# Market Crash Prediction Model for Markets in A Rational Bubble

HyeonJun Kim
UG in Dept.Finance, Soongsil University
<EMAIL_ADDRESS>
###### Abstract

Renowned method of log-periodic power law(LPPL) is one of the few ways that a
financial market crash could be predicted. Alongside with LPPL, this paper
propose a novel method of stock market crash using white box model derived
from simple assumptions about the state of rational bubble. By applying this
model to Dow Jones Index and Bitcoin market price data, it is shown that the
model successfully predicts some major crashes of both markets, implying the
high sensitivity and generalization abilities of the model.

keywords: [Rational Bubble], [Market Crash]

##  1 Introduction

It is well known that the efficient market price tends to follow the
characteristics of Brownian Motion that strictly depends on stochastic
nature[3]. Thus predicting the information about the short-term return of a
financial portfolio is known to be difficult, but there are few exceptions.
Firstly, there is a widely held belief among traders that it is possible for
some computational deterministic patterns to emerge for short period of time.
This belief, combined with econophysics, led the creation of algorithmic
trading funds such as Prediction Company and Renaissance Technologies.
Secondly, there is a concept of rational bubble that illustrates the rational
behavior of people who invest in markets that clearly creating an economic
bubble[1].

With these two facts in mind, it is possible to imagine a model with
reasonable assumptions to predict rational bubble’s key behaviors, such as the
stock market crash after a bubble economy. While there are more sophisticated
way of prediction such as log-periodic power law, there seems to be no clear
white-box model that also have high prediction abilities. In this paper, by
using some reasonable assumptions about the investors, a white-box model about
the stock market state will be derived. The model will then be applied to two
different cases, the Dow Jones Index and Bitcoin market prices to assess the
ability of the model.

###  1.1 Previous Study on Rational Bubble and its Predictions

Theories and methodologies of complex systems helped studies and individuals
to make attempts to solve the prediction problems. Log-periodic power
law(LPPL) is one of them[4], illustrating the dynamics of two conflicting
group of investors, value investors and trend followers[2]. There are some
successful Implementation cases of this model as well, demonstated by Yan,
Woodarda, Sornette[5].

##  2 Model and its Assumptions

###  2.1 Definition of Rational Bubble

A rational bubble in this paper’s context is the state of a market that the
majority of the new investors are investing because of the past market price
or any major index. For example, if investors of NASDAQ invest in NASDAQ only
because the past NASDAQ index was rising, it is called a rational bubble in
this paper. There is more formerly known definition about this particular
word. However for the convenience of understanding the model’s key points, the
definition will be little vague for now.

###  2.2 Assumptions

Because investing in markets with rational bubble is caused by the price
change, these investors are only interested in price or similar indices. This
causes the feedback loop in the market, as a self-fulfilling enthusiasm caused
by the investors themselves[5]. There, if the price point $p_{\alpha}$ where
average return of the investors is same as the market return is higher than
the current market price point, it is somewhat true that most investors’
portfolio is showing less return than the market presents. This is because not
all investors invested in the lowest price point, making the average return of
the portfolios grow slower than the market return during the bubble even
though the market price change is same to every investor. Thus, if the market
price exceeds price point $p_{\alpha}$, the investors expectation for return
caused by the superficial market return is not satisfied. This makes the
investors behave more irrationally, or doubt the market, as the root
assumption of their investment is not met. This potential tendency could cause
massive market crash just with somewhat minor price change, causing the doubt
to become a real action. All of this potential threat is possible after the
minimum price point where market return is same as the average return of the
investors.

###  2.3 Mathematical Derivation of the Model

For convenience, it is assumed that every investors invest in the market
portfolio. If a certain price point is $p_{x}$ and the price before the bubble
is $p_{0}$ and assuming that investors of the bubble market started investing
after the start of the bubble, the average return at price point $p_{x}$ of
every single stock in a continuous form is as follows.

