††thanks:  Professor of Physics; Ph.D. in Physics from the Landau Institute
for Theoretical Physics in 1987
ORCID https://orcid.org/0000-0003-3754-1794
Google Scholar https://scholar.google.com/citations?user=pEnxwCMAAAAJ
Web page https://physics.umd.edu/~yakovenk/econophysics/
E-mail<EMAIL_ADDRESS>
# Statistical Physics Perspective on Economic Inequality

Victor M. Yakovenko  JQI, Department of Physics, University of Maryland,
College Park, Maryland 20742, USA

(2 July 2023)

###### Abstract

This article is a supplement to my main contribution to the Routledge Handbook
of Complexity Economics (2023). On the basis of three recent papers, it
presents an unconventional perspective on economic inequality from a
statistical physics point of view. One section demonstrates empirical evidence
for the exponential distribution of income in 67 countries around the world.
The exponential distribution was not familiar to mainstream economists until
it was introduced by physicists by analogy with the Boltzmann-Gibbs
distribution of energy and subsequently confirmed in empirical data for many
countries. Another section reviews the two-class structure of income
distribution in the USA. While the exponential law describes the majority of
population (the lower class), the top tail of income distribution (the upper
class) is characterized by the Pareto power law, and there is no clearly
defined middle class in between. As a result, the whole distribution can be
very well fitted by using only three parameters. Historical evolution of these
parameters and inequality trends are analyzed from 1983 to 2018. Finally,
global inequality in energy consumption and CO2 emissions per capita is
studied using the empirical data from 1980 to 2017. Global inequality, as
measured by the Gini coefficient $G$, has been decreasing until around 2010,
but then saturated at the level $G=0.5$. The saturation at this level was
theoretically predicted on the basis of the maximal entropy principle, well
before the slowdown of the global inequality decrease became visible in the
data. This effect is attributed to accelerated mixing of the world economy due
to globalization, which brings it to the state of maximal entropy and thus
results in global economic stagnation. This observation has profound
consequences for social and geopolitical stability and the efforts to deal
with the climate change.

“Money, it’s a gas.” Pink Floyd, Dark Side of the Moon (1973)

econophysics, economic inequality, entropy, social classes, carbon emissions

##  I Introduction

This article is a concise follow-up to my paper Yakovenko (2016) reproduced in
this Routledge Handbook of Complexity Economics (2023). The update is based on
three papers Tao et al. (2019); Semieniuk and Yakovenko (2020); Ludwig and
Yakovenko (2022) published after Yakovenko (2016) and summarized in three
sections below. All papers are available at
https://physics.umd.edu/~yakovenk/econophysics/.

By analogy with the Boltzmann-Gibbs
distribution111https://en.wikipedia.org/wiki/Boltzmann_distribution of energy
in statistical physics, Drăgulescu and Yakovenko (2000) proposed that the
stationary probability distribution of money in an ensemble of interacting
economic agents follows an exponential law. They performed agent-based
modeling222https://physics.umd.edu/~yakovenk/econophysics/animation.html to
illustrate how the exponential Boltzmann-Gibbs distribution of money develops
from an initially equal distribution due to monetary transactions between the
agents, which are analogous to collisions between molecules in a gas. The same
computer simulations also shows that the entropy of the system increases and
saturates at its maximal value in statistical equilibrium, in agreement with
the second law of thermodynamics.

Rosser (2016) highlighted distinction between two flavors of entropy:
“ontological” and “metaphorical.” The former is the thermodynamic entropy
introduced by Carnot, Clausius, and Boltzmann to physics in the 19th century.
It governs flow of energy in physical systems via the second law of
thermodynamics. The latter is the informational entropy introduced by Shannon
and von Neumann in the 20th century as a measure of combinatorial complexity
in any statistical ensemble, not limited to a physical ensemble. The
statistical physics perspective presented below is guided by application of
the second, generalized concept of entropy to an ensemble of economic agents,
either at national or global scale. This perspective is the opposite to the
representative-agent approach of traditional economics, which, by
construction, reduces an ensemble to a single agent and thus ignores
statistical and heterogeneous aspects of the economy, such as entropy and
inequality Yakovenko (2022).

##  II Exponential distribution of income in 67 countries

Figure 1: Cumulative probability distribution in logarithmic scale on the
vertical axis versus normalized income in linear scale on the horizontal axis
for the European Union and its neighboring countries in 2014 (left panel) and
for select non-EU countries for various years (right panel). [Reproduced from
Tao et al. (2019)]

In their next paper, Drăgulescu and Yakovenko (2001a) found empirical evidence
for the exponential distribution of income in the USA by analyzing the data
from the U.S. Census Bureau. Further analysis of the data from the Internal
Revenue Service (IRS, the U.S. tax agency) confirmed the exponential shape for
the lower part of income distribution, where the overwhelming majority of
population belongs, whereas the tail of income distribution for the top few
percent of population was found to follow the Pareto power law Drăgulescu and
Yakovenko (2001b, 2003). The observation that income distribution for the
majority of population is described by the exponential law was virtually
unknown to the economists and statisticians, but was eventually recognized in
a special issue The Science of Inequality of the Science magazine in a one-
page article by Cho (2014).

Over time, the exponential distribution of income was found in numerous papers
for different countries, e.g., by Banerjee, Yakovenko, and Di Matteo (2006)
for Australia. A collaborative team with Chinese economists and data
scientists Tao et al. (2019) studied income distribution for 67 countries
around the world (where it was possible to obtain reasonably reliable data).
The results are shown in Fig. 1 for the European Union (EU) and its
neighboring countries in the left panel and for select non-EU countries in the
right panel. The graphs are plotted in log-linear scale, where normalized
income on the horizontal axis is in linear scale, whereas cumulative
probability on the vertical axis is in logarithmic scale. Data points
following straight lines in log-linear scale confirm that the middle part of
income distribution for these countries is, indeed, exponential. The upper
tail deviates from the exponential shape because of the Pareto law, but is not
well sampled in the survey data, so it is truncated. Income distribution at
very low income may also deviate from the exponential shape because of welfare
policies varying from country to country. These low-end deviations are also
truncated in Fig. 1. Overall, Fig. 1 demonstrates overwhelming empirical
evidence that that the low and middle part of income distribution follows a
universal exponential law for many countries. For some countries, the
exponential regime extents over a particularly wide range, e.g., for China
(CHN) in the right panel of Fig. 1.

Besides the empirical analysis, Tao et al. (2019) also reformulated the
arguments in favor of the exponential distribution from statistical physics to
the language of economics using the standard Arrow-Debreu’s general
equilibrium model, combined with Rawls’ fairness principle for free
competition.

