathbb{R}^{2}$. In general, we can define regions of
higher type respect to the fixed points of the $k$-th iteration of $g$, i.e.
the periodic points of prime period $k$.

######  Definition 2.4 (Region of Type $k$).

If the family of maps $x_{n+1}=g(x_{n},\lambda)$, not necessarily unimodal,
has an isolated periodic point of prime period $k\geq 1,$,
$x_{p}^{k}(\lambda)$, then we can define the _$k$ -th eigenvalue function_
$\phi_{k}:\mathbb{R}\rightarrow\mathbb{R}$ as

$\phi_{k}(\lambda)=\frac{\partial g^{(k)}}{\partial x}(x_{p}^{k}(\lambda)).$
(2.2)

Notice that $\frac{\partial g^{(k)}}{\partial x}$ refers to the _first_
derivative of the $k$-th iteration of $g$ and not the $k$-th derivative of
$g$, as in standard calculus notation. The set
$\\{(\lambda,\phi_{k}(\lambda))\in\mathbb{R}^{2}|x_{p}^{k}(\lambda)\text{ is
stable}\\}$ is denoted as the _region of type $k$ for $x_{p}^{k}$_.

The region of type $k$ of the periodic point $x_{p}^{k}$ is simply the region
of type 1 of the function $g^{(k)}$, of which $x_{p}^{k}$ is a fixed point.
Moreover, by the chain rule,

$\phi_{k}(\lambda)=\frac{\partial g^{(k)}}{\partial
x}=g^{\prime}_{\lambda}(x_{p}^{k})g^{\prime}_{\lambda}(g_{\lambda}(x_{p}^{k}))g^{\prime}_{\lambda}(g_{\lambda}^{(2)}(x_{p}^{k}))\cdots
g^{\prime}_{\lambda}(g_{\lambda}^{(k-1)}(x_{p}^{k}))$

so that it suffices that the absolute value of the product of the multipliers
along the orbit of a periodic point $x_{p}^{k}$ is less than one in order to
guarantee the stability of $x_{p}^{k}$. Nonetheless, the existence of periodic
points of period $k>1$ is not assured even for a unimodal map in general.
Thereafter, the periodic point, $x=x_{p}^{k}(\lambda)$, when it exists, is
stable in an interval of the parameter $\lambda$ with ends given by
$\phi_{k}^{-1}(1)$ and $\phi_{k}^{(-1)}(-1)$. We suppose here that the
function $\phi_{k}^{-1}$ exists, at least locally, i.e.
$\phi^{\prime}_{k}(\lambda)\neq 0$ for all $\lambda$ in an interval around a
value of interest $\lambda_{0}$ associated with the periodic point
$x_{p}^{k}(\lambda_{0})=x_{0}^{k}$, according to the Inverse Function Theorem
[Protter(1998)]. A simpler way of saying the latter is that $x_{p}^{k}$ will
be attracting as long as $-1<\phi_{k}(x_{p}^{k})<1$, but notice that depending
on the form of the graph of $\phi_{k}$, the eigenvalue function may never
leave this region therefore undefining $\phi_{k}^{-1}$ for $\lambda=\pm 1$.
Another way of looking at the region of $k$ is that the type $k$ of a given
region is given by the order $k$ of the attracting $k$-cycle existing in that
region.

Now, since a point $(\lambda,\phi(\lambda))\in\mathbb{R}^{2}$ can only belong
to a unique and specific region of a given periodic point, we can state the
following definition.

######  Definition 2.5 (Regular and Reversal Maps).

Let $\mathcal{A}\subset\mathbb{R}$ be an interval. A map
$x_{n+1}=g(x_{n},\lambda)$ is called a _regular map_ (respectively,
_reversal_) in $\mathcal{A}$ if it has an isolated periodic point such that
its associated eigenvalue function $\phi$ has the property that if
$\lambda_{1},\,\lambda_{2}\in\mathcal{A}$, with $\lambda_{1}<\lambda_{2}$,
then the type of the region containing the point
$(\lambda_{2},\phi(\lambda_{2}))$ is _higher_ (respectively, _lower_) or equal
than the type of the region containing the point
$(\lambda_{1},\phi(\lambda_{1}))$.

In a regular map, as the value of $\lambda$ increases, new attracting periodic
points of progressively higher periods are created and, therefore, the type of
the regions associated with these points equally rises (or at least does not
decrease) progressively. On the other hand, in a _reversal map_ , as the value
of the parameter $\lambda$ increases, higher order attracting periodic points
disappear and only progressively lower order attracting periodic points
remain, so that the type of the associated regions decreases correspondingly.

######  Example 2.1 (The Logistic Map).

The logistic map $f_{\lambda}(x)=\lambda x(1-x)$ is a classical example of a
regular map, since it satisfies the definition 2.5 in the interval
$[0,\lambda_{\infty}]$, with $\lambda_{\infty}\approx 3.570$. Its eigenvalue
function is

$\phi(\lambda)=\left.\lambda(1-2\lambda x)\right|_{x=x_{p}(\lambda)},$

for the nonzero fixed point $x_{p}(\lambda)$. In table 2.1 some type $k$
regions are shown for the logistic map.

Type of the region ($k$) | Interval [$(\lambda_{k-1},\lambda_{k})$]
---|---
1 | (0, 3)
2 | (3, 3.449489…)
3 | (3.449489…, 3.544090…)
4 | (3.544090…, 3.564407…)
5 | (3.564407…, 3.568759…)
6 | (3.568759…, 3.569692…)
7 | (3.569692…, 3.569891…)
Table 2.1: Table of regions of type $k$, $k=1,2,...,7$, for the logistic map.
The logistic map is a regular map (see definition 2.5). Data reproduced from
[Elaydi(2000), Table 1.1, p.41]

######  Example 2.2 (The Exponential Map).

Consider the one-parameter family of maps $E_{\lambda}(x)=\lambda e^{x}$, with
$\lambda<0$. In figure 2.1 are shown three important cases of the graphs of
the family$E_{\lambda}$. When $\lambda<-e$ the map has a repelling fixed point
because of theorem 1.1 (see figure 2.1c); when $\lambda=-e$,
$E_{\lambda}(-1)=-1$ and $E^{\prime}_{\lambda}(-1)=-1\neq 1$, so that -1 is a
non-hyperbolic fixed point and, by theorem 1.11, there is an interval around
$\lambda=-e$ in which this fixed point exists and is unique; and when
$\lambda>-e$ the fixed point is attracting also because of theorem 1.1.
Therefore the fixed point of $E_{\lambda}$ undergoes a change in stability
from unstable to stable as the value of $\lambda$ increases from $\lambda<-e$
to $\lambda>e$, with the _bifurcation value_ being exactly $\lambda=-e$. Also,
it is easy to see that $E^{2}_{\lambda}$ is an increasing function that is
concave upward if $E_{\lambda}(x)<-1$ and concave downward if
$E_{\lambda}(x)>-1$ (see figure 2.2). Thus $E^{2}_{\lambda}$ has two fixed
points at the points $q_{1}$ and $q_{2}$ shown in figure 2.2 when $\lambda<-e$
and they disappear when $\lambda$ increases to $-e$. Since we know that
$E_{\lambda}$ has a single fixed point, these must in fact be periodic points
of period 2. Note that, as these period 2 points lose their “attractiveness”,
the fixed point acquires it, so that there is an “exchange of stability”. Also
note that, at $\lambda=-e$,
$\frac{\partial}{\partial\lambda}\left(E^{2}_{\lambda}\right)^{\prime}(-1)\neq
0$, so that we can apply theorem 1.12 formally prove that a period doubling
bifurcation effectively takes place. The bifurcation diagram of this example
is shown in figure 2.3, where it is easy to see that it satisfies the
definition of a reversal map.

Figure 2.1: The graphs of the one-parameter family of maps
$E_{\lambda}(x)=\lambda e^{x}$ for (a) $-e<\lambda<0$, (b) $\lambda=-e$, and
(c) $\lambda<-e$. Reproduced from [Devaney(1989)]. Figure 2.2: The graphs of
$E^{2}_{\lambda}(x)$ for (a) $-e<\lambda<0$, (b) $\lambda=-e$, and (c)
$\lambda<-e$. Reproduced from [Devaney(1989)]. Figure 2.3: The bifurcation
diagram of $E_{\lambda}(x)=\lambda e^{x}$. Notice that the “vertical” curve is
the curve of period 2 attracting points. Reproduced from [Devaney(1989)].

######  Definition 2.6 (Regular-Reversal Map).

Let $\mathcal{A}$ be a nonempty interval in $\mathbb{R}$. A map
$x_{n+1}=g(x_{n},\lambda)$ is called a _regular-reversal map_ in $\mathcal{A}$
if the interval $\mathcal{A}$ can be decomposed in two nonempty disjoint
intervals $A_{1}$ and $A_{2}$ such that the map is regular in $A_{1}$ and
reversal in $A_{2}$.

In a regular-reversal map, there is a value of the parameter $\lambda$, say
$\lambda_{0}$, that divides the interval of the family of maps
$x_{n+1}=g(x_{n},\lambda)$ in two, in one of which it is regular and another
one in which it is reversal. A regular-reversal map is interesting because it
shows exactly the parametric range where periodicity takes place. So far, it
has been difficult to determine for which maps this phenomenon occurs. It is
for this reason that precedents have focused only in quadratic polynomial maps
[Solís and Jódar(2004)], but also because quadratic polynomials give a
complete family of unimodal maps with negative Schwarzian derivative and they
represent the simplest example of unimodal maps. In this work, we will focus
in higher order polynomial maps, particularly cubic, and we will be able to
determine a simple way to create regular-reversal polynomial maps of any
degrees. We can also define _reversal-regular maps_ in a similar way.

###  2.2 Quadratic maps

Consider the family of _quadratic discrete dynamical systems_ given by

$y_{n+1}=\alpha(\lambda)y_{n}^{2}+(\beta(\lambda)+1)y_{n}+\gamma(\lambda),$
(2.3)

where $\alpha$, $\beta$ and $\gamma$ are smooth functions of the parameter
$\lambda$ and $\alpha(\lambda)\neq 0$ for all $\lambda$.

The fixed points of this system are obtained by solving the general quadratic
equation

$\alpha(\lambda)y^{2}+\beta(\lambda)y+\gamma(\lambda)=0$

and they are given by the general quadratic formula

$y_{0}=(2\alpha)^{-1}\left(-\beta+\sqrt{\beta^{2}-4\alpha\gamma}\right),\quad
y_{1}=(2\alpha)^{-1}\left(-\beta-\sqrt{\beta^{2}-4\alpha\gamma}\right).$

We will assume that $\beta^{2}-4\alpha\gamma>0$ in order to have two distinct
real fixed points. We can simplify the system introducing the change of
variable

$x_{n}=Ay_{n}+B$

where $A=\alpha^{-1}$ and
$B=(2\alpha)^{-1}\left(-\beta-\sqrt{\beta^{2}-4\alpha\gamma}\right)=y_{1}$.
Note that $y=B$ is a fixed point of the system (2.3). We obtain the following
modified discrete dynamical system [Solís and Jódar(2004)]

$x_{n+1}=g(x_{n})\equiv x_{n}^{2}-b(\lambda)x_{n}=x_{n}(x_{n}-b(\lambda))$
(2.4)

where $b(\lambda)=-\beta(\lambda)-2\alpha(\lambda)B$. The map
$x_{n+1}=g(x_{n})$ is unimodal. Without loss of generality, we will restrict
to the case of positive and continuous $b(\lambda)$ for $\lambda>0$.