$E(R)={\int_{p_{0}}^{p_{x}}R_{p}V_{p}\,dp\over\int_{p_{0}}^{p_{x}}V_{p}\,dp}$

$R_{p}$ is the current return of a market portfolio in interest which is
bought at price $p$, and $V_{p}$ is the volume of a market portfolio in
interest which is bought at price $p$. As for the volume, the trading volume
will be used for approximation. Also, as the exact data of trading volume
could not be used because the discrete nature of trading volume, the trading
volume will be approximated by a regression line which shows the relationship
between price and trading volume. The formula of the regression line is as
follows.

$v=ap+b={r_{vp}S(v)\over S(p)}p+\left(E(v)-{rS(v)\over S(p)}E(p)\right)$

$v$ is volume of trades, $p$ is the price, $S(v)$ and $S(p)$ are the standard
deviation of $v$ and $p$, $E(v)$ and $E(p)$ are the average of $v$ and $p$,
and $r_{vp}$ is the correlation coefficient of the two variable $v$ and $p$.
By using this regression line, the model could be simplified to the following
form.

$E(R)={\int_{p_{0}}^{p_{x}}{p_{x}\over
p}(ap+b)\,dp\over\int_{p_{0}}^{p_{x}}ap+b\,dp}={ap_{x}(p_{x}-p_{0})+p_{x}b\ln\left({p_{x}\over
p_{0}}\right)\over 0.5ap_{x}^{2}-0.5ap_{0}^{2}+b(p_{x}-p_{0})}$

The resulting average return is in $1+R$ format. As the stability of the
market is removed when the average return of the investors becomes
substantially less than the market return during the bubble, The minimal price
point where market becomes unstable will be the price where market return
$R_{M}(p_{x})$ is same as the average return $E(R)$. In other words, we are
finding for $p_{\alpha}$ where

$M_{R}(p_{\alpha})={p_{\alpha}\over
p_{0}}=E(R)={ap_{\alpha}(p_{\alpha}-p_{0})+p_{\alpha}b\ln\left({p_{\alpha}\over
p_{0}}\right)\over 0.5ap_{\alpha}^{2}-0.5ap_{0}^{2}+b(p_{\alpha}-p_{0})}.$

Lastly, the average return $E(R)$ is compared with the current market return
$M_{R}$ for further evaluation.

##  3 Practical Implementation

As for any model, practical implementation is as important as its thorough
mathematical structure. Designing an implementation trial for this model is
quite difficult task. Because the model is only designed for assessing markets
in some form of the rational bubble, it is hard to determine the sample of the
market data that fits the terminology ”rational bubble”. Also, the trial
should be capable of assessing specificity as well. To satisfy all of this
requirements, testing for every possible time span of 50 trading days had been
considered to be the best option. The model’s performance is assessed
visually. Also, an implementation trial include non-traditional markets as
well, for testing the generalization capabilities of the model.

Unlike the clarity of the model’s assumptions and structure, most of the
model’s components are not set for practical usage. Therefore, some carefully
designed methodology should be used for implementation of the model. Although
the specific methodology that been used for this paper’s trial is
confidential, some of the key problems will be introduced, and the trial’s
result will be shown and discussed.

###  3.1 Data Preparation

For the stock market data, the Dow Jones Index from January 1999 to August of
2021 is been used, and for the non-traditional market data, the Bitcoin market
price data from January 2016 to August of 2021 is been used.

###  3.2 Problems with Implementation

There are some problems to solve before applying the model to the real world
data. First, the regression line between trade volume and price should be
calculated without problematic issues like assigning discrete data of volume
to the continuous interval of price. Also, as the model includes logarithm,
some solvers are incapable of solving the equation $M_{R}(p_{\alpha})=E(R)$
for $p_{\alpha}$ .

However these problems could be solved by simple practical solutions. The
method that used for this trial is confidential.

###  3.3 Implementation Results and Discussion

When applied to Dow Jones, the model showed significant prediction about 2008
financial crisis and 2020 stock market crash, which are located at around
index 2100 and 5300. In Other cases, the model come out with many
contradicting results during short time span or show weak predictions.
However, the model seems to consider the time span mid 2020 to mid 2021 as the
precursor of another market crash. This might means that the model’s
specificity is lower than expected, or just the error occurred while dealing
with wider price gap. Other explanation is that the stock market is really in
the rational bubble, but the bubble is sustained by the stimulus plan of US
government. Also, the model seems to also consider the short time span after
the market crash as the market crash itself. However this error can be easily
adjusted by human evaluator.