##  III Two-class structure of income distribution

Figure 2: Left panel: Cumulative distribution function of rescaled income in
the USA for 1983–2018 in log-log scale (main panel) and log-linear scale
(inset) [reproduced from Ludwig and Yakovenko (2022)]. Right panel: (a) The
Gini coefficient $G$ for 1983–2013 (connected line), compared with the formula
$G=(1+f)/2$ (open circles). (b) The exponent $\alpha$ of the power-law tail.
(c) The mean income $\langle r\rangle$ of the whole system, the mean income
$T$ of the lower class (the temperature of the exponential part), and the
fraction of income $f=1-T/\langle r\rangle$ going to the upper class.
[Reproduced from Yakovenko (2016)]

The papers reviewed in Sec. II considered income distribution for a given
year. Silva and Yakovenko (2005) extended this analysis to the multiyear
period of 1983–2001 for the USA. In subsequent papers Banerjee and Yakovenko
(2010); Yakovenko (2016) the range of years was expanded even further. The
latest paper Ludwig and Yakovenko (2022) studied the two-class income
distribution in the USA for 36 years from 1983 to 2018. The cumulative
distribution function versus rescaled income is shown for these years in the
left panel of Fig. 2. It clearly shows a two-class structure of income
distribution. The inset shows that the lower part of the distribution (about
96% of population in 2018) is well described by an exponential function, as
indicated by the straight lines in log-linear scale. On the other hand, the
main panel shows that the upper tail (about 4% of population in 2018) is well
described by a power law, as indicated by the straight lines in log-log scale.
Importantly, the data reveal only two classes: upper and lower, whereas middle
class does not exist. In fact, there is no commonly accepted definition of the
“middle class” in the literature, and each author who writes about it invents
their own definition.

Mathematically, the two-class distribution can be derived as a
(quasi)stationary solution of the Fokker-Planck
equation333https://en.wikipedia.org/wiki/Fokker–Planck_equation (also known as
the forward Kolmogorov equation in mathematics) for stochastic dynamics of
income with coexisting additive and multiplicative components Silva and
Yakovenko (2005); Banerjee and Yakovenko (2010); Yakovenko and Rosser (2009).
The stationary distribution interpolates between an exponential law at the low
end and a power law at the high end Ludwig and Yakovenko (2022) and is known
as the Pearson Type IV distribution Pearson (1895).

Using the two-class decomposition, income distribution can be very well fitted
by using only three parameters: the mean income $T$ of the exponential bulk
(which is analogous to temperature in physics), the exponent $\alpha$ of the
power-law tail, and the crossover income $r_{*}$ separating the lower and
upper classes. Historical evolution of the first two parameters is shown in
the right panel of Fig. 2. Panel (b) shows that the Pareto exponent $\alpha$
has been decreasing since 1983 (albeit flattening after 2000), which indicates
“fattening” of the upper tail, i.e., rich getting richer.

Panel (c) of Fig. 2 shows the income temperature $T$ in comparison with the
mean income $\langle r\rangle$ of the whole distribution. The normalized
difference between these two parameters represents the share of income
$f=(\langle r\rangle-T)/\langle r\rangle$ going to the upper class. The spikes
on the red curve for $f$ in Panel (c) indicate sharp increases in inequality
due to the enhanced share of income going to the upper class. The first spike
coincides with the .com bubble in stock market, the second spike with the
housing bubble, and the third spike with the Quantitative Easing (QE) pursued
by the Federal Reserve. Clearly, inequality peaks during speculative bubbles
in the financial markets. The third spike gives direct evidence that the
bailout of the financial system by the Fed resulted in increase of inequality.
The parameter $f$ has increased even further in subsequent years Ludwig and
Yakovenko (2022).

Panel (a) of Fig. 2 shows the Gini coefficient $G$ in comparison with the
theoretical formula $G=(1+f)/2$ derived from the two-class decomposition by
Silva and Yakovenko (2005). A very good agreement after 1995 indicates that
the historical dynamics of the Gini coefficient is completely determined by
the share of income $f$ going to the upper “superthermal” class, whereas
relative inequality within the lower “thermal” class remains essentially
constant over the span of 36 years Ludwig and Yakovenko (2022). Furthermore,
Shaikh and Ragab (2023) proposed a metric for ranking of countries, called the
Vast Majority Income, on the basis of this formula for the Gini coefficient.

Ludwig and Yakovenko (2022) also studied the shares of tax revenue coming from
the lower and upper classes, according to the IRS data. By 2018, the income
share of the top 1% of the population has increased to 21%, which is almost
twice the total income share of 12% going the bottom half (50%) of the
population. At the same time, the tax share paid by the bottom 50% of the
population has decreased to 3% and became almost negligible, because their
total income share is so low. In contrast, the tax share of the top 1% of the
population has increased to 40%. Thus, the majority of tax revenue now comes
from the top few percent of the population, where most of income is
concentrated.

Not only does the upper-class income share increase, but the fraction of the
population belonging to the upper class increases too Ludwig and Yakovenko
(2022). In a relative sense, the upper-class population expands (while still
remaining a small fraction at 4%), while the lower-class population shrinks
(while still remaining the overwhelming majority). Ludwig and Yakovenko (2022)
speculated that it is due to digitization of the economy in the last 40 years.
There was a rapid proliferation of personal computers during the 1980s,
followed by the spread of the Internet and the World Wide Web in the 1990s,
and then by ubiquitous personal mobile devices and a shift to cloud computing
at the present time. This transformation enabled the creation and relatively
easy scaling-up of digital platforms for highly non-local business operations.
In the past, many businesses were local: taxi companies, book stores, video
rental stores, etc. Now they are largely displaced by a small number of
national and global network platforms, such as Uber and Lyft for riding,
Amazon for books initially and then for all kinds of goods, Netflix for DVD
rentals and video streaming, etc. The founders and owners of such network
platforms become super-rich, because these platforms serve a huge number of
customers, in contrast to the old-fashioned local businesses. Thus the growth
of the upper class may be a reflection of the ongoing transformation of
business network topology from local clusters to highly connected
superclusters and global hubs.

##  IV Global Inequality in Energy Consumption and CO2 Emissions

Figure 3: Lorenz plots of the global energy consumption (left panel) and $\rm
CO_{2}$ emissions (right panel) per capita for 1980, 1990, 2000, and 2010
(colored curves), compared with an exponential distribution (black curve).
[Reproduced from Lawrence, Liu, and Yakovenko (2013)]

Sections II and III focus on income inequality within a given country.
However, there is also global inequality between rich and poor countries
around the world. Studying global monetary inequality is complicated by
different currencies, whose nominal conversion rates are not particularly
representative. One way around this problem is to use the purchasing power
parity444https://en.wikipedia.org/wiki/Purchasing_power_parity Milanović
(2012). Lawrence, Liu, and Yakovenko (2013) took another approach and
investigated global inequality in energy consumption and CO2 emissions per
capita using the data from the U.S. Energy Information Administration (EIA)
for 1980–2010. Energy consumption is a physical measure of inequality in
standards of living, whereas CO2 emissions are closely correlated with it,
because most energy is produced from fossil fuels.

The corresponding Lorenz curves are shown in Fig. 3. Computer animation of the
time evolution of these Lorenz curves is also
available.555https://physics.umd.edu/~yakovenk/econophysics/global.html In
both graphs, small circles indication various countries, whereas big circles
indicate some labeled countries. Not surprisingly, the Lorenz curves for
energy consumption and CO2 emissions look quite similar.