Under the above transformation, the fixed points of the new system are

$x_{0}=0,\quad x_{1}=1+b(\lambda)$ (2.5)

so that the eigenvalue function for each of them is given by

$\phi(x_{0})=\frac{\partial g}{\partial
x}(x_{0})=-b(\lambda),\quad\phi(x_{1})=\frac{\partial g}{\partial
x}(x_{1})=2+b(\lambda).$

The corresponding regions of type 1 are

$\displaystyle x_{0}$ $\displaystyle:$
$\displaystyle\\{(\lambda,b(\lambda))|-1<b(\lambda)<1\\}$ $\displaystyle
x_{1}$ $\displaystyle:$
$\displaystyle\\{(\lambda,b(\lambda))|-3<b(\lambda)<-1\\}.$

There are also two periodic points of period two, which we can calculate by
solving

$x=g^{(2)}(x)=g(g(x))=x^{4}-2bx^{3}+b(b-1)x^{2}+b^{2}x.$

We can factor $x(x-(b+1))$ from the expression since it corresponds to the
fixed points (order 1) and we then solve for the roots of

$x^{2}+(1-b)x+(1-b)=0;$

and, again by the general quadratic formula, we have

$\displaystyle x_{1}^{2}$ $\displaystyle=$
$\displaystyle(2)^{-1}\left(b-1+\sqrt{b^{2}+2b-3}\right)$ $\displaystyle
x_{2}^{2}$ $\displaystyle=$
$\displaystyle(2)^{-1}\left(b-1-\sqrt{b^{2}+2b-3}\right)$

with eigenvalue function

$\phi_{2}(\lambda)=\frac{\partial g^{(2)}}{\partial
x}(x)=\left.4x^{3}-6bx^{2}+2b(b-1)x+b^{2}\right|_{x=x_{j}^{2}},\quad j=1,2$

and, therefore, corresponding type 2 region given by

$\\{(\lambda,b(\lambda))|1<b(\lambda)<\sqrt{6}-1\approx 1.45\\}.$

Moreover, there are $2^{N}$ periodic points of period $2^{N}$ with region of
type $N$ given by

$\\{(\lambda,b(\lambda))|b_{N-1}<b(\lambda)<b_{N}\\}$

where the sequence $\\{b_{k}\\}_{k=0}^{\infty}$ is convergent, and converges
to the limiting value $b_{\infty}=1.569945\cdots$ [Solís and Jódar(2004)]. The
sequence $\\{b_{k}\\}_{k=0}^{\infty}$ is related to the one described by
[Feigenbaum(1978)] in [Feigenbaum(1978)] and converges exponentially. Table
2.2 summarizes this results.

Type of the region ($k$) | Interval [$(b_{k},b_{k+1})$]
---|---
1 ($x_{0}$) | (-3, -1)
1 ($x_{1}$) | (-1, 1)
2 ($x_{1}^{2},x_{2}^{2}$) | (1, $\sqrt{6}-1\approx 1.45)$
$\vdots$ | $\vdots$
$\infty$ | ($b_{\infty}=1.56994567...,\infty$)
Table 2.2: Regions of type $k$ for the quadratic map
$x_{n+1}=x_{n}^{2}-b(\lambda)x_{n}$.

The values of the original Feigenbaum sequence,
$\\{\lambda_{k}\\}_{k=0}^{\infty}$, correspond to $\lambda$ values of the
points of intersection of the function $b(\lambda)$ with the the values of the
sequence $\\{b_{k}\\}_{k=0}^{\infty}$. Stated otherwise, the sequence
$\\{b_{k}\\}_{k=0}^{\infty}$ results from evaluating the Feigenbaum sequence
$\\{\lambda_{k}\\}_{k=0}^{\infty}$ in the function $b(\lambda)$. When the
function $b(\lambda)$ is the identity function, we can recover the original
Feigenbaum’s sequence.

_Feigenbaum’s sequence_ , $\\{\lambda_{k}\\}_{k=0}^{\infty}$, describes the
values of the parameter $\lambda$ for which the _period doubling bifurcation_
takes place in a unimodal map. The sequence converges exponentially. The
values $\\{\lambda_{k}\\}_{k=0}^{\infty}$ for a given iteration function $g$
–of the following form– are given implicitly by the definition of
[Feigenbaum(1978)] in its original paper [Feigenbaum(1978)], for a map $f$ as

$x_{p}^{2^{k}}=g^{2^{k}}(x_{p}^{2^{k}})=(\lambda_{k}f)^{2^{k}}(x_{p}^{2^{k}}),\quad\lambda\in\mathbb{N},$

for the periodic point of period $2^{k}$, $x_{p}^{2^{k}}$, where we must solve
for $\lambda_{k}$. In table 2.4 the values of a Feigenbaum sequence are shown
for a specific iteration function.

A simple way to estimate the values of Feigenbaum sequences is starting from
Feigenbaum’s constant, $\delta=4.669201609...$, and the first calculated
values of the sequence $\\{\lambda_{k}\\}_{k\in\mathbb{N}}$, through the
approximation

$\lambda_{k+1}=\lambda_{k}+\frac{\lambda_{k}-\lambda_{k-1}}{\delta}.$ (2.6)

This estimate is used as a predictor for the next value of the sequence and
tends to be more precise as $k\rightarrow\infty$.

Figure 2.4: Feigenbaum’s sequence for $f(x)=1-2x^{2}$. $N$ denotes the cycle
of order $2^{N}$. The parameter is denoted as $\lambda$.
$\delta_{N}=(\lambda_{N+1}-\lambda_{N})/(\lambda_{N+2}-\lambda_{N+1})$ and
$\delta_{N}^{\prime}=(\delta_{N+1}-\delta_{N})/(\delta_{N+2}-\delta_{N+1})$.
Reproduced from [Feigenbaum(1978)]

###  2.3 Types of periodic points and chaos

By the known properties of the sequence $\\{b_{k}\\}_{k=0}^{\infty}$ and
Sarkovskii’s theorem, we can state the following propositions [Solís and
Jódar(2004)]:

######  Proposition 2.1.

If the function $b(\lambda)$ is bounded from above by $b_{N}$, the map
$x_{n+1}=g(x_{n})\equiv x_{n}^{2}-b(\lambda)x_{n}$ can only have periodic
points of period $2^{m}$ with $m<N$. Moreover, if the bound is given by
$b_{\infty}$, the system is not chaotic and it can only have periodic points
of periods of powers of two.

It is worth remarking that there are unimodal maps that have no periodic
points with periods greater than one, in which case the bifurcation diagram
consists of a single branch given by the fixed point.

######  Proposition 2.2.

Suppose that the function $b(\lambda)$ is bounded. If the supreme of the
function $b(\lambda)$ lies within the interval $(b_{n},\,b_{n+1})$, then the
map $x_{n+1}=x_{n}^{2}-b(\lambda)x_{n}$ only has periodic points of period
$2^{k}$ with $k\in\\{1,\,2,\,...,\,n\\}$ and therefore, it is _not_ chaotic.

In order for the system $x_{n+1}=x_{n}^{2}-b(\lambda)x_{n}$ to be a regular
reversal map for $\lambda$ in some interval
$\mathcal{A}=(\lambda_{I},\,\lambda_{F})$, it is necessary that the function
$b(\lambda)$ has the following property: there exist
$\lambda_{1}<\lambda_{2}<\lambda_{3}$ in $\mathcal{A}$ such that
$b(\lambda_{1})<b(\lambda_{3})=b_{i}<b(\lambda_{2})$ for some $i\in\mathbb{N}$
where $b_{i}$ is the $i$-th element of the sequence
$\\{b_{k}\\}_{k=0}^{\infty}$. It is relatively easy to construct functions
$b(\lambda)$ with the aforementioned property, which means there is a large
class of unimodal maps that are _not_ chaotic.

Beyond the value $b_{\infty}$, there are other regions that divide the
positive quadrant of $\mathbb{R}^{2}$. The upper region of $\mathbb{R}^{2}$,
that is, the region $\\{(\lambda,b(\lambda))|b(\lambda)>b_{\infty}\\}$, can be
defined as the region of type $\infty$ and corresponds to divergent orbits. It
is not clear how this region is subdivided and, in general, this is an open
question.

###  2.4 Quadratic examples

The goal of the following examples is to show that regular reversal maps are a
very broad family and that it is relatively simple to construct maps with
specific – _ad hoc_ – bifurcation diagrams with or without chaos.

######  Example 2.3.

Let $b(\lambda)$ be a quadratic unimodal function given by

$b(\lambda)=\frac{4\alpha_{0}}{\beta_{0}^{2}}\lambda(\beta_{0}-\lambda),$

with $\alpha_{0},\,\beta_{0}>0$. Note that $\alpha_{0}$ is the maximum value
and $\beta_{0}$ is a root of $b(\lambda)$ (the other root is $\lambda=0$). We
are interested in knowing the periodic point structure of the map
$x_{n+1}=x_{n}^{2}-b(\lambda)x_{n}$ as we vary the parameter $\alpha_{0}$.
Fixing $\beta_{0}=2$, we then have the following cases:

  * •

If $0<\alpha_{0}<b_{0}=1$, we have a bifurcation diagram consisting of a
single straight line coincident with the $\lambda$ axis.

  * •

If $1<\alpha_{0}<b_{2}=\sqrt{6}-1$, we obtain a regular reversal map with a
bifurcation diagram consisting of a straight line, as the previous case, but
with a closed asymmetrical loop homeomorphic to a circle (see figure 2.6a on
page 2.6a). The lack of symmetry stems from the fact that the upper branch of
the loop corresponds the the periodic point $x_{1}^{2}$ and the lower branch
to $x_{2}^{2}$. The diameter of the loop is calculated by finding the two
intersections of $b(\lambda)$ with the line $b=b_{1}=1$ and it is equal to
$2\beta_{0}\sqrt{\alpha_{0}^{2}-1}\alpha_{0}$. The graph of $b(\lambda)$ is
shown in figure 2.5a at page 2.5a.

  * •

If $b_{1}=1<\alpha_{0}<\sqrt{6}-1=b_{2}$, two new loops appear: one that comes
out of the lower branch of the previous case and another one from the upper
branch, corresponding to the periodic points of period $2^{2}$ (see figure
2.6b). The matching graph of $b(\lambda)$ is shown in figure 2.5b.

  * •

If $b_{n}<\alpha_{0}<b_{n+1}$ for some $n\in\mathbb{N}$, the graph of
$b(\lambda)$ looks like in figure 2.5c and we have a bifurcation diagram
consisting of a straight line with a set of nested loops as shown in figure
2.6c.

  * •

If $\alpha_{0}=1.56994567...=b_{\infty}$, then we have branches that bifurcate
to form infinite nested loops, as is shown in figure 2.6d and whose graph of
$b(\lambda)$ is shown in figure 2.5d.

(a) $\alpha_{0}=1.2$ and $\beta_{0}=2$.

(b) $\alpha_{0}=1.5$ and $\beta_{0}=2$.

(c) $\alpha_{0}=1.562$ and $\beta_{0}=2$.

(d) $\alpha_{0}=b_{\infty}\approx 1.57$ and $\beta_{0}=2$.

Figure 2.5: $b(\lambda)$ functions of the quadratic regular reversal maps of
example 2.3. The horizontal lines correspond to the values of $b_{k}$, for
$k=1,2$ and $\infty$.

(a) $\alpha_{0}=1.2$ y $\beta_{0}=2$.

(b) $b_{2}<\alpha_{0}=1.5<b_{3}$ and $\beta_{0}=2$.

(c) $b_{3}<\alpha_{0}=1.562<b_{4}$ y $\beta_{0}=2$.

(d) $\alpha_{0}=1.57\approx b_{\infty}$ and $\beta_{0}=2$.

Figure 2.6: Bifurcation diagrams of the quadratic regular reversal maps of
example 2.3.

######  Example 2.4.

If we modify $b(\lambda)$ choosing it as periodic “unimodal”222Formally, this
cannot be a unimodal function, since it is a requirement for a unimodal
function to have a single maximum of the form

$b(\lambda)=\alpha\sin{2\pi(\lambda+3/2)}+1,$ (2.7)

with $1+\alpha<b_{\infty}$), the new bifurcation diagrams will be very similar
to those of the last case, except that they will be periodic. In figure 2.7 at
page 2.7, the graphs of $b(\lambda)$ are shown, and in figure 2.8, are the
corresponding bifurcation diagrams, both for the corresponding parameter cases
of figures 2.5 and 2.6, respectively.

(a) $\alpha=0.2$.

(b) $\alpha=0.5$.

(c) $\alpha=0.562$.

(d) $\alpha=0.57$. ($1+\alpha\approx b_{\infty}$.)

Figure 2.7: Functions $b(\lambda)$ of the regular reversal maps of example 2.4
given by the equation 2.4. The horizontal lines correspond to the values of
$b_{k}$, for $k=1,2$ and $\infty$.

(a) $\alpha=0.2$.

(b) $b_{2}<1+\alpha=1.5<b_{3}$.

(c) $b_{3}<1+\alpha=1.562<b_{4}$.

(d) $1+\alpha=1.57\approx b_{\infty}$.

Figure 2.8: Bifurcation the regular reversal maps of example 2.4.