Figure 1: Application of the model to Dow Jones Index. Black line indicates
Dow Jones index, and blue color line over or under the black line indicates
the model’s prediction of price change direction.

As for the bitcoin market, the model successfully predicted two major crashes,
each in 2017 and 2021.

Figure 2: Application of the model to Bitcoin makert price. Black line
indicates Dow Jones index, and blue color line over or under the black line
indicates the model’s prediction of price change direction.

##  4 Conclusion

While having some implementation issues and prediction errors, the model
presented in this paper suggest the logical explanation and prediction for
irrational economic behaviors. Although its capability of clear prediction is
limited at this point, we expect to improve the model and its implementation
methodology and use the model as the potential index for understanding the
irrational nature of our economy.

## References

  * [1] Behzad T Diba and Herschel I Grossman.  The theory of rational bubbles in stock prices.  The Economic Journal, 98(392):746–754, 1988.
  * [2] Kayo Ide and Didier Sornette.  Oscillatory finite-time singularities in finance, population and rupture.  Physica A: Statistical Mechanics and its Applications, 307(1-2):63–106, 2002.
  * [3] Robert C Merton.  Optimum consumption and portfolio rules in a continuous-time model.  In Stochastic optimization models in finance, pages 621–661. Elsevier, 1975.
  * [4] Didier Sornette.  Why stock markets crash.  Princeton University Press, 2009.
  * [5] Wanfeng Yan, Ryan Woodard, and Didier Sornette.  Diagnosis and prediction of tipping points in financial markets: Crashes and rebounds.  Physics Procedia, 3(5):1641–1657, 2010.


# Coherence and stochastic resonances in a noisy van der Pol-type circadian
pacemaker model driven by light

F. L. Tsafack Tayong Fundamental Physics Laboratory, Physics of Complex System
group, Department of Physics, Faculty of Science, University of Douala, Box 24
157 Douala, Cameroon.  R. Yamapi (Corresponding author) Fundamental Physics
Laboratory, Physics of Complex System group, Department of Physics, Faculty of
Science, University of Douala, Box 24 157 Douala, Cameroon.  G. Filatrella
Department of Sciences and Technologies and INFN Gruppo collegato Salerno,
University of Sannio, Via de Sanctis, I-82100 Benevento, Italy.

(August 28, 2024)

###### Abstract

Daylight plays a major role in the wake/sleep cycle in humans. Indeed, the
wake/sleep system stems from biological systems that follow a circadian rhythm
determined by the light/dark alternation. The oscillations can be modeled by
the higher order non-linearity van der Pol -type equation driven by a term
that mimics the light cycle. In this work noise in the illumination is
introduced to investigate its effect on the human circadian cycle. It is found
that the presence of noise is detrimental for the sleep/wake rhythm, except
for some special values for which it may favor regular oscillations. Depending
for system parameters, noise induces regularities, such as stochastic
resonance: if the natural light is turned off, it emerges that there is an
optimal value of intensity noise which most deteriorates the regularity of the
cycle, it is the phenomenon of anti-coherent resonance. Also, the phenomenon
of stochastic resonance occurs: in the presence of the drive of natural light,
there is an optimal noise intensity which improves the evolution of the wake /
sleep system. However, there is a critical value of the noise beyond which the
system becomes chaotic; indeed, for sufficiently high noise levels (how high
depends upon the parameter of the system), the sleep/wake cycle evolves in a
random and unpredictable manner, for whatever parameters of the external
light.

Keywords: van der Pol oscillator; noise; stochastic resonance; circadian
oscillations.