The Lorenz curves in 1980 exhibit a sharp slope change in the middle, which
separates two groups of countries. One group in the top-right sector has high
slope, indicating high energy consumption and CO2 emissions per capita. This
group consists of “developed” countries, mostly in North America and Europe.
Another group in the bottom-left sector has low slope, indicating low energy
consumption and CO2 emissions per capita. This group consists of “developing”
countries, such as China, India, and Brazil.

In the subsequent years since 1980, the Lorenz curves move up, indicating that
global inequality decreases. By 2010, the cusp in the middle of the Lorenz
curves has smoothed out. Now there is no sharp boundary between “developed”
and “developing” countries anymore. This is largely due to China moving to the
middle of the curve between these two groups of countries.

Figure 4: Historical evolution of the global Gini coefficient for CO2
emissions per capita. Black circles show the data for 1980–2000 as published
by Lawrence, Liu, and Yakovenko (2013), whereas red squares show the new EIA
data for 2011–2017. The blue dashed line is a linear extrapolation from 2000
to 2010. The horizontal line at $G=0.5$ corresponds to a theoretical
exponential distribution. [Reproduced from Semieniuk and Yakovenko (2020)]

The black curve in Fig. 3 represents the (analytically) calculated Lorenz
graph for an exponential distribution. In statistical physics, the latter
corresponds to the Boltzmann-Gibbs distribution, which maximizes entropy in
thermal equilibrium. We observe that the empirical Lorenz curves move toward
the calculated black exponential curve, approaching it closely from below.
Lawrence, Liu, and Yakovenko (2013) attributed this behavior to globalization
of the world economy, which mixes the world and brings it closer to the state
of maximal entropy. They predicted that global inequality will soon stop
decreasing and will saturate at the Gini coefficient $G=0.5$ corresponding to
the exponential distribution.

This prediction was spectacularly confirmed in the follow-up paper by
Semieniuk and Yakovenko (2020), when the data for subsequent years up to 2017
became available. Figure 4 shows the Gini coefficient from 1980 to 2017 for
CO2 emissions per capita. Black circles are the same data points for 1980–2010
as analyzed in Lawrence, Liu, and Yakovenko (2013). They manifest a decreasing
trend and no sign of saturation yet. In contrast, the new data points for
2011–2017 (red squares) exhibit saturation at the level of $G=0.5$, indicting
that global inequality stopped decreasing soon after 2010. This observation
confirms the prediction made by Lawrence, Liu, and Yakovenko (2013) on the
basis of the maximal entropy principle. Remarkably, this prediction was made
at the time when global inequality has been steadily decreasing, and there was
no indication of saturation in the available data yet. The advanced prediction
in Lawrence, Liu, and Yakovenko (2013) and the subsequent confirmation in
Semieniuk and Yakovenko (2020) strongly support the proposition that economic
globalization is, indeed, governed by the principle of maximal entropy.

This observation has profound consequences for strategies and scenarios
dealing with the climate change Semieniuk and Yakovenko (2020). Various calls
have been made for either lifting billions of people from poverty, which
implies increased consumption and carbon emissions, or capping the level of
per-capita CO2 emissions at the top of the distribution above a certain
threshold. Semieniuk and Yakovenko (2020) constructed the Lorenz curves
implied by these redistributive proposals and demonstrated that they would
require an unprecedented reduction in global inequality, far below historical
levels. The recent saturation of the global inequality decrease further
undermines feasibility of such redistributive scenarios.

A decreasing trend was also observed in global income inequality Milanović
(2012, 2020, 2023); Tharoor (2023); Salmon (2023), but saturation has not been
recognized yet.

##  V Conclusion

The term “econophysics” was introduced in 1995 at a conference in Kolkata by
the theoretical physicist Eugene Stanley for a new interdisciplinary research
field applying methods of statistical physics to economics and finance
Chakrabarti (2005). The paper by Stanley et al. (1996) presented a manifesto
of the new field, arguing that “behavior of large numbers of humans (as
measured, e.g., by economic indices) might conform to analogs of the scaling
laws that have proved useful in describing systems composed of large numbers
of inanimate objects.” About 30 years later, econophysics is now well
recognized in physics Yakovenko and Rosser (2009) and is gradually penetrating
into mainstream economics Shaikh (2017). Infusion of new ideas from a
different field often results not in answering old questions within old
framework, but in establishing new framework with new questions. Much of
misunderstanding and miscommunication between economists and physicists
happens not because they are getting different answers, but because they are
answering different questions Yakovenko (2022).

The empirical demonstration of the two-class structure of income distribution,
described by the simplest mathematical functions: exponential and power law,
is a significant accomplishment of econophysics. The Pareto power law for the
upper tail has been known for a long time, but the exponential distribution,
describing the majority of population, was not known before. The ability to
fit the whole income distribution by using only three parameters is somewhat
reminiscent of Kepler’s discovery that the vast amount of accumulated data for
positions of planets can be reduces to a few parameters for their elliptic
orbits. Such data compression is a signature of scientific progress, which
subsequently led Newton his discovery of the gravity law as an explanation for
the elliptic orbits. While social classes have been known in political economy
since Karl Marx, realization that they are described by simple mathematical
distributions is quite new. As a follow-up to Drăgulescu and Yakovenko (2000),
Wright (2005, 2009) demonstrated emergence of two classes in more
sophisticated agent-based simulations. This work was further developed by
Isaac (2019) and in the book by Cottrell, Cockshott, Michaelson, Wright, and
Yakovenko (2009), integrating economics, computer science, and physics.

The prediction Lawrence, Liu, and Yakovenko (2013) and confirmation Semieniuk
and Yakovenko (2020) of the saturation of the global inequality decrease is
another significant accomplishment of econophysics. As mentioned in the former
paper, “in physics, theories that not only explain known experiments, but also
make successful predictions about future observations are particularly
valuable.” At the International Energy
Workshop666https://www.internationalenergyworkshop.org/meetings-10.html in
2017 at College Park, I chatted with the chief energy modeler of EIA. He said
in his talk that they have dynamical models, which are complicated and have
many unknown parameters, so predictions cannot be made in practice. I said
that I do not have such models, but can make a valid prediction on the basis
of a general principle (of entropy maximization).

There is discussion in the media about a global economic stagnation, sometimes
called the “economic ice age,” pointing to various specific reasons Bott
(2013). In contrast, Lawrence, Liu, and Yakovenko (2013) suggested that the
actual underlying reason is entropic. Globalization was driving the world
economy toward the state of maximal entropy, which has been achieved now, thus
resulting in global stagnation. Paradoxically, this advance toward the global
statistical equilibrium may bring on social and geopolitical unrest because of
slowdown in upward mobility for the lower part of global distribution and a
downward slide from the previously privileged positions for the upper part
Milanović (2020, 2023). On the other hand, the global economic stagnation may
help to slow down the climate change. The relentless growth, fueled by fossil
fuels for the last couple centuries since the beginning of Industrial
Revolution, must decelerate because of limited environmental capacity of the
Earth. However, the saturation of the global inequality decrease greatly
complicates any agreement on limiting carbon emissions because of vast
differences between the countries at the opposite ends of global distribution.
The urgently needed transition from redistributable hydrocarbon fuels toward
decentralized locally-generated and locally-consumed renewable energy could
help to both lower carbon emissions and reduce global inequality Lawrence,
Liu, and Yakovenko (2013).