######  Example 2.5.

If we introduce an asymmetry factor to the function 2.7 we can achieve an
asymmetrical bifurcation diagram with the different lobes having different
amplitudes. One way to accomplish the above is defining $b(\lambda)$ as

$b(\lambda)=\left(1-e^{-\lambda}\right)\sin^{2}\left(\pi\lambda/4\right)\\{\alpha\sin\left[\pi\left(\lambda+3/2\right)\right]+4/5\\}+4/5.$
(2.8)

Varying the parameter $\alpha$ we obtain graphs of $b(\lambda)$ like the ones
shown in figure 2.9 at page 2.9, with its respective bifurcation diagrams,
shown in figure 2.10 at page 2.10. For small values of $\alpha$ we have two
lobes (or “humps”) in the function $b(\lambda)$, with its local maxima under
the value $b_{2}$, which translates into a bifurcation diagram with four
asymmetrical lobes and of different amplitude (see figure 2.10a at page
2.10a). However, due to asymmetry, rising the value of $\alpha$ to $0.4$ will
only produce the appearance of nested loops in two of the right lobes of the
bifurcation diagram (see figure 2.10b), which repeats itself as we continue to
raise the value of $\alpha$ (see figure 2.10c).

(a) $\alpha=0.2$.

(b) $\alpha=0.4$.

(c) $\alpha=0.5$.

Figure 2.9: Graphs of $b(\lambda)$, given by equation 2.8, for the regular
reverse maps of example 2.8.

(a) $\alpha=0.2$.

(b) $\alpha=0.4$.

(c) $\alpha=0.5$.

Figure 2.10: Bifurcation diagrams of the regular reversal maps of example 2.5.

######  Example 2.6.

A similar effect to the one of the last example can be achieved by taking
$b(\lambda)$ as a polynomial of fourth grade in $\lambda$, with its
coefficients chosen in such way that it is not _unimodal_. For example,

$b(\lambda)=-\left(\alpha\lambda^{4}-1.533\lambda^{3}+4.1083\lambda^{2}-4.166\lambda\right)$
(2.9)

with $\alpha\in(0.18,0.192)$. The function $b(\lambda)$ has three local
extremes (see figure 2.11 at page 2.11). In this case, the bifurcation diagram
is shown for different values of the parameter in figure 2.12 at page 2.12
where, for progressively higher values of $\alpha$ we progressively obtain
lobes that stem from the branches corresponding to the periodic points. In
this case the diagram is no symmetrical nor periodic because the function
$b(\lambda)$ is neither.

(a) $\alpha=0.19166$.

(b) $\alpha=0.191$.

(c) $\alpha=0.19$.

Figure 2.11: Graphs of $b(\lambda)$, given by equation 2.9, for the quadratic
regular reversal maps of example 2.6.

(a) $\alpha=0.19166$.

(b) $\alpha=0.191$.

(c) $\alpha=0.19$.

Figure 2.12: Bifurcation diagrams of the regular reversal maps of example 2.6.

With the above examples we have shown we can design discrete dynamical systems
in order to have specific bifurcation diagrams “on demand” with desired
properties. The fundamental fact that allows us to do the latter is to note
where the function $\phi(\lambda)$ intersects the values of the sequence
$\\{b_{k}\\}_{k=1}^{\infty}$. The detailed form of $\phi(\lambda)$ is
irrelevant as long as it does not cross a value of the sequence, for it only
affects the specific form of the “branches” of the periodic points in the
bifurcation diagram, but it does not alter the fundamental structure of the
set of periodic points.

##  Chapter 3 Quadratic maps revisited

Now, we will restate and expand the work of [Solís and Jódar(2004)] in a way
that will allow us to gain greater insight into the way we can control the
appearance of chaos in a parametric family of quadratic maps and we will be
able to generalize these results in a natural way for cubic maps and, later
on, for general polynomial maps of $n$-th degree.

###  3.1 General quadratic form

Consider the quadratic iteration function $f(y)$ as stated by the

######  Definition 3.1 (General Quadratic Map).

The _General Quadratic Map (GQM)_ is defined by

$f_{2}(y):=\alpha+(\beta+1)y+\gamma y^{2}=y-P_{f_{2}}(y),$ (3.1)

where

$P_{f_{2}}(y):=-(\alpha+\beta y+\gamma y^{2})$ (3.2)

and $\alpha$, $\beta$ and $\gamma$ are real coefficients that depend upon the
parameter $\lambda$ and are $C^{1}$. $P_{f_{2}}$ is called the Fixed Points
Polynomial (of the general quadratic map).

It is clear that any quadratic map can be put in this form. Although the minus
sign in the definition of $P_{f_{2}}$ may now seem “unnatural”, its function
will be clear shortly111Check page 3.2 below..

Evidently, the zeros of $P_{f_{2}}$ are the fixed points of $f_{2}$ –thence
the name of $P_{f_{2}}$– since, if $y_{0}$ is a root of $P_{f_{2}}$, then
$P_{f_{2}}(y_{0})=0$ implies $f_{2}(y_{0})=y_{0}$.

In the following, for simplicity, we will drop the subscript “2” from $f$ to
denote the CQM. Since the derivatives of $f$ are

$\displaystyle f^{\prime}(y)$ $\displaystyle=2\gamma y+\beta+1,$ (3.3)
$\displaystyle f^{\prime\prime}(y)$ $\displaystyle=2\gamma,$

straightforward calculations lead us to

######  Lemma 3.1.

$f$ has the following two roots and critical point, respectively,

$\displaystyle y_{0}^{r}$
$\displaystyle=-\frac{\beta+1+\sqrt{(\beta+1)^{2}-4\alpha\gamma}}{2\gamma},$
(3.4) $\displaystyle y_{1}^{r}$
$\displaystyle=-\frac{\beta+1-\sqrt{(\beta+1)^{2}-4\alpha\gamma}}{2\gamma},$
$\displaystyle y_{c}$ $\displaystyle=-\frac{\beta+1}{2\gamma},$

with corresponding extreme value

$f_{ext}=\frac{4\alpha\gamma-(\beta+1)^{2}}{4\gamma}=\alpha-\gamma y_{c}.$

And, finally, it has the fixed points

$\displaystyle y_{0}^{*}$
$\displaystyle=-\frac{\beta+\sqrt{{\beta}^{2}-4\,\alpha\,\gamma}}{2\,\gamma},$
(3.5) $\displaystyle y_{1}^{*}$
$\displaystyle=-\frac{\beta-\sqrt{{\beta}^{2}-4\,\alpha\,\gamma}-\beta}{2\,\gamma}.$

###  3.2 Linear factors form

Suppose we know both roots of $P_{f_{2}}$ and they are
$y_{0},\,y_{1}\in\mathbb{C}$, which exist by the fundamental theorem of
algebra. Then, by the factor theorem, we know that $(y-y_{i}),\,i\in\\{0,1\\}$
is a factor of $P_{f_{2}}$ and we can rewrite it as

$P_{f_{2}}(y)=M(y-y_{0})(y-y_{1}),\quad M\in\mathbb{R}\setminus\\{0\\}.$ (3.6)

Therefore, we can also rewrite $f_{2}$ as

$f_{2}(y)=y-M(y-y_{0})(y-y_{1}).$ (3.7)

Since $M\in\mathbb{R}\setminus\\{0\\}$, then we can define $M:=s\tilde{M}$,
where

$s:=\mathrm{sgn}(M)=\begin{cases}1,&\mathrm{if}\,M>0,\\\
-1,&\mathrm{if}\,M<0,\end{cases}$ (3.8)

is the sign function and $\tilde{M}:=|M|$; this definition will be useful for
our purposes further below.

With the above, we have

######  Definition 3.2 (Linear Factors Form of the Quadratic Map).

The GQM can be rewritten as

$f_{2}(y)=y-s\tilde{M}(y-y_{0})(y-y_{1}):=h_{2}(y,\,\lambda)$ (3.9)

where $M,\,y_{1}$ and $y_{2}$ are smooth functions of the parameter $\lambda$.
We call $h_{2}$ the _Linear Factors Form of the Quadratic Map (LFFQM)_.

Again, straightforward calculations lead us to

######  Lemma 3.2.

(3.9) has first and second derivatives with respect to $y$ given by

$\displaystyle h_{2}^{\prime}(y,\,\lambda)$
$\displaystyle=1-s\tilde{M}\left(2y-y_{0}-y_{1}\right),$ (3.10) $\displaystyle
h_{2}^{\prime\prime}(y,\,\lambda)$ $\displaystyle=-2\,s\tilde{M}.$

The roots of $h_{2}$ are

$\displaystyle y_{0}^{r}$
$\displaystyle=-\frac{\sqrt{s^{2}\tilde{M}^{2}(y_{1}-y_{0})^{2}+2s\tilde{M}(y_{1}+y_{0})+1}-s\tilde{M}(y_{1}+y_{0})-1}{2s\tilde{M}}$
(3.11) $\displaystyle y_{1}^{r}$
$\displaystyle=\frac{\sqrt{s^{2}\tilde{M}^{2}(y_{1}-y_{0})^{2}+2s\tilde{M}(y_{1}+y_{0})+1}+s\tilde{M}(y_{1}+y_{0})+1}{2s\tilde{M}}$

and the only critical point is at

$y_{c}=\frac{s\tilde{M}(y_{1}+y_{0})+1}{2s\tilde{M}},$

with corresponding extreme value

$h_{ext}=y_{0}+\frac{[s\tilde{M}(y_{1}-y_{0})+1]^{2}}{4s\tilde{M}}.$

Here the apparently “unnatural” minus sign of the definition 3.1 of the fixed
points polynomial $P_{f_{2}}$ will show its usefulness: for _purely aesthetic
reasons_ , in the definition of the LFFQM, we want that
$h_{2}^{\prime}(y_{0})>0$ whenever $0<y_{0}<y_{1}$ and $M>0$ which, without
loss of generality, is accomplished by arbitrarily introducing the mentioned
sign.

###  3.3 Canonical form

We will introduce a _canonical form_ of the quadratic map by requiring both
fixed points in the linear factors form (3.2) to be real or, equivalently,
$\beta^{2}-4\alpha\gamma>0$ in the general form (3.1); also, we will require
an “amplitude”, $M(\lambda)$, equal to unity222Recall $M(\lambda)$ is the
function preceding the product of the linear factors of the fixed points in
the LFFQM defined in Eq. (3.2)., a fixed point mapped to zero333This is the
reason we require both fixed points to be real, otherwise we would need a
complex transformation to map the complex fixed point to zero, which is beyond
the scope of this work. and the other fixed point being mapped to

$x_{1}(\lambda):=s\tilde{M}(y_{1}-y_{0}).$ (3.12)

The latter is accomplished by taking one of the following linear
transformations

$\displaystyle y=T_{0}(x)$ $\displaystyle:=y_{0}+s\tilde{M}^{-1}x,$ (3.13)
$\displaystyle y=T_{1}(x)$ $\displaystyle:=y_{1}+s\tilde{M}^{-1}x.$

Without loss of generality, we will take transformation $T_{0}$ and call it
simply $T$. Since $T$ is evidently linear, its inverse can be calculated
directly as

$x=T^{-1}(y)=s\tilde{M}(y-y_{0}).$ (3.14)

Performing the corresponding calculations on the LFFQM we obtain

######  Definition 3.3 (Canonical Quadratic Map).

The map defined by

$g_{2}(x,\lambda):=x-x(x-x_{1}(\lambda)),$ (3.15)

where $\lambda\in\mathbb{R}$ is a parameter, is called the _Canonical
Quadratic Map (CQM)_.

We have thus mapped the real fixed points $y_{0}\mapsto 0$ and $y_{1}\mapsto
x_{1}$. This form is much simpler than the previous ones and it has the
advantage of putting the whole parametric dependence onto the single non-
constant fixed point $x_{1}=x_{1}(\lambda)$ through (3.12). Implicit in this
dependence is the parametric dependence of the corresponding original roots of
the LFFQM (3.9), so we can explicitly give this dependence if desired.
Moreover, the fact that $T$ is actually a _topological conjugacy_ between
$h_{2}$ and $g_{2}$ will be proved in chapter 5, by which we can state that
_the study of stability and chaos in quadratic maps with real fixed points can
be summarized by the study of the Canonical Quadratic Map_ , which yields its
importance.