##  I Introduction

A biological rhythm is defined as a series of statistically significant
physiological variations that appear as time oscillations of reproducible
form. Circadian rhythms, characterized by a period close to 24 hours, are
observed in almost all living organisms, bacteria, mammals, plants and insects
Lit1 ; Lit2 . Of all the organisms studied, the human organism in particular
is synchronized thanks to an internal clock located at the level of the brain,
in the supra-chiasmatic nuclei of the hypothalamus Lit33 ; these nuclei are
organs located on either side of the third ventricle just above the chiasma,
as their name suggests.

Deterministic models based on self-sustaining van der Pol oscillators such as
the Kronauer Kronauer99b  and Jewett Jewett98  models account for the
existence of circadian rhythms under constant environmental conditions. The
Jewett’s variation Jewett98  of the higher order nonlinearity van der Pol
oscillator that is highly tractable also in presence of noise Yamapi10 ;
Yang18  and periodic drive Mbakob17  that is added to mimic the external light
tsaf , as the changes in light have an influence on the internal circadian
clock. Thus, light gives the circadian clock an indication about the time of
the day.

In the biological system, noise is naturally present and essentially
unavoidable, for biological signals always act on a background of
disturbances. Noise generally describing the undesirable fluctuations and
being able to change the dynamics of a system, it is thus a principal question
to study the chaotic behavior consequent to the presence of noise. The aim of
the paper is to determine the conditions of stability in the system and
subsequently to know if noise can have a constructive influence on the
evolution of the studied system (as e.g. in the noise induced enhanced
stability Spagnolo15 ); in other words, can noise contribute to the correct
process of the sleep-wake cycle in living beings? Of course, noise in the
illumination is a phenomenon to living organisms, for it has appeared with the
appearance of artificial light.

The name of a phenomenon which is a prime example of the counter-intuitive
idea of non monotonic consequences of a random term is Stochastic Resonance
(SR) Gammaitoni98 ; Lit3 . Thus, we shall emphasize the phenomenon of
resonance (coherent or stochastic) on the wake-to-sleep cycle. Stochastic
Resonance, although a term originally used in the specific context of noisy
and driven bistable systems, is now widely applied to describe any phenomenon
where the right amount of noise in a nonlinear system improves the quality of
the response Lit4 . Stochastic resonance has been widely observed in nature;
it has been reported and quantified in systems as diverse as electronic
circuits Lit5 , neural models Lit6 , chemical reactions Lit7 , cell biology
Lit8  and even organic chemistry semiconductors Lit9  and superconducting
elements Addesso16 .

The paper is organized as follows: in section II, we briefly describe the
biological model of the van der Pol oscillator, the added external force and
the noise introduced into the system to highlight some digital tools useful
for our study; in Section III we study the chaotic behavior of the system, in
presence of random fluctuations, as a function of the parameters and of the
external excitation; in Section IV we bring out the phenomenon of coherent
resonance in the presence of noise; next we focus on stochastic resonance in
Section V for the study of resonance; finally, Section VI leads to the
conclusions.

##  II The Van der Pol-type circadian model driven by light and noise

For several years, some mathematical models of the circadian stimulator of the
human body have been made, among which we can refer to the two-process model
model1  and the model based on van der Pol’s oscillators Jewett98 . In this
Section, it will be given a complete description of the model used, a
variation of the van der Pol type oscillators to account for some realistic
features of the light drive, and some tools necessary for its numerical
solution will be highlight.

###  II.1 The van der Pol-type circadian model driven by light

The wake/sleep cycle model used is the basic model van der Pol Kronauer82 ;
Kronauer90 ; Jewett99  self-sustaining oscillator with order seven
nonlinearity described by the following differential equation:

$\ddot{x}+\mu\left(-\frac{1}{3}-4x^{2}+\frac{256}{15}x^{6}\right)\left(\frac{\pi}{12}\right)\dot{x}+\left(\frac{2\pi}{\tau_{x}}\right)^{2}x=0,$
(1)

where $x$ represents the amplitude of the circadian stimulator which is
closely linked to all the mechanisms of the human body (internal body
temperature, the desire to sleep, level of vigilance, and so forth) having a
daily period Kronauer82 , $\tau_{x}$ the period of the pacemaker which is
approximately $24.4h$ and $\mu$ the stiffness of the oscillator chose in the
interval $]0;0.45]$. Since the pulses of light applied at the right time in a
circadian cycle have a direct effect on the internal clock, one can add to Eq.
(1) the luminous flux $I(t)$ as:

$\ddot{x}+\mu\left(-\frac{1}{3}-4x^{2}+\frac{256}{15}x^{6}\right)\left(\frac{\pi}{12}\right)\dot{x}+\left(\frac{2\pi}{\tau_{x}}\right)^{2}x=I(t),$
(2)

we note that $I(t)$ consisting of a mixture of an empty function representing
total darkness during the night and a sinusoidal function representing the
evolution of sunlight during the day tsaf . A model of this type of
illumination $I(t)$ is defined as follows:

$\displaystyle I(t)$
$\displaystyle=\theta\left(D_{L}-t_{24}\right)I_{0}\sin{\left(\omega
t_{24}+\phi\right)}$ (3) $\displaystyle\omega$
$\displaystyle=\frac{\pi}{D_{L}},\quad\quad t_{24}={\textrm{m}od}(t,24);$

here $I_{0}$ is the amplitude of the light intensity, $24h$ is the cycle
period, $D_{L}$ is the duration of the daylight, $\theta$ the Heaviside
function and $\phi$ the difference of phase between light and circadian
stimulator. The model so far introduced employs arbitrary units. However, for
the sake of intuition, time is normalized to $24$ rather than to $1$.

###  II.2 The stochastic van der Pol-type circadian model

Any set of unwanted elements may be considered noise to be added to a signal
altering its characteristics. Noise can be introduced in a multiplicative or
additive way Nagumo ; moreover it can white or colored Zaikin . To make the
system more realistic, a random excitation can be thus added to the model of
the sleep-wake cycle of Eq. (7), which amounts to the following equation:

$\ddot{x}+\left(\frac{2\pi}{\tau_{x}}\right)^{2}x+\mu\left(-\frac{1}{3}-4x^{2}+\frac{256}{15}x^{6}\right)\left(\frac{\pi}{12}\right)\dot{x}=\theta\left(I(t)+\eta(t)\right)\left[I(t)+\eta(t)\right].$
(4)

$\eta(t)$ here characterizes the additive Gaussian white noise blanc ; blanc1
which is an elementary noise model used to mimic many random processes that
occur in nature. The Heaviside function is introduced to ensure that the total
illumination is positive.

Noise follows the normal laws of mean and variance of data, with the following
properties:

$\displaystyle\langle\eta(t)\rangle$ $\displaystyle=0,$ (5a)
$\displaystyle\langle\eta(t),\eta(t^{{}^{\prime}})\rangle$
$\displaystyle=2D\delta(t-t^{{}^{\prime}}),$ (5b)

where $D$ governs the magnitude of the noisy external force, $\delta$ is the
Dirac function. Several natural sources produce white noise that can be
assumed uncorrelated, one can cite thermal vibrations linked to atoms in
conductors, bodies or objects at high temperature, celestial sources and the
black body radiation of the Earth blanc2 . We here assume that artificial
light is of this type.

###  II.3 Algorithm of numerical simulations

Numerical simulations are based on the implementation of mathematical
techniques in order to solve real and complex physical problems which are most
of the time difficult to solve analytically. Thus, to generate the white noise
we are going to use the Box-Mueller algorithm fox  through two random numbers
uniformly distributed over a unit interval, that gives the following
algorithm:

$\displaystyle a,b$ $\displaystyle=\mathrm{random\,numbers,}$ (6a)
$\displaystyle\eta_{h}$ $\displaystyle=\sqrt{[-4Dh\log(a)]}\cos(2\pi b),$ (6b)
$\displaystyle x_{t+h}$ $\displaystyle=x_{t}+hu_{t},$ (6c) $\displaystyle
u_{t+h}$
$\displaystyle=u_{t}+h\left[-\mu\left(-\frac{1}{3}-4x_{t}^{2}+\frac{256}{15}x_{t}^{6}\right)\left(\frac{\pi}{12}\right)u_{t}-\left(\frac{2\pi}{\tau_{x}}\right)^{2}x_{t}\right]+h\quad\theta\left(I(t)+\frac{\eta_{h}}{h}\right)\left[I(t)+\frac{\eta_{h}}{h}\right],$
(6d)

where $h$ is the integration time step, generally very small to ensure the
stability of the system ($h=10^{-3}$ in the simulations presented here).