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# Study of all the periods of a Neuronal Recurrence Equation

Serge Alain Ebélé, René Ndoundam 111Corresponding author<EMAIL_ADDRESS>
University of Yaounde I, LIRIMA, Team GRIMCAPE, P.o.Box 812 Yaounde, Cameroon
CETIC, Yaounde, Cameroon
IRD, UMI 209, UMMISCO, IRD France Nord, F-93143, Bondy, France;
Sorbonne Unversités, Univ. Paris 06, UMI 209, UMMISCO, F-75005, Paris, France
E.mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We characterize the structure of the periods of a neuronal recurrence
equation. Firstly, we give a characterization of k-chains in 0-1 periodic
sequences. Secondly, we characterize the periods of all cycles of some
neuronal recurrence equation. Thirdly, we explain how these results can be
used to deduce the existence of the generalized period-halving bifurcation.

Keywords. Neuronal recurrence equation, cycle, period, generalized period-
halving bifurcation.

##  1 Introduction

The human brain can be viewed as a set of interconnected neurons. Caianiello
[3, 4] suggested to model the brain using the following threshold automata
network:

$x_{i}(t+1)={\bf
1}\Big{(}\sum_{j=1}^{n}\sum_{s=1}^{k}a_{ij}(s)x_{j}(t+1-s)-\theta_{i}\Big{)}\
\ \ \ 1\ \leq i\ \leq n,\ \ t\geq k-1$ (1)

where:

$x_{j}(t+1-s)$ is the state of the neuron $j$ at time $t+1-s$,
---
$a_{ij}(s)$ represents the influence of the neuron $j$ at time $t+1-s$ on the
neuron $i$ at time $t+1$,
$\theta_{i}$ is the threshold of the excitation of the neuron $i$,
$\sum_{j=1}^{n}\sum_{s=1}^{k}a_{ij}(s)x_{j}(t+1-s)$ is the potential of the
neuron $i$ at time $t$,
$n$ is the number of the neurons of the network,
$k$ is the size of the memory,
1[$u$] = 0 if $u\ <\ 0$, and 1[$u$] = 1 if $u\geq 0$.

The dynamics of this model has been studied in some particular cases:

  1. 1.

In the Equation (1), when $k=1$, we obtain the following equation:

$x_{i}(t+1)={\bf 1}\Big{(}\sum_{j=1}^{n}a_{ij}x_{j}(t)-\theta_{i}\Big{)}\ \ \
\ 1\ \leq i\ \leq n$ (2)

which models the dynamic behavior of $n$ interconnected neurons of memory size
$1$. These networks were introduced by McCulloch and Pitts [9], and are quite
powerful.

  2. 2.

In the Equation (1), when $n=1$, we obtain the following equation:

$\displaystyle x(n)={\bf 1}\Biggl{(}\sum_{j=1}^{k}a_{j}x(n-j)-\theta\Biggl{)}$
(3)

introduced by Caianiello and De Luca [5] which models the dynamic behavior of
a single neuron with a memory, that does not interact with other neurons.

Neural networks are usually implemented by using electronic components or are
simulated by a software on a digital computer. One way in which the collective
properties of a neural network may be used to implement a computational task
is through the energy minimization concept. The Hopfield network is a well-
known example of such an approach. It has attracted a wide attention in
literature as a content-addressable memory [2].
Caianiello networks have been studied by Goles [13] and Ndoundam [18]. Many
studies have been devoted to the McCulloch and Pitts’ neural networks [1, 14,
11, 12, 15, 10, 2]. Matamala [15] studied McCulloch and Pitts’ reverberation
neural networks (i.e. neural networks of McCulloch and Pitts where each state
of the system, after a finite number of steps comes back to itself, hypercube
permutation).

Cosnard, Moumida, Goles and T. de St. Pierre [6] showed the following result
in the case of palindromic memory:

###### Proposition 1

[6] If the interacting coefficients $(a_{1},a_{2},\cdots,a_{k})$ verify

$a_{i}\ =\ a_{k+1-i}\ \ \forall i\in\mathbb{N},\ 1\leq i\leq k$

Then the length of each cycle is a divisor of $k+1$.

$\blacksquare$
In the case of j-palindromic memory, they also showed:

###### Proposition 2

[6] If the interacting coefficients $(a_{1},a_{2},\cdots,a_{k})$ are
j-palindromic, i.e. verify

  * •

$a_{1}\ =\ a_{2}\ =\ \cdots\ a_{j}\ =\ 0$

  * •

$a_{i}\ =\ a_{k+j+1-i}\ \ \ \ \forall i\in\mathbb{N},\ j+1\leq i\leq k$

Then the length of each cycle is a divisor of $k+j+1$.

$\blacksquare$
When the memory are geometric sequence, they showed the following result:

###### Proposition 3

[6] If the interacting coefficients verify $a_{i}\ =\ -(b^{i})$ with
$b\in]0,\frac{1}{2}]$ Then the length of each cycle is less or equal to $k+1$.

$\blacksquare$
In the case of positive geometric sequence, they showed

###### Proposition 4

[6] If the interacting coefficients verify $a_{i}\ =\ (b^{i})$ with
$b\in]0,\frac{1}{2}]$ Then the length of each cycle is $1$.

$\blacksquare$
Another results have been established on neuronal recurrence equations
modeling neurons with memory [1, 6, 7, 8, 24, 16, 17, 20, 23, 25]. From the
point of view of the period:

  * •

in [7, 23, 16, 17, 20], the authors didn’t study all the cycles generated by
the neuronal recurrence equation;

  * •

in this paper, we are studying all the cycles generated by the neuronal
recurrence equation $\\{y(n)\ :\ n\geq 0\\}$.

From the point of view of bifurcation:

  * •

in [19], we studied the dynamics of the sequence $\\{z(n)\ :\ n\geq 0\\}$ from
one and only one initial configuration. We characterized only one cycle of the
sequence $\\{z(n)\ :\ n\geq 0\\}$;

  * •

in [21], for any $d$ ( $0\leq d\leq\rho(m)-1$ ), we studied the dynamics of
the sequence $\\{z(n,d)\ :\ n\geq 0\\}$ from one and only one initial
configuration. We characterized only one cycle of the sequence $\\{z(n,d)\ :\
n\geq 0\\}$;

  * •

in this paper, for any $d$ ( $0\leq d\leq\rho(m)-1$ ), we are showing how to
study the dynamics of the sequence $\\{z(n,d)\ :\ n\geq 0\\}$ from any initial
configuration. We are showing how to characterize the length of all cycles of
the sequence $\\{z(n,d)\ :\ n\geq 0\\}$.

Our work, from some point of view, is similar to the work of Matamala [15] in
the sense that we study all the periods.

The paper is organized as follows: in Section 2, some previous results are
presented. Section 3 presents a characterization of k-chains in 0-1 periodic
sequences. Section 4 is devoted to the characterization of the period length
of all the cycles. In section 5, we study a bifurcation. Concluding remarks
are stated in Section 6.