With some straightforward calculations, we easily arrive at the following

######  Lemma 3.3.

The Canonical Quadratic Map has the roots

$\displaystyle x_{0}^{r}$ $\displaystyle=0,$ (3.16) $\displaystyle x_{1}^{r}$
$\displaystyle=x_{1}+1,$

and has derivatives

$\displaystyle g_{2}^{\prime}(x,\lambda)$ $\displaystyle=-2x+x_{1}+1,$ (3.17)
$\displaystyle g_{2}^{\prime\prime}(x,\lambda)$ $\displaystyle=-2,$

so that its critical point is at

$x_{c}=\frac{x_{1}+1}{2}$

with value

$x_{m}=\frac{(x_{1}+1)^{2}}{4}=x_{c}^{2}.$ (3.18)

We see then that the second root of the canonical map, $x_{1}^{r}$, is always
one unit to the right of the non-constant fixed point $x_{1}$. Also, the
critical point is always a _maximum_ and is located in the middle point
between 1 and $x_{1}$, with its value being simply the square of the point.

###  3.4 Stability and chaos in the canonical quadratic map

####  3.4.1 Fixed points

The stability of the fixed points is given by the absolute value of its
multiplier, which must be less than one. Making use of the defined eigenvalue
function, we have from (3.17) that for the fixed point $x_{0}=0$,

$\phi_{0}(\lambda)=g_{2}^{\prime}(0)=x_{1}(\lambda)+1$ (3.19)

so that it is straightforward to see that if

$-2<x_{1}(\lambda)<0$ (SC0)

then $x_{0}=0$ is a stable fixed point. We will call inequality SC0 the
_stability condition_ for the fixed point $x_{0}=0$. We see then that the
stability of $x_{0}$ actually depends on the value of $x_{1}$. In terms of the
original fixed points of the quadratic map, $y_{0}$ and $y_{1}$, what counts
is the separation between them scaled by the factor $s\tilde{M}$, as can be
seen from equation (3.12).

On the other hand, for $x_{1}$,

$\phi_{1}(\lambda)=g_{2}^{\prime}(x_{1})=1-x_{1}$ (3.20)

so that $x_{1}$ is stable as long as

$0<x_{1}(\lambda)<2.$ (SC1)

We call the value $b_{1}=2$ our first _bifurcation value_. Notice that
$-b_{1}=-2$ is also a bifurcation value since it is the limit of the stability
condition for $x_{0}=0$, given by (SC0). We summarize this results in the
following

######  Proposition 3.4.

Let $x_{0}=0$ and $x_{1}$ be the two fixed points of the canonical quadratic
map as defined above. Then

  * •

if $-2<x_{1}<0$, zero will be an asymptotically stable fixed point;

  * •

if $0<x_{1}<2$, $x_{1}$ will be an asymptotically stable fixed point.

####  3.4.2 Periodic points of period two

To determine the period two periodic points we must of course solve

$g_{2}^{2}(x)=x.$

Some basic algebra shows that

$g_{2}^{2}(x)=(x_{1}+1)^{2}x-(x_{1}+1)(x_{1}+2)x^{2}+2(x_{1}+1)x^{3}-x^{4}.$

Therefore, we must solve

$x_{1}(x_{1}+2)x-(x_{1}+1)(x_{1}+2)x^{2}+2(x_{1}+1)x^{3}-x^{4}=0.$

Clearly, we must factor out $x=0$ and $x=x_{1}$ since we know those are the
fixed points of the original map, and we are looking (only) for period two
points. Performing the factorization, we get

$x^{2}-(x_{1}+2)x+(x_{1}+2)=0.$ (3.21)

Solving the latter equation we get to the following

######  Lemma 3.5.

The canonical quadratic map has a unique 2-cycle given by

$x_{0}^{2}=\frac{1}{2}\left[x_{1}+2+\sqrt{(x_{1})^{2}-4}\right],\quad
x_{1}^{2}=\frac{1}{2}\left[x_{1}+2-\sqrt{(x_{1})^{2}-4}\right],$

where the superscript will indicate the period of the periodic point.

###### Proof.

It is straightforward to see that $g_{2}(x_{0}^{2})=x_{1}^{2}$ and
$g_{2}(x_{1}^{2})=x_{0}^{2}$, therefore proving that both $x_{0}^{2}$ and
$x_{1}^{2}$ are period 2 periodic points of $g_{2}$ and the orbit
$\\{x_{0}^{2},\,x_{1}^{2}\\}$ is indeed a 2-cycle. The uniqueness comes from
the fact that both are algebraic roots of the second degree equation (3.21)
and every such period 2 periodic point must be so. ∎

To analyze the stability of this 2-cycle, we must calculate its own eigenvalue
function. The derivative of $g_{2}^{2}$ is

$\frac{\partial g_{2}^{2}}{\partial
x}(x)=(x_{1}+1)^{2}-2(x_{1}+1)(x_{1}+2)x+6(x_{1}+1)x^{2}-4x^{3},$

so that evaluating in $x=x_{0}^{2}(\lambda)$ and $x=x_{1}^{2}(\lambda)$ we
have

$\left(g^{2}_{2}\right)^{\prime}(x_{0}^{2}(\lambda))=\left(g^{2}_{2}\right)^{\prime}(x_{1}^{2}(\lambda))=\phi_{2}(\lambda)=5-x_{1}(\lambda)^{2},$
(3.22)

so that, not surprisingly, both points of the 2-cycle have the same stability
criteria. The stability region is then determined by

$|5-(x_{1})^{2}|<1$

whose solutions lead us to

######  Lemma 3.6.

The unique 2-cycle of the canonical quadratic map is asymptotically stable
whenever one of the following inequalities is met

$\displaystyle-\sqrt{6}$ $\displaystyle<x_{1}$ $\displaystyle<-2,$
$\displaystyle 2$ $\displaystyle<x_{1}$ $\displaystyle<\sqrt{6},$

where $x_{1}$ is the nonzero fixed point of the CQM.

So, we see that the stability region is divided in two in terms of the value
of the original fixed point $x_{1}$. As a reference, $\sqrt{6}\approx
2.44949$. We call this value $b_{2}=\sqrt{6}$ the second bifurcation value of
the parameter $\lambda$ for the canonical quadratic map. Notice again that
$-b_{2}=-\sqrt{6}$ is also a bifurcation value.

It is worth noting that in the case of the first inequality of lemma 3.6, it
is the zero fixed point that produces a bifurcation, while for the second one,
it is $x_{1}$ that bifurcates to produce the 2-cycle. The latter can be seen
from lemma 3.5 since, for $|x_{1}|<2$, the 2-cycle is not real and it only
begins to be so at the bifurcation values $\pm b_{1}$; when $x_{1}=-2$, both
$x_{0}^{2}$ and $x_{1}^{2}$ are zero (and equal to the zero fixed point) and
thereon separate as the two branches of the square root function (positive and
negative). On the other hand, when $x_{1}=2$, both $x_{0}^{2}$ and $x_{1}^{2}$
are also 2 in value and thereon separate in the same way as before.

We can now prove that, indeed, what happens at the bifurcation values of $\pm
b_{1}=\pm 2$ are period doubling bifurcations.

######  Proposition 3.7.

Let444The dot represents derivative with respect to the parameter $\lambda$ to
avoid confusion with the apostrophe representing derivative with respect to
$x$. $x_{1}(\lambda_{0})=-2$ (respectively, $x_{1}(\lambda_{0})=2$) and
$\dot{x_{1}}(\lambda_{0})\neq 0$. Then the zero fixed point (respectively, the
$x_{1}$ fixed point) of the CQM undergoes a period doubling bifurcation
precisely when $\lambda=\lambda_{0}$.

###### Proof.

According to theorem 1.12 we must prove that:

  1. 1.

$g_{2}(0)=0$ for all $\lambda$ in an interval around $\lambda=-2$. This is
trivially true since the zero fixed point does not vary with the parameter.

  2. 2.

$g_{2}^{\prime}(0)=-1$. Which follows directly from eq. 3.19.

  3. 3.

$\left.\frac{\partial\left(g^{2}_{2}\right)^{\prime}}{\partial\lambda}\right|_{\lambda=-2}(0)\neq
0$. From (3.22) we see that this is true since $\dot{x_{1}}(\lambda)\neq 0$ by
hypothesis.

The case for the period doubling bifurcation of the other fixed point when
$x_{1}(\lambda)=2$ is analogous. ∎

The requirement $\dot{x_{1}}(\lambda)\neq 0$ will be understood more clearly
in the examples below. Likewise, we can prove that, in turn, the fixed point
$x_{0}^{2}$ (or $x_{1}^{2}$) of $g_{2}^{2}$ —periodic point of period 2 for
$g_{2}$— also undergoes a period doubling bifurcation associated with the
second bifurcation value when $x_{1}(\lambda)=\pm b_{2}=\pm\sqrt{6}$.

######  Proposition 3.8.

Let $x_{1}(\lambda_{0})=-\sqrt{6}$ and $\dot{x_{1}}(\lambda_{0})\neq 0$. Then
the fixed points $x_{0}^{2}$ and $x_{1}^{2}$ of the $g_{2}^{2}$ undergo a
period doubling bifurcation precisely when $\lambda=\lambda_{0}$.

###### Proof.

Consider first the case for $x_{0}^{2}$ given by lemma 3.5. According to
theorem 1.12 we must prove that:

  1. 1.

$\left(g_{2}^{2}\right)^{\prime}(x_{0}^{2})=-1$. Which follows directly from
eq. 3.22 since by hypothesis $x_{1}(\lambda_{0})=-\sqrt{6}$.

  2. 2.

$g_{2}^{2}(x_{0}^{2})=x_{0}^{2}$ for all $\lambda$ in an interval around
$\lambda=-\sqrt{6}$. Theorem 1.11 allows us to affirm this since by the above
point $\left(g_{2}^{2}\right)^{\prime}(x_{0}^{2})\neq 1$.

  3. 3.

$\left.\frac{\partial\left(g^{4}_{2}\right)^{\prime}}{\partial\lambda}\right|_{\lambda=\lambda_{0}}(x_{0}^{2})\neq
0$. Since, by the chain rule,

$\displaystyle\left(g_{2}^{4}(x_{0}^{2})\right)^{\prime}$
$\displaystyle=\left(g_{2}^{2}\left(g_{2}^{2}(x_{0}^{2})\right)\right)^{\prime}$
$\displaystyle=\left(g_{2}^{2}\right)^{\prime}\left(g_{2}^{2}(x_{0}^{2})\right)\,\left(g_{2}^{2}(x_{0}^{2})\right)^{\prime}$
$\displaystyle=\left(g_{2}^{2}(x_{0}^{2})\right)^{\prime}\left(1-(x_{1})^{2}\right)$
$\displaystyle=\left(1-(x_{1})^{2}\right)^{2}$

using eq. (3.22) and the fact that $x_{0}^{2}$ is a fixed point of
$g_{2}^{2}$. The partial derivative with respect to $\lambda$ is then

$\displaystyle\frac{\partial\left(g^{4}_{2}\right)^{\prime}}{\partial\lambda}(x_{0}^{2}(\lambda_{0}))$
$\displaystyle=-4\left(1-(x_{1}(\lambda_{0}))^{2}\right)(x_{1}(\lambda_{0}))^{2}\dot{x_{1}}(\lambda_{0})$
$\displaystyle=120\,\dot{x_{1}}(\lambda_{0})\neq 0$

since $x_{1}(\lambda_{0})=-\sqrt{6}$ and $\dot{x_{1}}(\lambda_{0})\neq 0$ by
hypothesis.