##  2 Previous Results

Given a finite neural network, the configuration assumed by the system at time
$t$ is ultimately periodic. As a consequence, there is an integer $p\ >\ 0$
called the period (or the length of a cycle) and another integer $T\geq 0$
called the transient length such that:

  * $\bullet$

$Y(p+T)=Y(T)$

  * $\bullet$

$\nexists\ T^{\prime}\ and\ p^{\prime}\ \ (T^{\prime},p^{\prime})\neq(T,p)\ \
\ T\geq T^{\prime}\ and\ p\geq p^{\prime}\ \ {\rm such\ that}\
Y(p^{\prime}+T^{\prime})\ =\ Y(T^{\prime})$

where $Y(t)=(x(t),x(t-1),\dots,x(t-k+2),x(t-k+1))$. The period and the
transient length of the sequences generated are good measures of the
complexity of the neuron. A bifurcation occurs when a small smooth change made
to the parameter values (the bifurcation parameters) of a system, causes a
sudden ’qualitative’ or topological change in its behaviour. A period-halving
bifurcation in a dynamic system, is a bifurcation in which the system switches
to a new behaviour with half the period of the original system from some
initial configuration. A generalized period-halving bifurcation is a period-
halving bifurcation from any initial configurations.

Cosnard, Tchuente and Tindo [7] show the following lemma:

###### Lemma 1

[7]
If there is a neuronal recurrence equation with memory length $k$ that
generates sequences of periods $p_{1},p_{2},\dots,p_{r}$, then there is a
neuronal recurrence equation with memory length $kr$ that generates a sequence
of period $r\times lcm(p_{1},\cdots,p_{r})$.

$\blacksquare$
Lemma 1 does not take into account the study of the transient length. One can
amend Lemma 1 to obtain the following lemma:

###### Lemma 2

[21] If there is a neuronal recurrence equation with memory length $k$ that
generates a sequence $\\{x^{\jmath}(n):n\geq 0\\}\ ,1\leq\jmath\leq g$ of
transient length $T_{\jmath}$ and of period $p_{\jmath}$, then there is a
neuronal recurrence equation with memory length $kg$ that generates a sequence
of transient length $g\times max(T_{1},T_{2},\dots,T_{g})$ and of period
$Per$. $Per$ is defined as follows:
First case: $\exists\ j,\ 1\leq\ j\ \leq\ g$ such that $p_{j}\geq 2$

$Per\ =\ g\times lcm(p_{1},\cdots,p_{g}).$

Second case: $p_{j}=1$ ; $\forall\ j,\ 1\leq\ j\ \leq\ g$.

$Per{\rm\ is\ a\ divisor\ of\ }g.$

$\blacksquare$

Cosnard and Goles [8] studied the bifurcation in two particular case of
neuronal recurrence equations.
Case 1: Geometric coefficients and bounded memory
Cosnard and Goles completely described the structure of the bifurcation of the
following equation:

$x_{n+1}={\bf 1}\Biggl{(}\theta-\sum_{i=0}^{k-1}b^{i}x_{n-i}\Biggl{)}$

when $\theta$ varies. They showed that the associated rotation number is an
increasing number of the parameter $\theta$.
Case 2: Geometric coefficients and unbounded memory.

Cosnard and Goles completely described the structure of the bifurcation of the
following equation:

$x_{n+1}={\bf 1}\Biggl{(}\theta-\sum_{i=0}^{n}b^{i}x_{n-i}\Biggl{)}$

when $\theta$ varies. They showed that the associated rotation number is a
devil’s staircase.

The next section is devoted to the study of k-chains.

##  3 Characterization of k-chains in 0-1 periodic sequences

We recall the concept of k-chains in 0-1 periodic sequences [1] which is
useful in the study of the limit orbits. Let $Y\ =\ (y(t)\ :\ t\in\mathbb{N})$
be a periodical sequence of 0’s and 1’s; suppose that the period $\gamma(Y)$ (
which is a priori unknown ) divides $T$. Thus $y(t)\in\\{0\ ,1\\}$ for any
$t\in\mathbb{Z}$ and $y(t)=y(t^{\prime})$ when $t\equiv t^{\prime}\pmod{T}$.
In studying period lengths, we shall deal with sets invariant under
translations[1], so the following notation will be useful: if
$\Gamma\subset\mathbb{Z}_{T}$, $l\in\mathbb{Z}$, we write:

$\Gamma+l\ =\ \\{\ t+l\ (\ mod\ T\ )\ :\ t\ \in\ \Gamma\\}$

Let us partition the set $\mathbb{Z}_{T}$ into $\Gamma^{0}(Y)\ =\ \\{\ t\ \in\
\mathbb{Z}_{T}\ :\ y(t)=0\\}$ and $\Gamma^{1}(Y)\ =\ \\{\ t\ \in\
\mathbb{Z}_{T}\ :\ y(t)=1\\}$ which is called the support of Y. The period of
the set $\Gamma^{1}(Y)$ is the smallest positive number $\gamma$ such that
$\Gamma^{1}(Y)+\gamma\ =\ \Gamma^{1}(Y)$. The following result was established
in [1] : the period of the sequence ( i.e. $\gamma(Y)$ ) is equal to the
period of $\Gamma^{1}(Y)$. It is shown in [1] that:

$\gamma(Y)\ {\rm divides\ k\ if\ and\ only\ if}\ \Gamma^{1}(Y)\ +k=\
\Gamma^{1}(Y)$

Now let us define k-chains ( for $k\geq 1$ ) contained in the support
$\Gamma^{1}(Y)$. A subset $C\subset\Gamma^{1}(Y)$ is called a k-chain if and
only if it is of the form $C\ =\ \\{\ t+kl\ (\ {\rm mod\ T}\ )\ :\ 0\leq l\leq
s-1\\}$ for some $s\geq 1$. So a k-chain is a subset $C\ =\ \\{\ t+kl\ \in\
\mathbb{Z}_{T}\ :\ 0\leq\ l\ \leq s-1\ \\}$ such that $y(t^{\prime})\ =\ 1$
for any $t^{\prime}\ \in\ C$.

We characterize the 0-1 sequence which contains two different chains.

###### Lemma 3

If a 0-1 sequence $\\{u(n)\ :\ n\geq 0\\}$ contains:

  * •

an $\ell_{1}$-chain,

  * •

an $\ell_{2}$-chain,

  * •

such that $\ell_{1}$ and $\ell_{2}$ are relatively prime.

Then
$\exists\ t\ \in\mathbb{N}$ such that:

  * •

$u(t)\ =\ 1$,

  * •

$u(t+\ell_{1})\ =\ 1$,

  * •

$u(t+\ell_{2})\ =\ 1$.

$\blacksquare$

We use the lemma 3 to characterize all the periods of all the attractors.