The cases for $x_{1}^{2}$ and $\lambda_{0}=-\sqrt{6}$ are analogous. ∎

####  3.4.3 Periodic points of higher period

Some more algebra shows us that

$g_{2}^{3}(x)=x\,\left({x_{1}}^{3}+3\,{x_{1}}^{2}+3\,x_{1}+1\right)+{x}^{2}\,\left(-{x_{1}}^{4}-5\,{x_{1}}^{3}-10\,{x_{1}}^{2}-9\,x_{1}-3\right)\\\
+{x}^{3}\,\left(2\,{x_{1}}^{4}+10\,{x_{1}}^{3}+20\,{x_{1}}^{2}+18\,x_{1}+6\right)+{x}^{4}\,\left(-{x_{1}}^{4}-10\,{x_{1}}^{3}-25\,{x_{1}}^{2}-25\,x_{1}-9\right)\\\
+{x}^{5}\,\left(4\,{x_{1}}^{3}+18\,{x_{1}}^{2}+24\,x_{1}+10\right)+{x}^{6}\,\left(-6\,{x_{1}}^{2}-14\,x_{1}-8\right)+{x}^{7}\,\left(4\,x_{1}+4\right)-{x}^{8}$
(3.23)

with derivative

$\left(g_{2}^{3}(x)\right)^{\prime}=\left({x_{1}}^{3}+3\,{x_{1}}^{2}+3\,x_{1}+1\right)+x\,\left(-2\,{x_{1}}^{4}-10\,{x_{1}}^{3}-20\,{x_{1}}^{2}-18\,x_{1}-6\right)\\\
+{x}^{2}\,\left(6\,{x_{1}}^{4}+30\,{x_{1}}^{3}+60\,{x_{1}}^{2}+54\,x_{1}+18\right)+{x}^{3}\,\left(-4\,{x_{1}}^{4}-40\,{x_{1}}^{3}-100\,{x_{1}}^{2}-100\,x_{1}-36\right)\\\
+{x}^{4}\left(20\,{x_{1}}^{3}+90\,{x_{1}}^{2}+120\,x_{1}+50\right)+{x}^{5}\,\left(-36\,{x_{1}}^{2}-84\,x_{1}-48\right)+{x}^{6}\,\left(28\,x_{1}+28\right)-8\,{x}^{7}$
(3.24)

So that, in order to find the periodic points of period three we must, in
principle, explicitly solve

$-x^{7}+4(x_{1}+1)\,x^{6}-(6\,x_{1}^{2}+14\,x_{1}+8)\,x^{5}+(4\,x_{1}^{3}+18\,x_{1}^{2}+24\,x_{1}+10)\,x^{4}\\\
-(x_{1}^{4}+10\,x_{1}^{3}+25\,x_{1}^{2}+25\,x_{1}+9)\,x^{3}+(2\,x_{1}^{4}+10\,x_{1}^{3}+20\,x_{1}^{2}+18\,x_{1}+6)\,x^{2}\\\
-(x_{1}^{4}+5\,x_{1}^{3}+10\,x_{1}^{2}+9\,x_{1}+3)\,x+(x_{1}^{3}+3\,x_{1}^{2}+3\,x_{1}+1)=0$
(3.25)

which is an seventh degree polynomial in $x$. Even factoring out the other
known fixed point of $g_{2}$, $x_{1}$, we are still left with a sixth degree
polynomial on $x$ which is, in general, impossible to solve explicitly by
algebraic means. Nevertheless, for specific polynomials we can determine the
roots numerically.

Much less can be said about determining higher period periodic points.
However, we may still approximate the stability regions of these $k$-cycles by
numerically determining the bifurcation value of the parameter $\lambda$,
which in this case we have done by plotting very high-precision bifurcation
diagrams (see Appendix A). One way to determine an approximation to the value
of the infinite period bifurcation value, after which the onset of chaos takes
place, is through equation (2.6), which yields a very rough estimate of
$b_{\infty}\approx 2.57$. Some bifurcation values for the CQM found in this
work are shown in table 3.1; moreover, table 3.2 shows the the stability
conditions for each periodic point,up to period $2^{7}$; the boundary of the
stability bands correspond to the $b_{k}$ bifurcation values.

$k$ | $b_{k}$
---|---
0 | 0
1 | 2
2 | $\sqrt{6}$
3 | $2.5440\pm 0.0005$
4 | $2.5642\pm 0.0002$
5 | $2.56871\pm 4\times 10^{-5}$
6 | $2.56966\pm 1\times 10^{-5}$
7 | $2.569881\pm 5\times 10^{-6}$
$\vdots$ | $\vdots$
$\infty$ | $\sim 2.569941\pm 5\times 10^{-7}$
Table 3.1: Bifurcation values for the Canonical Quadratic Map. $b_{0}=0$ is included only as a reference, although it is not a bifurcation value. Period | Periodic Points | Stability Condition
---|---|---
1 | $x_{0}^{1}=0$ | $-b_{1}<x_{1}<b_{0}$
1 | $x_{1}^{1}=x_{1}$ | $b_{0}<x_{1}<b_{1}$
2 | $x_{0}^{2}=\frac{1}{2}\left[x_{1}+2+\sqrt{(x_{1})^{2}-4}\right]$ | $b_{1}<|x_{1}|<b_{2}$
| $x_{1}^{2}=\frac{1}{2}\left[x_{1}+2-\sqrt{(x_{1})^{2}-4}\right]$ |
4 | $x_{0}^{4},\,x_{1}^{4},\,x_{2}^{4},\,x_{3}^{4}$ | $b_{2}<|x_{1}|<b_{3}$
8 | $x_{0}^{8},\,x_{1}^{8},\,...,x_{7}^{8}$ | $b_{3}<|x_{1}|<b_{4}$
16 | $x_{0}^{16},\,x_{1}^{16},\,...,x_{15}^{16}$ | $b_{4}<|x_{1}|<b_{5}$
32 | $x_{0}^{32},\,x_{1}^{32},\,...,x_{31}^{32}$ | $b_{5}<|x_{1}|<b_{6}$
64 | $x_{0}^{64},\,x_{1}^{64},\,...,x_{63}^{64}$ | $b_{6}<|x_{1}|<b_{7}$
⋮ | ⋮ | ⋮
$\infty$ | - | $|x_{1}|>b_{\infty}$
Table 3.2: Periodic points and corresponding stability conditions for the
canonical quadratic map.

####  3.4.4 Fixed points with multiplicity

In the CQM, the only way we can have multiplicity in the fixed points is when
$x_{1}(\lambda)=x_{0}=0$. In this case, $g_{2}$ takes the form

$g_{2}(x)=x\,(1-x).$ (3.26)

Its derivative is then

$g_{2}^{\prime}(x)=1-2x,$ (3.27)

and, therefore, $g_{2}^{\prime}(0)=1$, so that it is a nonhyperbolic fixed
point, according to definition 1.6.

######  Proposition 3.9.

Let $g_{2}$ be the CQM with a single fixed point with multiplicity of two, as
in eq. (3.26). Then the single fixed point $x_{0}=x_{1}=0$ is semistable from
the right.

###### Proof.

It is straightforward to see that $g_{2}^{\prime\prime}(x)=-2$ and then
$g_{2}^{\prime\prime\prime}(x)=0$ for all $x$. Therefore, in particular,
$g_{2}^{\prime\prime}(0)=-2\neq 0$ and then $g_{2}^{\prime\prime\prime}(0)=0$,
so that, by the latter, we cannot use theorem 1.7. However, theorem 1.9 tells
us that the fixed point is semistable from the right. Indeed, as $g_{2}$ is
concave downward at $x=0$, then $g_{2}^{\prime}$ is decreasing in a small
neighborhood about zero and, since $g_{2}^{\prime}(0)=1$ and $g_{2}^{\prime}$
is continuous, we have that there exists $\delta>0$, so that
$g_{2}^{\prime}(x)>1$ for $x\in(-\delta,\,0)$ and $g_{2}^{\prime}(x)<1$ for
$x\in(0,\,\delta)$. Consequently, $x_{0}=x_{1}=0$ is unstable from the left
and semistable from the right, according to definition 1.9 and the proof of
theorem 1.1 [Elaydi(2000)]. ∎

It is worth noting that, although the case of multiplicity in the fixed point
of the CQM is possible, the case of _complex_ fixed points is not, since the
CQM requires one fixed point to be zero, then its complex conjugate —which
should also be a fixed point— is itself.

####  3.4.5 Regular-reversality and chaos in the canonical quadratic map

Let us recall, that the definition of $x_{1}$ transforms, through $T$, the
parametric dependence of the original fixed points of the general quadratic
map into a single parameter-dependent function $x_{1}=x_{1}(\lambda)$. Recall
then that,

$x_{1}(\lambda)=M(\lambda)\,\left(y_{1}(\lambda)-y_{0}(\lambda)\right).$

Therefore, assuming a constant value of $M(\lambda)=M_{0}=1$, we see that the
value of $x_{1}$ is exactly the difference between the two original fixed
points, $y_{1}$ and $y_{0}$, of the General Quadratic Map (which we forced to
be real). We can then interpret the found values of the sequence
$\\{b_{k}\\}_{k\in\mathbb{N}}$ as determining the

######  Definition 3.4 (Stability Bands of a Quadratic Map).

Let $y_{i}:\mathcal{A}\subseteq\mathbb{R}\rightarrow\mathbb{R}$,
$i\in\\{1,\,2\\}$ be the two fixed points of the family of quadratic maps
$f_{\lambda}$, as given by 3.1, and $\\{b_{k}\\}_{k\in\mathbb{N}}$ the
sequence of bifurcation values of table 3.1. The union of open intervals

$\left(y_{i}(\lambda)-b_{k+1},\,y_{i}(\lambda)-b_{k}\right)\bigcup\left(y_{i}(\lambda)+b_{k},\,y_{i}(\lambda)+b_{k+1}\right),\quad\lambda\in\mathcal{A}$
(3.28)

is called the $k$-th _stability band_ of $y_{i}$.

Graphically, we see then that stability bands run along the values of
$y_{1}(\lambda)$ and $y_{0}(\lambda)$ as functions of the parameter and, as
long as the value of one fixed point does not cross a the limit of a band of
the other fixed point, there will be no changes in the periodic points
structure of the map, i.e. there will only be bifurcations when the value of
one fixed point crosses the limit of a stability band of the other fixed
point. If $M$ were not held constant, it would simply act as a “modulator” for
the width of the bands along the parameter $\lambda$.

Stability bands for $x_{0}=0$ and $x_{1}(\lambda)$ in the CQM are defined
analogously.

######  Definition 3.5 (Stability Bands of the CQM).

Let $x_{1}:\mathcal{A}\subseteq\mathbb{R}\rightarrow\mathbb{R}$ be the nonzero
fixed point of the family of quadratic maps $g_{2}$, as given by definition
3.3, and $\\{b_{k}\\}_{k\in\mathbb{N}}$ the sequence of bifurcation values of
table 3.1. The union of open intervals

$\left(-b_{k+1},\,-b_{k}\right)\bigcup\left(b_{k},\,b_{k+1}\right),\quad\lambda\in\mathcal{A}$
(3.29)

is called the $k$-th _stability band_ of the CQM.

Notice that that the part of the $k$-th stability band in the upper
(respectively, lower) semi-plane corresponds to the stability condition of
period $2^{k}$ attracting periodic points associated with period doubling
bifurcation of the $x_{1}$ (respectively, zero) fixed point.

Finally, we must remark that the width of these stability bands —given by the
values of the sequence $\\{b_{k}\\}_{k\in\mathbb{N}}$— is not equal for
polynomial maps of different degrees, as we shall see in the next chapters.

By the above, the big picture is that depending on the absolute value of
$x_{1}(\lambda)$ the one-parameter family of canonical quadratic maps can
present or not attracting periodic points of different periods or even become
chaotic. In particular, as long as $|x_{1}(\lambda)|<b_{N}$ we will only have
attracting periodic points of period less or equal to $2^{N}$. We can thus
restate the propositions 2.1 and 2.2 given in section 2.3 of chapter 2,
originally by [Solís and Jódar(2004)] ([Solís and Jódar(2004)]).

######  Proposition 3.10.

If the absolute value of the fixed point $x_{1}^{*}(\lambda)$ is bounded from
above by $b_{N}$, the GQM map $x_{n+1}=g_{2}(x_{n})\equiv
x_{n}-x_{n}(x_{n}-x_{1}^{*}(\lambda))$ is not chaotic and can only have
periodic points of period $2^{m}$ with $m<N$. Moreover, if the bound is given
by $b_{\infty}$, the system is still not chaotic and it can only have periodic
points of periods of powers of two.

And more specifically,

######  Proposition 3.11.

Suppose that the absolute value of the fixed point $x_{1}^{*}(\lambda)$ is
bounded. If the supreme of the function $|x_{1}^{*}(\lambda)|$ lies within the
interval $(b_{n},\,b_{n+1})$, then the map $x_{n+1}=g_{2}(x_{n})\equiv
x_{n}-x_{n}(x_{n}-x_{1}^{*}(\lambda))$ only has periodic points of period
$2^{k}$ with $k\in\\{1,\,2,\,...,\,n\\}$ and therefore, it is _not_ chaotic.

In order for the system
$x_{n+1}=x_{n}-x_{n}\,\left(x_{n}-x_{1}(\lambda)\right)$ to be a regular
reversal map for $\lambda$ in some interval
$\mathcal{A}=(\lambda_{I},\,\lambda_{F})$, it is necessary that the continuous
function $x_{1}(\lambda)$ has the following property: there exist
$\lambda_{1}<\lambda_{2}<\lambda_{3}$ in $\mathcal{A}$ such that
$x_{1}(\lambda_{1})<x_{1}(\lambda_{3})=b_{i}<x_{1}(\lambda_{2})$ for some
$i\in\mathbb{N}$ where $b_{i}$ is the $i$-th element of the sequence of
bifurcation values $\\{b_{k}\\}_{k=0}^{\infty}$. It is easy to construct
functions $x_{1}(\lambda)$ with the aforementioned property, which means that
constructing regular-reversal canonical quadratic maps is also easy.
Furthermore, knowing beforehand the properties of a desired bifurcation
diagram, i.e. the fixed points, the $\lambda$ values for desired bifurcations,
etc. we can construct $x_{1}(\lambda)$ functions that accomplish the desired
properties in the bifurcation diagram.

###  3.5 Quadratic examples with the canonical map

Recall that the CQM is

$g_{2}(x,\lambda)=x-x\left(x-x_{1}(\lambda)\right),$

where

$x_{1}(\lambda)=M(\lambda)\left(y_{1}(\lambda)-y_{0}(\lambda)\right).$ (3.30)

Therefore, we can “parametrize” the map $g_{2}(x,\lambda)$ by defining either
$x_{1}(\lambda)$ directly or going back to the parametric dependence of the
original LFFQM $h_{2}(x,\lambda)$ from which $x_{1}$ was defined using the
transformation $T$, i.e. defining $M(\lambda)$, $y_{0}(\lambda)$ and
$y_{1}(\lambda)$. We will look at both interpretations at the same time in the
following examples.

####  3.5.1 Linear fixed points

We will begin with the simplest case: linearly varying $y_{0}(\lambda)$ and
$y_{1}(\lambda)$ and $M(\lambda)=1$. Consider,

$y_{0}(\lambda):=\lambda,\quad y_{1}(\lambda):=2\,\lambda.$

The graphs of these parametrizations of the fixed points are shown in figure
3.1. Clearly, $x_{1}(\lambda)=\lambda$ and its graph is also shown in figure
3.1.

(a) Graphs of linear $y_{0}(\lambda)$ and $y_{1}(\lambda)$ along the selected
values of $\lambda$. Also, the stability conditions (constrained by the
bifurcation values $b_{1}$, $b_{2}$ and $b_{\infty}$) of $y_{0}$ are shown
around it.

(b) Graph of corresponding $x_{1}(\lambda)$ along with the bifurcation values
$b_{1}$, $b_{2}$ and $b_{\infty}$.

Figure 3.1: Linear parametrization of fixed points (example 3.5.1).

The bifurcation diagram for this specific parametrization of the fixed points
of the CQM is shown in figure 3.2. In this bifurcation diagram we can see that
it is precisely at the values of $\lambda$ for which $y_{1}$ crosses limits of
the stability conditions of $y_{0}$ (or, equivalently, where $x_{1}$ crosses
the bifurcation values) that bifurcations take place. Notice that, since
$x_{1}>0$, it is $x_{1}$ the fixed point that undergoes period doubling
bifurcation.

Figure 3.2: Bifurcation diagram for the linear parametrization of the fixed
points example.

In fact, it does not matter what the actual values of the slopes of
$y_{0}(\lambda)$ and $y_{1}(\lambda)$ are: as long as their difference is the
same (has the same slope) they will produce exactly the same bifurcation
diagram, since the parametric dependence of $x_{1}(\lambda)$ will be unchanged
(see eq. 3.30). Making precise calculations of the bifurcations diagram near
$\lambda$-values of interest we can determine numerically the rest (or at
least some more) of the bifurcation values forming the sequence
$\\{b_{k}\\}_{k\in\mathbb{N}}$, but this is best accomplished using the
parametrization of example 3.5.5 below.

####  3.5.2 Quadratic fixed point

Redefining the parametric dependencies of $y_{0}$ and $y_{1}$ we can achieve
_regular-reversal maps_ as described in definition 2.6 in page 2.6. For
example, leaving $y_{0}$ in the above example 3.5.1, but redefining $y_{1}$
with a quadratic parametric dependence on $\lambda$ as

$\displaystyle y_{0}(\lambda)$ $\displaystyle:=\lambda$ (3.31) $\displaystyle
y_{1}(\lambda)$ $\displaystyle:=\lambda-p_{1}\lambda(\lambda-p_{2}),\quad
p_{1},\,p_{2}\in\mathbb{R}.$

We immediately see that zero is a root of the parametric form of $y_{1}$, and
the other root is $p_{2}$. Since it is exactly like the defined _Linear
Factors Form_ defined for the quadratic map in subsection 3.2 on page 3.2, we
know that its maximum lies at

$\lambda_{c}=\frac{p_{1}\,p_{2}+1}{2\,p_{1}},$

with corresponding value

$y_{1}^{max}=\lambda_{c}^{2}=\frac{(p_{1}\,p_{2}+1)^{2}}{4\,p_{1}^{2}}.$

Using, for example, the values, $p_{1}=1$ and $p_{2}=3$ we know then that
$\lambda_{c}=2$ and, therefore, $y_{1}^{max}=4$ (this functions are shown in
figure 3.3a. As for $x_{1}$ it is then

$x_{1}(\lambda)=\lambda(3-\lambda),$

where the maximum is clearly located at $\lambda=3/2$ with a value of
$x_{1}^{max}=9/4$. The graph of $x_{1}$ is shown in figure 3.1b. Also, $x_{1}$
crosses the first bifurcation value ($b_{1}=2$) two times at $\lambda=1$ and
$\lambda=2$, and it is easy to see that $\dot{x_{1}}(\lambda)\neq 0$ for those
values of $\lambda$, so that by proposition 3.7, we expect period doubling
bifurcations at this values. Also, since $x_{1}^{max}=2.25<2.45\approx b_{2}$,
according to proposition 3.11 we know that we will only get one such
bifurcation leading to stable 2-cycles in the interval $(1,\,2)$, and then
going back to a unique stable fixed point (i.e. $x_{1}$ stays in the
1-stability band in the interval $(1,\,2)$). Thus, this example complies with
our definition of a _regular-reversal map_. The latter predictions are
confirmed in the bifurcation diagram shown in figure 3.4.

(a) Graphs of linear $y_{0}(\lambda)$ and quadratic $y_{1}(\lambda)$ along the
selected values of $\lambda$. Also, the stability bands ($b_{1}$, $b_{2}$ and
$b_{\infty}$) of $y_{0}$ are shown around it.

(b) Graph of corresponding $x_{1}(\lambda)$ along with its stability bands.

Figure 3.3: Quadratic parametrization of fixed points with $p_{1}=1$ and
$p_{2}=3$. Figure 3.4: Bifurcation diagram for the quadratic parametrization
of the fixed points with $p_{1}=1$ and $p_{2}=3$..

Now, increasing the maximum value of $x_{1}$ we can accomplish more
bifurcations in the diagram. This can be achieved by increasing the slope of
$y_{1}$ to $p_{1}=1.1422$, since for this value $x_{0}^{max}\approx
b_{\infty}$. The corresponding graph of $x_{1}$ is shown in figure 3.5a. And,
also, the respective bifurcation diagram is shown in figure 3.5b, where we can
see that the system undergoes a cascade of period doubling bifurcations in the
central part. We thus produce another regular-reversal map with more nested
regions of different types.

(a) Graph of corresponding $x_{1}$. In this case $x_{1}^{max}\approx
b_{\infty}$. Also, the stability bands ($b_{1}$, $b_{2}$ and $b_{\infty}$) of
$x_{0}$ are shown above it.

(b) Bifurcation diagram. We have set here $x_{1}^{max}\approx b_{\infty}$.

Figure 3.5: Quadratic parametrization of the fixed points with $p_{1}=1.1422$
and $p_{2}=3$.

If we continue increasing the maximum value of $x_{1}$ we can achieve a
chaotic region within the regular-reversal map. Any value above the used last
above will suffice. Say we take $p_{1}=1.2$. The graph of $x_{1}$ for this
case is shown in figure 3.6 and we then get the bifurcation diagram from
figure 3.7, which displays a chaotic region in the central part, as stated.
This is still a regular-reversal map.

Figure 3.6: Graph of corresponding $x_{1}$ for linear $y_{0}(\lambda)$ and
quadratic $y_{1}(\lambda)$ with $p_{1}=1.2$ and $p_{2}=3$. The maximum value
of $x_{1}(\lambda)$ is set to be slightly above $b_{\infty}$. Also, the
bifurcation values $b_{1}$, $b_{2}$ and $b_{\infty}$ are shown. Figure 3.7:
Bifurcation diagram for the quadratic parametrization of the fixed points with
$p_{1}=1.2$ and $p_{2}=3$. We have set here $x_{1}^{max}>b_{\infty}$.

####  3.5.3 Sinusoidal fixed point

We can construct a periodic regular-reversal map by making one of the fixed
points have a sinusoidal parametric dependence on $\lambda$. That is, we leave
$y_{0}$ as before and define

$y_{1}(\lambda):=\lambda+a\,(1+\sin{\pi\,\lambda}),$ (3.32)

where we can tweak the sinusoidal form of $y_{1}$ by means of the constant $a$
which is just an amplitude (and translation, at the same time). Choosing
$a=1.3\gtrsim\frac{b_{\infty}}{2}$ we can get a periodic regular-reversal map
with chaotic regions in the periodic central regions. The graphs of the roots
are shown in figure 3.8 and the bifurcation diagram is shown in figure 3.9.

(a) Graphs of linear $y_{0}(\lambda)$ and sinusoidal $y_{1}(\lambda)$ along
the selected values of $\lambda$. Also, the stability bands ($b_{1}$, $b_{2}$
and $b_{\infty}$) of $y_{0}$ are shown around it.

(b) Graph of corresponding $x_{1}(\lambda)$ along with its stability bands.

Figure 3.8: Sinusoidal parametrization of fixed points. Figure 3.9:
Bifurcation diagram for the sinusoidal parametrization of the fixed points
example.

####  3.5.4 When zero loses its stability

It is important to point out that if $x_{1}$ becomes negative, i.e. the
difference between $y_{1}$ and $y_{0}$ becomes negative (assuming again
$M=1$), then we enter the stability region of the fixed point at zero,
$x_{0}$, in the canonical quadratic map (and $y_{0}$ in the general quadratic
map). Thus decreasing values of $x_{1}$ will cause bifurcations to appear this
time from the zero fixed point, as the stability bands are crossed by $x_{1}$
(correspondingly, $y_{1}$ in the general quadratic map). We can achieve the
latter by defining again a linear $y_{1}$ as

$y_{1}(\lambda):=p_{1}+p_{2}\,\lambda$

with, say $p_{1}=3$ and $p_{2}=-1$. Then we get $y_{1}$ to cross through _all_
the stability bands of $y_{0}$ in the interval $(0,\,3)$ of the parameter
$\lambda$. This is shown graphically in figure 3.10 along with its
corresponding graph of $x_{1}$.

The bifurcation diagram in this case starts, from left to right, being
chaotic, then reversal, then switching to stability of $x_{0}$
(correspondingly, $y_{0}$) and then being regular until chaos is reached
again, as $x_{1}$ crosses the stability bands. This is shown in figure 3.11.

(a) Graphs of linear $y_{0}(\lambda)$ and linear $y_{1}(\lambda)$ crossing all
stability bands.

(b) Graph of corresponding $x_{1}(\lambda)$ crossing all stability bands.

Figure 3.10: Linear parametrization of fixed points crossing through all
stability bands. Figure 3.11: Bifurcation diagram for the linear
parametrization of the fixed points example that crosses through all stability
bands.

####  3.5.5 Asymptotic fixed point

Since we know that the onset of chaos occurs at $b_{\infty}\approx 2.57$ (as
calculated through approximation formula 2.6), we can force the moving $x_{1}$
fixed point of the CQM to asymptotically approach this value in order to
visualize more clearly the bifurcation process that takes place very near this
value. Now, for symmetry reasons, it will be best if we make the $x_{0}=0$
fixed point to be the one that bifurcates, instead of $x_{1}$; we do this by
making $x_{1}\rightarrow-b_{\infty}$ (remember $(-b_{\infty},\,0)$ is the
interval where the period doubling bifurcations cascade occurs for $x_{0}=0$).
So, we define

$x_{1}(\lambda):=(-1)^{p_{1}}b_{\infty}+(-1)^{p_{1}+1}\exp(-p_{2}\,\lambda),$
(3.33)

where $p_{1}\in\\{0,\,1\\}$ and $p_{2}>0$. We choose for this example, for the
above reasons, $p_{1}=p_{2}=1$. The corresponding bifurcation diagram for this
case is shown in figure 3.12, upper panel, where we can clearly see the
bifurcations take place up to the attracting periodic point of period $2^{6}$
(64-cycle); the lower panel in the same figure shows the value of the fixed
points, which shows the asymptotically varying $x_{1}$. The scale of the
parameter value is the same in both panels to allow for comparison; notice
that, when the fixed point curve crosses the marked $b_{k}$ values, period
doubling bifurcation takes place in the bifurcation diagram, as expected. By
making precise calculations based on this diagrams we could determine the
bifurcation values up to an arbitrary $2^{N}$-cycle, as we did for up to $N=7$
in this work.