##  4 Characterization of the periods of all the cycles

Let us consider a positive integer $m$ and a positive real number $\theta\geq\
2m$, we note:

###### Notation 1

  * •

$p_{0},p_{1},\cdots,p_{s-1}$ are prime numbers taken between $2m$ and $3m$
such that $p_{i}\ <\ p_{i+1}$,

  * •

$\alpha_{i}\ =\ 3m-p_{i}$, $0\ \leq\ i\ \leq s-1$,

  * •

$k=6m$,

and we define the coefficients as follows:

$coef_{1}(j)=\begin{cases}(\theta/2)-\alpha_{i}&\text{if $j\ =\
3m-\alpha_{i},\ 0\leq\ i\ \leq s-1$;}\\\ (\theta/2)+\alpha_{i}&\text{if $j\ =\
2(3m-\alpha_{i}),\ 0\leq\ i\ \leq s-1$;}\\\
-k(\theta+m)&\text{otherwise}.\end{cases}$ (4)

The coefficients defined in Equation (4) are analog to those defined in [23].
For each $i$, $0\leq i\leq s-1$, the first $k$ terms of the sequence
$\\{x^{\alpha_{i}}(n)\ :\ n\geq 0\\}$ are defined as follows:

$x^{\alpha_{i}}(0)x^{\alpha_{i}}(1)\cdots x^{\alpha_{i}}(k-1)\ =\
\underbrace{00\dots 0}_{2\alpha_{i}}\underbrace{100\dots
0}_{3m-\alpha_{i}}\underbrace{100\dots 0}_{3m-\alpha_{i}}$ (5)

$\forall\ n\ \geq\ k$, the term $x^{\alpha_{i}}(n)$ of the sequence
$\\{x^{\alpha_{i}}(n)\ :\ n\in\mathbb{N}\\}$ is defined as follows:

$x^{\alpha_{i}}(n)\ =\ {\bf
1}(\sum_{i=1}^{k}coef_{1}(i)x^{\alpha_{i}}(n-i)-\theta)$

By using the technique developped by Tchuente and Tindo [23], it is easy to
prove the following lemma:

###### Lemma 4

The sequence $\\{x^{\alpha_{i}}(n)\ :\ n\in\mathbb{N}\\}$ describes a cycle of
length $3m-\alpha_{i}$ of the following form:

$\underbrace{00\dots 0}_{2\alpha_{i}}\underbrace{100\dots
0}_{3m-\alpha_{i}}\underbrace{100\dots 0}_{3m-\alpha_{i}}\underbrace{100\dots
0}_{3m-\alpha_{i}}\cdots\underbrace{100\dots 0}_{3m-\alpha_{i}}\cdots$

$\blacksquare$
In [23], the authors didn’t study all the cycles generated by the neuronal
recurrence equation. One of our aims is to study all the cycles generated by
some neuronal recurrence equation.

We construct the sequence $\\{u(n)\ :\ n\geq 0\\}$ generated by the neuronal
recurrence equation

$u(n)\ =\ {\bf 1}\Bigl{(}\sum_{j=1}^{k}coef_{1}(j)u(n-j)-\theta\Bigl{)}\ ,\
n\geq k$ (6)

such that the initial terms are defined as follows:

$u(0)u(1)\cdots u(k-1)\ \in\\{0,1\\}^{k}.$ (7)

Let us characterize the period of the sequence $\\{u(n)\ :\ n\geq 0\\}$ by
showing the following proposition:

###### Proposition 5

The sequence $\\{u(n)\ :\ n\geq 0\\}$ converges

  * •

to the null sequence i.e. to $0\ 0\cdots 0\ 0\cdots 0\ 0\cdots$ , or

  * •

to one of the sequences $\\{x^{\alpha_{i}}(n):n\ \geq\ 0\\}\ ,\ 0\ \leq\ i\
\leq\ s-1$.

$\blacksquare$
Example: In the aim to give an idea of the basin of attraction of the sequence
$\\{u(n)\ :\ n\geq 0\\}$, we choose the following parameters:

$m=5,\ \theta=12,\ p_{0}=11,\ p_{1}=13,\ {\rm and}\ k=30.$

We build from the preceding parameters the following neuronal recurrence
equation

$u(n)\ =\ {\bf 1}\Bigl{(}\sum_{j=1}^{30}coef_{1}(j)u(n-j)-\theta\Bigl{)}\ ,\
n\geq 30$ (8)

where:

$coef_{1}(j)=\begin{cases}2\ ,&\text{if $j=11$ }\\\ 4\ ,&\text{if $j=13$ }\\\
10\ ,&\text{if $j=22$ }\\\ 8\ ,&\text{if $j=26$ }\\\ -510\
,&\text{otherwise.}\end{cases}$ (9)

Let us note:

$config(i)\ =\ \\{\,u(0)u(1)\cdots u(28)u(29)\ :\ \forall i,\ 0\leq i\leq 29\
,\ u(i)\in\\{0,1\\}\\\ \rm{and\ the\ neuronal\ recurrence\ equation\ defined\
by\ equation}(\ref{kk0:kk8})\ {\rm from\ the}\\\ {\rm initial\ terms}\
u(0)u(1)\cdots u(28)u(29)\ {\rm converges\ to\ a\ cycle\ of\ length}\ i\,\\}.$
(10)

We also note $\chi(i)\ =\ card(config(i))$. By numerical simulations, the
values of the sequence $\chi(i)$ are:

$\displaystyle\chi(i)=0\ \ if\ i\ \notin\ \\{1,\ 11,\ 13\\}$
$\displaystyle\chi(1)=1073713157$ $\displaystyle\chi(11)=4094$
$\displaystyle\chi(13)=24573.$

###### Notation 2

Let us define the memory length of some neuronal recurrence equations as
follows:

$h\ =\ s\times k=6m\times s.$

Let $\\{y(n):n\geq 0\\}$ be the sequence whose first $h$ terms are defined as
follows:

$y\left(i\right)\ \in\ \\{0,1\\},\ \ 0\leq i\leq-1+h,$ (11)

and the other terms are generated by the following neuronal recurrence
equation:

$y(n)={\bf 1}\left[\sum_{f=1}^{h}coef_{2}(f)y(n-f)-\theta_{1}\right]\ ;\ n\geq
h$ (12)

where

$coef_{2}(f)=\begin{cases}coef_{1}(j)\ ,&\text{if $f=s\times j,\ \ 1\leq j\leq
k$ }\\\ 0\ ,&\text{otherwise.}\end{cases}$ (13) $\theta_{1}=\theta.$ (14)

The parameters $coef_{1}(j)$ are those defined in Equation (4).

###### Remark 1

(a) The first $h$ terms of the sequence $\\{y(n):n\geq 0\\}$ are obtained by
taking any element of the set $\\{0,1\\}^{h}$.
(b) The coefficients $coef_{2}(f)$ of neuronal recurrence Equation (12) are
obtained by applying the construction of Lemma 1 to the parameters defined by
Equation (4).

Our aim is to characterize the structure of all the periods of the sequence
$y(n)$ from a qualitative point of view. The next theorem gives the period of
the sequences $\\{y(n):n\geq 0\\}$.

###### Theorem 1

From any initial term, the sequence $\\{y(n)\ :\ n\geq 0\\}$ converges to a
cycle of length :

  * •

$s\times lcm(elt_{1},elt_{2},\dots,elt_{s})$ where $elt_{i}\ \in\
\\{p_{0},p_{1},\dots,p_{s-1}\\}$ for any $i\in\\{0,1,2,\dots,s-1\\}$, or

  * •

$p$ where $p$ is equal to $1$.

$\blacksquare$

In the next section, we show how to apply the previous technique to the study
of bifurcation of the neuronal recurrence equation $z(n,d)$.

##  5 Generalized Bifurcation of the neuronal recurrence equation

Let us define the neuronal recurrence equation $\\{\tilde{x}(n)\ :\ n\ \geq\
0\ \\}$ by the following recurrence:

$\tilde{x}(t)={\bf
1}\Bigl{(}\sum_{j=1}^{k_{2}}coef_{3}(j)\tilde{x}(t-j)-\bar{\theta}\Bigl{)}\ ;\
t\geq k_{2}$ (15)

where $coef_{3}(j)$ is defined as follows:

First case: $s$ is even and $\forall\ i_{2}\in\mathbb{N},\ 0\leq i_{2}\leq
s-1$

$coef_{3}(j)=\begin{cases}2&\text{if $j\in R1(\alpha_{i_{2}})\ {\rm and}\
j\leq\frac{3\times s\times p_{i_{2}}}{2}\ ,$}\\\ -2&\text{if $j\in
R1(\alpha_{i_{2}})\ {\rm and}\ j\ >\frac{3\times s\times p_{i_{2}}}{2}\ ,$}\\\
-4\times k_{2}&\text{otherwise. }\end{cases}$ (16)

Second case: $s$ is odd, $s\geq 3$ and $\forall\ i_{2}\in\mathbb{N},\ 0\leq
i_{2}\leq s-1$

$coef_{3}(j)=\begin{cases}2&\text{if $j\in R1(\alpha_{i_{2}})\ {\rm and}\
j\leq\frac{(3\times s)-1}{2}\times p_{i_{2}}\ ,$ }\\\ -2&\text{if $j\in
R1(\alpha_{i_{2}})\ {\rm and}\ \frac{(3\times s)+1}{2}\times p_{i_{2}}\leq
j\leq(2s-2)\times p_{i_{2}},$}\\\ -1&\text{if $j\in\\{(2s-1)\times p_{i_{2}}\
,\ 2s\times p_{i_{2}}\\}$ \ , }\\\ -4\times k_{2}&\text{otherwise.
}\end{cases}$ (17)

The parameters $R1(\alpha_{i})$, $\bar{\theta}$ and $k_{2}$ are defined as
follows:

$\displaystyle R1(\alpha_{i})$ $\displaystyle=\\{jp_{i}:j=1,\dots,2s\\}$ (18)
$\displaystyle=\\{p_{i},2p_{i},\dots,(-1+2s)p_{i},2\times s\times p_{i}\\},\ \
0\ \leq i\ \leq-1+s;$ (19) $\displaystyle R2$
$\displaystyle=\\{i:i=1,\dots,k_{2}\\}=\\{1,2,\dots,-1+k_{2},k_{2}\\};$ (20)
$\displaystyle R3$ $\displaystyle=\bigcup_{i=0}^{-1+s}R1(\alpha_{i});$ (21)
$\displaystyle R4$ $\displaystyle=R2\setminus R3;$ (22)
$\displaystyle\bar{\theta}$ $\displaystyle=\ 2\times s;$ (23) $\displaystyle
k_{2}$ $\displaystyle=(6m-1)\times s.$ (24)

By applying the technique developped in sections 2 and 3, and the one
developped in [21] to the neuronal recurrence equation defined by Equation
(15), it is easy to construct a family of neuronal recurrence equations
$\\{z(n,d):n\geq 0\\}$ which verify the following theorem.

###### Theorem 2

$\forall m,d\in\mathbb{N}$ such that $m\geq e^{2}$ and $0\leq d\leq s-2$, we
construct a set of neuronal recurrence equations whose behaviour has the
following characteristics:

  * •

From any initial configuration, the neuronal recurrence equation
$\\{z(n,d):n\geq 0\\}$ converges to a cycle of length $s\times
lcm(elt_{1},elt_{2},\dots,elt_{s})$ where $elt_{i}\ \in\
\\{p_{d+1},p_{d+2},\dots,p_{s-1}\\}$ for any $i\in\\{1,2,\dots,s\\}$, or to a
cycle of length 1.

  * •

From any initial configuration, the neuronal recurrence equation
$\\{z(n,s-1):n\geq 0\\}$ converges to a fixed point (i.e. the period of a
cycle is 1).

$\blacksquare$

In other words, the first part of Theorem 2 can be interpreted as follows: in
some cases, the length of the cycles of the neuronal recurrence equation
$\\{z(n,d-1):n\geq 0\\}$ is divided by $p_{d}$ to obtain the length of cycles
of the neuronal recurrence equation $\\{z(n,d):n\geq 0\\}$.
By perturbation, we can build the neuronal recurrence equation
$\\{z(n,d):n\geq 0\\}$ from the neuronal recurrence equation
$\\{z(n,d-1):n\geq 0\\}$.

###### Remark 2

The new contributions in this paper with respect to the previous works are:
Firstly, from the point of view of the period:

  * •

in [7, 23, 16, 17, 20], the authors didn’t study all the cycles generated by
the neuronal recurrence equation;

  * •

in this paper, we studied all the cycles generated by the neuronal recurrence
equation $\\{y(n)\ :\ n\geq 0\\}$.

Secondly, from the point of view of bifurcation:

  * •

in [19], we studied the dynamics of the sequence $\\{z(n)\ :\ n\geq 0\\}$ from
one and only one initial configuration. We characterized only one cycle of the
sequence $\\{z(n)\ :\ n\geq 0\\}$;

  * •

in [21], for any $d$ ( $0\leq d\leq\rho(m)-1$ ), we studied the dynamics of
the sequence $\\{z(n,d)\ :\ n\geq 0\\}$ from one and only one initial
configuration. We characterized only one cycle of the sequence $\\{z(n,d)\ :\
n\geq 0\\}$;

  * •

in this paper, for any $d$ ( $0\leq d\leq\rho(m)-1$ ), we studied the dynamics
of the sequence $\\{z(n,d)\ :\ n\geq 0\\}$ from any initial configurations. We
characterized the length of all cycles of the sequence $\\{z(n,d)\ :\ n\geq
0\\}$.

##  6 Conclusion

We have given a characterization of k-chains in 0-1 periodic sequences. This
characterization allows us to determine the periods of all cycles of some
neuronal recurrence equations. From the structure of the periods of all
cycles, we show how to build the family of neuronal recurrence equation
$\\{z(n,d):n\geq 0\\}$ which admits a generalized period-halving bifurcation.
The structure of the configuration of neuronal recurrence equation can be used
in steganography (see Second Approach and Third Approach of the paper [22]).
Acknowledgements
We thank the reviewers for their suggestions.This work was supported by
UMMISCO , LIRIMA  and the University of Yaounde 1.