Figure 3.12: Bifurcation diagram (upper) and fixed points plot with stability
bands for the exponential parametrization of the fixed point example (lower).
In the lower panel, we have set $x_{1}\rightarrow-b_{\infty}$ (continuous
thick line), the zero fixed point is marked by a dash-dotted line and the
bifurcation values $-b_{1},\,-b_{2},...,\,-b_{5}$ are also shown as dashed
lines.

##  Chapter 4 Cubic maps

As stated before, in chapter 3 we furthered the results of chapter 2, mainly
based on the work of [Solís and Jódar(2004)], and restated them in the frame
of a new formulation that would allow for generalization. Well, the first such
generalization comes precisely for third degree polynomial maps, i.e. cubic
polynomials.

###  4.1 General Cubic Map

Consider a cubic map (iteration function) in its general form, as stated by
the

######  Definition 4.1 (General Cubic Map).

The _General Cubic Map_ is defined by

$f_{3}(y):=y+P_{f_{3}}(y),$ (4.1)

where

$P_{f_{3}}(y)=\alpha+\beta\,y+\gamma\,y^{2}+\delta\,y^{3}$ (4.2)

is called the _Fixed Points Polynomial_ of $f_{3}$. All the coefficients
$\alpha,\,\beta,\,\gamma$ and $\delta$ are functions of the parameter
$\lambda$.

It is evident that any cubic map can be put in this form by adjusting the
corresponding values of the coefficients in the fixed points polynomial. By
the fundamental theorem of algebra, we know that (4.2) has three roots, by
which the GCM has three fixed points. The roots of $P_{f_{3}}$ are then
[Spiegel et al.(2009)Spiegel, Lipschutz, and Liu]

$\displaystyle y_{0}$ $\displaystyle=S+T-\frac{1}{3}\tilde{\gamma}$ (4.3)
$\displaystyle y_{1}$
$\displaystyle=-\frac{1}{2}(S+T)-\frac{1}{3}\tilde{\gamma}+\frac{1}{2}i\,\sqrt{3}(S-T)$
$\displaystyle y_{2}$
$\displaystyle=-\frac{1}{2}(S+T)-\frac{1}{3}\tilde{\gamma}-\frac{1}{2}i\,\sqrt{3}(S-T)$

where $\tilde{\alpha}=\alpha/\delta$, $\tilde{\beta}=\beta/\delta$,
$\tilde{\gamma}=\gamma/\delta$, and

$\displaystyle Q$ $\displaystyle=\frac{3\tilde{\beta}-\tilde{\gamma}^{2}}{9},$
$\displaystyle R$
$\displaystyle=\frac{9\tilde{\beta}\tilde{\gamma}-27\tilde{\alpha}-2\tilde{\gamma}^{3}}{54},$
(4.4) $\displaystyle D$ $\displaystyle=Q^{3}+R^{2},$ $\displaystyle S$
$\displaystyle=\sqrt[3]{R+\sqrt{D}},$ $\displaystyle T$
$\displaystyle=\sqrt[3]{R-\sqrt{D}}.$

The coefficients of eq. (4.2) and its fixed points are related by

$\displaystyle y_{0}+y_{1}+y_{2}$ $\displaystyle=-\tilde{\gamma},$
$\displaystyle y_{0}y_{1}+y_{1}y_{2}+y_{2}y_{0}$
$\displaystyle=\tilde{\beta},$ $\displaystyle y_{0}y_{1}y_{2}$
$\displaystyle=-\tilde{\alpha}.$ (4.5)

$D$ is called the discriminant and we have three cases:

  * •

If $D>0$ then one fixed point is real and the other two are complex
conjugates.

  * •

If $D=0$ then the three fixed points are real with at least two of them equal.

  * •

If $D<0$ then all fixed points are real and distinct.

It is the last two cases (real fixed points) that will interest us most for
the time being. Suppose in particular that $D<0$. Then, we can write [Spiegel
et al.(2009)Spiegel, Lipschutz, and Liu]

$\displaystyle y_{0}$
$\displaystyle=2\sqrt{-Q}\,\cos\left(\frac{\theta}{3}\right)-\frac{1}{3}\tilde{\beta},$
(4.6) $\displaystyle y_{1}$
$\displaystyle=2\sqrt{-Q}\,\cos\left(\frac{\theta+\pi}{3}\right)-\frac{1}{3}\tilde{\beta},$
$\displaystyle y_{2}$
$\displaystyle=2\sqrt{-Q}\,\cos\left(\frac{\theta+2\pi}{3}\right)-\frac{1}{3}\tilde{\beta},$

where $\cos\theta=R/\sqrt{-Q^{3}}$.

###  4.2 Linear Factors Product Form

Since we know that (4.2) has three roots, by the factor theorem we can rewrite
$P_{f_{3}}$ as

$\displaystyle P_{f_{3}}(y)$ $\displaystyle=M(y-y_{0})(y-y_{1})(y-y_{2}),$
(4.7) $\displaystyle=s\tilde{M}(y-y_{0})(y-y_{1})(y-y_{2})$

where $s=\mathrm{sign}(M)$, $\tilde{M}=|M|$; and then rewrite (4.1) as

######  Definition 4.2 (Linear Factors Form of the Cubic Map).

Let $f_{3}$ be a general cubic map with three fixed points, $y_{0}$, $y_{1}$
and $y_{2}\in\mathbb{C}$. We can write $f_{3}$ as

$\displaystyle h_{3}(y)$ $\displaystyle=y+M(y-y_{0})(y-y_{1})(y-y_{2})$ (4.8)
$\displaystyle=y+s\tilde{M}(y-y_{0})(y-y_{1})(y-y_{2}).$

where all $M,\,y_{0},\,y_{1}$ and $y_{2}$ are functions of the parameter
$\lambda$. We call $h_{3}$ the _Linear Factors Form of the cubic map (LFFCM)_.

The LFFCM has the derivatives

$\displaystyle h^{\prime}(y)$
$\displaystyle=1+M\left[(y-y_{1})(y-y_{2})+(y-y_{0})(y-y_{2})+(y-y_{0})(y-y_{1})\right],$
(4.9) $\displaystyle h^{\prime\prime}(y)$
$\displaystyle=-2\,M\,\left(y_{0}+y_{1}+y_{2}-3\,y\right).$

By equating the first derivative to zero we can obtain the two critical points

$\displaystyle y_{1}^{c}$
$\displaystyle=\frac{1}{3}\left[(y_{0}+y_{1}+y_{2})+\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M}\right],$
(4.10) $\displaystyle y_{1}^{c}$
$\displaystyle=\frac{1}{3}\left[(y_{0}+y_{1}+y_{2})-\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M}\right]$

and, evaluating these critical points in $h^{\prime\prime}$ we have

$\displaystyle h^{\prime\prime}(y_{1}^{c})$
$\displaystyle=2M\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M},$
(4.11) $\displaystyle h^{\prime\prime}(y_{2}^{c})$
$\displaystyle=-2M\sqrt{\frac{1}{2}\left[(y_{1}-y_{0})^{2}+(y_{2}-y_{1})^{2}+(y_{2}-y_{0})^{2}\right]-3/M}=-h^{\prime\prime}(y_{1}^{c}),$

so that necessarily one critical point is a maximum and the other one is a
minimum, unless they are both equal, in which case we have a saddle point.

###  4.3 Canonical Cubic Map

As in the case for the Canonical Quadratic Map, we will apply a linear
transformation to (4.8) so that one fixed point is mapped to zero and the
“amplitude” of the linear factors term is unity. We know that, since $f_{3}$
is cubic, at least one fixed point is real, so we can map this fixed point to
zero. This can be accomplished by one of the following transformations

$\displaystyle T_{0}(x)$ $\displaystyle=y_{0}\pm s\tilde{M}^{-1/2}x,$
$\displaystyle T_{1}(x)$ $\displaystyle=y_{1}\pm s\tilde{M}^{-1/2}x,$
$\displaystyle T_{2}(x)$ $\displaystyle=y_{2}\pm s\tilde{M}^{-1/2}x,$ (4.12)

by taking $y=T_{k}(x),\,k\in\\{0,\,1,\,2\\}$ if $y_{k}$ is real. Without loss
of generality, we will assume $y_{0}$ is real and apply $T_{0}$ with the plus
sign calling it simply $T$, so that, once the calculations are done, we get

######  Definition 4.3 (Canonical Cubic Map).

The _Canonical Cubic Map (CCM)_ is defined by

$g_{3}(x;\lambda)=x+sx(x-x_{1}(\lambda))(x-x_{2}(\lambda)),$ (4.13)

where it has been stressed out that both fixed points $x_{1}$ and $x_{2}$
depend upon the parameter $\lambda$.

So that, if $M>0$ then

$g_{3}(x)=x+x(x-x_{1})(x-x_{2}),$ (4.14)

and if $M<0$

$g_{3}(x)=x-x(x-x_{1})(x-x_{2}),$ (4.15)

From the easily verifiable calculations, it can be shown that the relation
between the roots of the linear factors form of the cubic map and the
canonical cubic map is given by

######  Lemma 4.1.

The fixed points of the linear factors form of the cubic map and the canonical
cubic map are related by

$\displaystyle x_{1}(\lambda)$
$\displaystyle=s\tilde{M}^{1/2}\left[y_{1}(\lambda)-y_{0}(\lambda)\right],$
$\displaystyle x_{2}(\lambda)$
$\displaystyle=s\tilde{M}^{1/2}\left[y_{2}(\lambda)-y_{0}(\lambda)\right].$
(4.16)

We have then again reduced the parametric dependence to only two functions of
the parameter $\lambda$: $x_{1}$ and $x_{2}$. Notice $T$ is a homeomorphism
between the _domains_ of both maps; this will help us in chapter 5 to prove
that the Linear Factors Form and the Canonical Form of polynomial maps are
actually topologically conjugate, which in turn means that the stability and
chaos properties are preserved between the maps, which allows us to determine
stability properties for any cubic map by analyzing only the CCM.

Now, one first immediate result is

######  Proposition 4.2.

The Canonical Cubic Map has three roots:

$\displaystyle x_{0}^{r}$ $\displaystyle=0,$ $\displaystyle x_{1}^{r}$
$\displaystyle=\frac{x_{1}+x_{2}+s\sqrt{(x_{2}-x_{1})^{2}-4s}}{2},$
$\displaystyle x_{2}^{r}=\frac{x_{1}+x_{2}-s\sqrt{(x_{2}-x_{1})^{2}-4s}}{2}.$
(4.17)

Moreover, if $M>0$

  * •

If $x_{2}\neq-x_{1}$ and $x_{2}-x_{1}=\pm 2$, we will have a single root
different from zero with multiplicity of two, and equal to $(x_{1}+x_{2})/2$,
i.e. the midpoint between the two non-zero fixed points.

  * •

If $x_{2}=-x_{1}$ and $x_{2}-x_{1}=\pm 2$, zero will be the only root, with
multiplicity of three.

  * •

If $|x_{2}-x_{1}|>2$, we will have two real and distinct non-zero roots.

  * •

If $|x_{2}-x_{1}|<2$, zero will be the only real root of the CCM and the other
two roots will be complex conjugates.

And if $M<0$ the three roots are always real and distinct, except when
$x_{1}=x_{2}=\pm 1$ when $x_{0}^{r}=0$ has multiplicity of two.