Appendix

Proof of Lemma 3
From the hypothesis, the sequence $\\{u(n)\ :\ n\geq 0\\}$ contains two
chains, we can deduce that:

  * •

$\exists\ a\ \in\mathbb{N}\ ,\ 0\leq a\ <\ \ell_{1}$

  * •

$\exists\ b\ \in\mathbb{N}\ ,\ 0\leq b\ <\ \ell_{2}$

  * •

$u(a+(i\times\ell_{1}))\ =\ 1\ ,\forall i\in\mathbb{N}$

  * •

$u(b+(j\times\ell_{2}))\ =\ 1\ ,\forall j\in\mathbb{N}$

By hypothesis, integers $\ell_{1}$ and $\ell_{2}$ are relatively prime and
from the definition of greatest common divisor, we can deduce:

$\exists\ n_{1},n_{2}\in\mathbb{Z}\ {\rm such\ that\
}n_{1}\times\ell_{1}+n_{2}\times\ell_{2}\ =\ 1$ (25)

From Equation (25), we can easily deduce:

$\displaystyle n_{1}\times(b-a)\times\ell_{1}+n_{2}\times(b-a)\times\ell_{2}\
=\ b-a$ (26a) $\displaystyle
a+(n_{1}\times(b-a)\times\ell_{1})=b-(n_{2}\times(b-a)\times\ell_{2})$ (26b)

From Equation (26b), it follows that $\exists\ i_{0},\ j_{0}\ \in\mathbb{N}$
defined as follows:

$\displaystyle i_{0}\ =\ n_{1}\times(b-a)+\Bigl{(}\bigl{(}1+\mid
n_{1}(b-a)\mid+\mid n_{2}(b-a)\mid\bigl{)}\times\ell_{2}\Bigl{)}$ (27a)
$\displaystyle j_{0}\ =\ -n_{2}\times(b-a)+\Bigl{(}\bigl{(}1+\mid
n_{1}(b-a)\mid+\mid n_{2}(b-a)\mid\bigl{)}\times\ell_{1}\Bigl{)}$ (27b)

such that:

$a+(i_{0}\times\ell_{1})\ =\ b+(j_{0}\times\ell_{2})$

It suffices to choose $t=a+(i_{0}\times\ell_{1})$.
$\blacksquare$

Proof of Proposition 5
Without loss of generality, let us choose the following initial terms

$u(0)u(1)u(2)\cdots u(k-1)\ \in\ \\{0,1\\}^{k}.$

We suppose that from the following initial terms

$u(0)u(1)u(2)\cdots u(k-1)\ \in\ \\{0,1\\}^{k}$

the sequence $\\{u(n)\ :\ n\geq 0\\}$ describes a transient of length $T_{1}$
and a cycle of length $P_{1}$. We define the sequence $\\{w(n)\ :\ n\geq 0\\}$
as follows:

$\forall\ n\in\mathbb{N}\ \ \ w(n)=u(n+T_{1}).$

In others words, the sequence $\\{u(n)\ :\ n\geq 0\\}$ converges to the
attractor $\\{w(n)\ :\ n\geq 0\\}$. The proof is divided into two parts:
Firstly, let us suppose that the sequence $\\{w(n)\ :\ n\geq 0\\}$ is not
equal to one of the following two sequences:

  * •

to the null sequence i.e. to $0\ 0\cdots 0\ 0\ 0\cdots$

  * •

to one of the sequences $\\{x^{\alpha_{i}}(n):n\ \geq\ 0\\}\ ,\ 0\ \leq\ i\
\leq\ s-1$.

We can extract from the sequence $\\{w(n)\ :\ n\geq 0\\}$ an $\ell$-chain such
that :

$\ell\ \not\equiv\ 0\bmod{p_{i}}\ ,\ 0\ \leq i\ \leq s-1.$ (28)

Without loss of generality, let us assume that:

$w(t_{1})\ =\ 1$ (29)

from the fact that the sequence $\\{w(n)\ :\ n\geq 0\\}$ admits an
$\ell$-chain, we can deduce that:

$w(t_{1}+\ell)\ =\ 1.$ (30)

From Equation (28), we can easily deduce that:

$coef_{1}(\ell)\ =\ -k(\theta+m).$ (31)

From the fact that:

  * •

$w(\ell+t_{1})\ =\ {\bf 1}(\sum_{j=1}^{k}coef_{1}(j)w(\ell+t_{1}-j)-\theta)$,

  * •

$coef_{1}(\ell)\ =\ -k(\theta+m)$,

  * •

$w(t_{1})\ =\ 1$.

We deduce that : $w(\ell+t_{1})\ =\ 0$. It follows that we have a
contradiction with Equation (30). We can deduce that there is no $\ell$-chain
in the sequence $\\{w(n)\ :\ n\geq 0\\}$ that verifies Equation (28).

Secondly, let us suppose that on the sequence $\\{u1(n)\ :\ n\geq 0\\}$, there
exist at least two different chains.

Without loss of generality, let us suppose that there exist on the sequence
$\\{w(n)\ :\ n\geq 0\\}$ :

  * •

an $\ell_{1}$-chain such that $\ell_{1}\ =\ p_{i1}\ ,\ 0\ \leq i1\ \leq s-1$;

  * •

an $\ell_{2}$-chain such that $\ell_{2}\ =\ p_{i2}\ ,\ 0\ \leq i2\ \leq s-1$;

  * •

$l_{1}\ <\ l_{2}$, i.e. $p_{i1}\ <\ p{i2}$.

From the fact that the sequence $\\{w(n)\ :\ n\geq 0\\}$ admits two chains :
$\ell_{1}$-chain and $\ell_{2}$-chain, we deduce by application of Lemma 3
that there exist $t_{1}\in\mathbb{N}$ which verifies:

$\displaystyle w(t_{1})=1;$ (32a) $\displaystyle w(t_{1}+\ell_{1})=1;$ (32b)
$\displaystyle w(t_{1}+\ell_{2})=1.$ (32c)

We have : $2\times m\leq p_{i1}\ <\ p_{i2}\leq 3\times m$. It follows that:

$\imath\ =\ \ell_{2}-\ell_{1}\ =\ p_{i2}-p_{i1}\ \leq\ m.$ (33)

From Equation (4) and Equation (33), we deduce that:

$coef_{1}(\imath)\ =-k(\theta+m)$ (34)

Based on the facts that:

  * •

$w(t_{1}+\ell_{1})\ =\ 1$,

  * •

$coef_{1}(\imath)\ =\ -k(\theta+m)$,

  * •

$w(t_{1}+\ell_{2})\ =\ {\bf
1}\Bigl{(}\sum_{j=1}^{k}coef_{1}(j)w(t_{1}+\ell_{2}-j)-\theta\Bigl{)}$.

We deduce easily that $w(t_{1}+\ell_{2})=0$. This is a contradiction with the
Equation (32c). We easily deduce that the sequence $\\{w(n)\ :\ n\geq 0\\}$
contains one and only one chain.
$\blacksquare$

Proof of Theorem 1
Based on Lemma 2 and Proposition 5, we deduce the result.
$\blacksquare$

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