As for the derivatives of the CCM, they are

$\displaystyle g_{3}^{\prime}(x)$
$\displaystyle=1+s\left[(x-x_{1})(x-x_{2})+x(x-x_{2})+x(x-x_{1})\right],$
(4.18) $\displaystyle g_{3}^{\prime\prime}(x)$
$\displaystyle=s\left[6x-2(x_{1}+x_{2})\right]$

from which we can determine that the critical points are

$\displaystyle x_{1}^{c}$
$\displaystyle=\frac{1}{3}\left[x_{1}+x_{2}+s\sqrt{x_{1}^{2}-x_{1}x_{2}+x_{2}^{2}-3s}\right],$
(4.19) $\displaystyle x_{2}^{c}$
$\displaystyle=\frac{1}{3}\left[x_{1}+x_{2}-s\sqrt{x_{1}^{2}-x_{1}x_{2}+x_{2}^{2}-3s}\right].$

So that, evaluating $g^{\prime\prime}$ at the critical values of $g,$
$x_{1}^{c}$ and $x_{2}^{c}$ we have thatwe have that

$g_{3}^{\prime\prime}(x_{1}^{c})=-g_{3}^{\prime\prime}(x_{2}^{c})=2\sqrt{x_{1}^{2}-x_{1}x_{2}+x_{2}^{2}-3s},$

and, then, we have as expected one maximum and one minimum unless both are
equal, in which case we have a saddle point. From this we have proved

######  Proposition 4.3.

Consider the CCM as defined above. Then, if $M>0$

  * •

If $x_{1}^{2}+x_{2}^{2}>3+x_{1}x_{2}$, both critical points are real and
distinct, one corresponding to a maximum and the other to a minimum.

  * •

If $x_{1}^{2}+x_{2}^{2}=3+x_{1}x_{2}$, both critical points are equal and
correspond to a saddle point.

  * •

If $x_{1}^{2}+x_{2}^{2}<3+x_{1}x_{2}$, the two critical points are complex
conjugates, so we do not have any real critical points.

And if $M<0$,

  * •

If $x_{1}^{2}+x_{2}^{2}+3>x_{1}x_{2}$, both critical points are real and
distinct, one corresponding to a maximum and the other to a minimum.

  * •

If $x_{1}^{2}+x_{2}^{2}+3=x_{1}x_{2}$, both critical points are equal and
correspond to a saddle point.

  * •

If $x_{1}^{2}+x_{2}^{2}+3<x_{1}x_{2}$, the two critical points are complex
conjugates, so we do not have any (real) critical points.

###  4.4 Stability and Chaos in the Canonical Cubic Map

Next, let us determine the stability of the periodic points of the CCM. As we
will see further below in chapter 5, this analysis will suffice for any cubic
map, by means of topological conjugacy. However, we can only explicitly give
this for the fixed points.

####  4.4.1 Fixed Points

We already know, by construction, that the fixed points of the CCM are
$x_{0}=0,\,x_{1}$ and $x_{2}$. While the first is constant always, the other
two fixed points are set to be functions of the parameter $\lambda$. By
evaluating in $g_{3}^{\prime}$ we get the eigenvalue functions. For $x_{0}=0$
we have

$\phi_{0}(\lambda)=g_{3}^{\prime}(0)=sx_{1}(\lambda)x_{2}(\lambda)+1,$

So that, the stability condition for this fixed point is

$-2<sx_{1}x_{2}<0.$ (4.20)

We can draw some conclusions from this. In order for zero to be a stable
(attracting) fixed point one must have:

######  Lemma 4.4.

The following are sufficient conditions for the asymptotic stability of the
zero fixed point of the Canonical Cubic Map:

  * •

in magnitude, $|x_{1}||x_{2}|<2$; and

  * •

if $M>0$, $x_{1}$ and $x_{2}$ must have _different_ signs; or

  * •

if $M<0$, $x_{1}$ and $x_{2}$ must have _the same_ sign.

Notice that the stability condition (4.20) states that the product of the
distances from the other two fixed points to the zero fixed point must within
the range $(-2,\,0)$, for positive $M$ [or $(0,\,2)$ for negative $M$], for
the zero fixed point to be asymptotically stable.

The case of $x_{k}=0$, $k\in\\{1,\,2\\}$, is not included in the discussion
here since this would represent repeated fixed points (multiplicity), which
will be discussed in section 4.6 below; likewise, in the remainder of this
section we will avoid dealing with multiplicity of the fixed points.

Now, for $x_{1}$, its eigenvalue function is

$\phi_{1}(\lambda)=g_{3}^{\prime}(x_{1}(\lambda))=1+sx_{1}(\lambda)(x_{1}(\lambda)-x_{2}(\lambda)),$

so that the stability condition for this fixed point is

$-2<sx_{1}(x_{1}-x_{2})<0.$ (4.21)

This gives us the following

######  Lemma 4.5.

The following are sufficient conditions for the asymptotic stability of the
$x_{1}$ fixed point:

If $M>0$ then

  * •

$x_{1}$ and $x_{2}$ must have _the same_ sign; and

  * •

$|x_{1}|<|x_{2}|<|x_{1}+2/x_{1}|$.

On the other hand, if $M<0$,

  * •

$|x_{2}|<|x_{1}|$; and

  * •

if $|x_{1}|\geq\sqrt{2}$, then $|x_{1}-2/x_{1}|<|x_{2}|<|x_{1}|$; or

  * •

if $0<x_{1}<\sqrt{2}$, then $x_{1}-2/x_{1}<x_{2}<x_{1}$; or

  * •

if $-\sqrt{2}<x_{1}<0$, then $x_{1}<x_{2}<x_{1}-2/x_{1}$.

Again, notice that the stability condition (4.21) for $x_{1}$ can be
translated as that the product of the distances between the other two fixed
points and $x_{1}$ must be within the range $(-2,\,0)$ for positive $M$ [or
$(0,\,2)$ for negative $M$]. Also notice that when $0<|x_{1}|<\sqrt{2}$ the
bound $x_{1}-2/x_{1}$ may be negative even if $x_{1}>0$ or positive even if
$x_{1}<0$, therefore the usefulness of the distinction.

Finally, for $x_{2}$ we have the eigenvalue function

$\phi_{2}(\lambda)=g_{3}^{\prime}(x_{2}(\lambda))=1+sx_{2}(\lambda)(x_{2}(\lambda)-x_{1}(\lambda)),$

which gives the stability condition

$-2<sx_{2}(x_{2}-x_{1})<0,$ (4.22)

which in turn produces the following

######  Lemma 4.6.

The following are sufficient conditions for the asymptotic stability of the
$x_{2}$ fixed point of the Canonical Cubic Map:

If $M>0$ then

  * •

$x_{1}$ and $x_{2}$ must have _the same_ sign; and

  * •

$|x_{2}|<|x_{1}|<|x_{2}+2/x_{2}|$.

On the other hand, if $M<0$,

  * •

$|x_{1}|<|x_{2}|$; and

  * •

if $|x_{2}|\geq\sqrt{2}$, then $|x_{2}-2/x_{2}|<|x_{1}|<|x_{2}|$; or

  * •

if $0<x_{2}<\sqrt{2}$, then $x_{2}-2/x_{2}<x_{1}<x_{2}$; or

  * •

if $-\sqrt{2}<x_{2}<0$, then $x_{2}<x_{1}<x_{2}-2/x_{2}$.

Likewise, we see that the stability condition (4.22) involves the product of
the distances from the other two fixed points to the one whose stability is of
interest. We will later generalize these “stability conditions” to functions
of the parameter which are different for each fixed point, but of whose value
depend the stability of not only the fixed points, but higher period periodic
points also, through period doubling bifurcations. From the stability
conditions for the three fixed points we have proved the following

######  Corollary.

A cubic polynomial map with three different real roots can only have a single
attracting fixed point.

###### Proof.

Compare the stability conditions for the three fixed points. ∎

And,

######  Theorem 4.7.

If $M>0$, then sufficient conditions for the stability of a fixed point of the
canonical cubic map are

  * •

The product of the distances between each unstable fixed point and the stable
one must be negative, which means one distance is positive and the other
negative, which leads us to

  * •

the fixed point that lies between the other two will be stable, while the
outer fixed points will be unstable, as long as

  * •

the product of the distances between each unstable fixed point and the stable
one must be greater than -2.

###  4.5 Higher period periodic points

Although, as previously stated, in general, we cannot calculate the values of
the periodic points of period 2 or higher, we can calculate for which values
of the stability conditions above the fixed points undergo period doubling
bifurcations. We will see in chapter 5 that these stability conditions can
actually be generalized to something called the “Product Distance Function”,
which depends on the parameter and is different for each fixed point. Example
4.4 below allowed us to determine the bifurcation values, $c_{k}$, of the
fixed points of the CCM up to some precision. The values obtained are shown in
table 4.1. When the stability conditions of each fixed point crosses these
values, bifurcations take place.

$k$ | $c_{k}$
---|---
0 | 0
1 | 2
2 | $3.0\pm 0.005$
3 | $3.236\pm 0.002$
4 | $3.288\pm 0.0005$
5 | $3.29925\pm 0.00025$
$\vdots$ | $\vdots$
$\infty$ | $\sim 3.30228\pm 5\times 10^{-6}$
Table 4.1: Bifurcation values for the Canonical Cubic Map. $c_{0}=0$ is
included only as a reference, although it is not a bifurcation value.

From these values, we can construct the analogue of the stability bands of the
CQM for the CCM.

######  Definition 4.4 (Stability Bands of the CCM).

Let $x_{1},\,x_{2}:\mathcal{A}\subseteq\mathbb{R}\rightarrow\mathbb{R}$ be the
two nonzero fixed points of the family of cubic maps $g_{3}$, as given by
definition 4.3, and $\\{c_{k}\\}_{k\in\mathbb{N}}$ the sequence of bifurcation
values of table 4.1. The open interval

$\left(-c_{k+1},\,-c_{k}\right),\quad\lambda\in\mathcal{A}$ (4.23)

is called the $k$-th _stability band_ of the CCM.

Notice, however, that in contrast with the stability bands of the CQM, the
stability bands of the CCM cannot be plotted along the fixed points plots, at
least not directly as just defined, but rather they must be represented in a
separate plot for the stability conditions, as we shall see in the examples of
section 4.7 further below.

###  4.6 Multiplicity of the fixed points

When multiplicity of the fixed points takes place in the CCM, without loss of
generality, $g_{3}$ can take the following forms

$g_{3}(x)=\begin{cases}x+sx^{2}(x-x_{1}),&\text{if }x_{2}=x_{0}=0,\,x_{1}\neq
0\\\ x+sx(x-x_{1})^{2},&\text{if }x_{1}=x_{2}\neq 0\\\ x+sx^{3},&\text{if
}x_{1}=x_{2}=0.\end{cases}$ (4.24)

with corresponding derivatives

$g^{\prime}_{3}(x)=\begin{cases}1+2sx(x-x_{1})+sx^{2},&\text{if
}x_{2}=x_{0}=0,\,x_{1}\neq 0\\\ 1+sx(x-x_{1})^{2}+2sx(x-x_{1}),&\text{if
}x_{1}=x_{2}\neq 0\\\ 1+3sx^{2},&\text{if }x_{1}=x_{2}=0.\end{cases}$ (4.25)

and therefore, $g^{\prime}_{3}(x_{k})=1,\,k\in\\{0,\,1,\,2\\}$, for all three
cases, so that we deal with nonhyperbolic fixed points, according to
definition 1.6.

######  Proposition 4.8.

The stability of the fixed points of the CCM when they present multiplicity
is, without loss of generality, as follows

  1. 1.

If $x_{2}=x_{0}=0,\,x_{1}\neq 0$, the zero fixed point is an unstable fixed
point with multiplicity of two; and

     * •

if $M>0$ and

       * –

if $x_{1}>0$ it is semiasymptotically stable from the right;

       * –

if $x_{1}<0$ it is semiasymptotically stable from the left;

     * •

or if $M<0$ and

       * –

if $x_{1}>0$ it is semiasymptotically stable from the left;

       * –

if $x_{1}<0$ it is semiasymptotically stable from the right.

  2. 2.

If $x_{1}=x_{2}\neq 0$, this fixed point has multiplicity of two and it is
_unstable_ ; moreover

     * •

if $M>0$ and

       * –

if $x_{1}>0$ it is semiasymptotically stable from the left;

       * –

if $x_{1}<0$ it is semiasymptotically stable from the right;

     * •

or if $M<0$ and

       * –

if $x_{1}>0$ it is semiasymptotically stable from the right;

       * –

if $x_{1}<0$ it is semiasymptotically stable from the left.

  3. 3.

If $x_{0}=x_{1}=x_{2}=0$, the zero fixed point has multiplicity of three and

     * •

if $M>0$, it is unstable;

     * •

if $M<0$, it is asymptotically stable.

###### Proof.

Notice that

$g^{\prime\prime}_{3}(x)=\begin{cases}2s(x-x_{1})+4sx,&\text{if
}x_{2}=x_{0}=0,\,x_{1}\neq 0\\\ 4s(x-x_{1})+2sx,&\text{if }x_{1}=x_{2}\neq
0\\\ 6sx,&\text{if }x_{1}=x_{2}=0.\end{cases}$ (4.26)

and