##  III Dynamical Analysis

Dynamical bifurcation is concerned with a family of random dynamical systems
which is differential and has the invariant measure $\mu_{\alpha}$. If there
exists a constant $\alpha_{D}$ satisfying in any neighbourhood of
$\alpha_{D}$, there exists another constant $\alpha$ and the corresponding
invariant measure $\nu_{\alpha}\neq\mu_{\alpha}$ satisfying
$\nu_{\alpha}\to\mu_{\alpha}$ as $\alpha\to\alpha_{D}$. Then, the constant
$\alpha_{D}$ is a point of dynamical bifurcation.

In this section, one finds the effects of noise in the higher nonlinearity van
der pol-type circadian pacemaker model driven by light. Dynamical bifurcation
(called D-bifurcation) is analysing using the largest Lyapunov exponent
$\lambda$ is defined as lyap1 ; lyap :

$\lambda=\lim_{t\to\infty}\frac{1}{t}\ln\left|\frac{\delta\mathbf{X}(t)}{\delta\mathbf{X}(0)}\right|=\lim_{t\to\infty}\frac{1}{t}\ln\left|D\Phi\right(\mathbf{X})|,$
(7)

where $\Phi$ is the flow of the  van der pol model  and $D\Phi(\mathbf{X})$,
the Jacobian of $\Phi$ defined at the state $\mathbf{X}$. The analysis of the
stability of the system is carried out in the presence of additive Gaussian
white noise, which is then related to the abrupt change of the sign of the
Lyapunov exponent. Figure 1 represents the phase portraits for some small
noise intensity amplitude, it is clear that the system becomes more and more
disordered as noise increases; which makes perfect sense as noise is supposed
to be harmful to physical ordered systems. We firstly investigate the
influence of the noise intensity $D$ on the small parameter $\mu$, seeking for
sudden changes of the maximum Lyapunov exponent, or D-Bifurcations. Indeed, in
Fig. 2 (i) it is shown the variation of the largest Lyapunov exponent versus
the noise intensity for $I_{0}=0$ for different values of the nonlinear
damping $\mu$ of the system. In this figure one can notice that there is a
value of the noise beyond which the system becomes chaotic. Indeed, as the
noise reaches the value $D_{c}$, the Lyapunov number becomes positive and
therefore the system becomes chaotic. This critical value depends on the
stiffness $\mu$ of the system. To be more detailed, we have drawn in Fig. 2
(ii) the map ($\mu$, $D_{c}$) representing the critical noise as a function of
the stiffness of the system. It turns out that when the value of $\mu$
increases, the interval of noise amplitude $]0;D_{c}]$ in which the system is
stable decreases; thus the increase in stiffness contributes to the a loss of
stability of the wake/sleep system in the presence of noise.

Figures 3 show the variation of the largest Lyapunov exponent in the presence
of the noise $D$ and the light intensity $I_{0}$ of the external force for
three values of the stiffness ($\mu=0.13$ i , $\mu=0.23$ ii and $\mu=0.48$
iii). The light intensity has an impact on the critical noise $D_{c}$; indeed
the increase of $I_{0}$ causes the decrease of $D_{c}$. This critical noise is
a function of the stiffness $\mu$ of the system. In fact, we have drawn in
figure 4 the map ($I_{0};D_{c}$) of the critical noise as a function of the
light intensity for three values of the stiffness. It is therefore clear that
the light intensity and the stiffness have an impact on the critical noise.

$I_{0}$ $\mu$ | 0.023 | 0.069 | 0.115 | 0.23 | 0.39 | 0.46
---|---|---|---|---|---|---
0.5 | 3.7 $\times$ $10^{-3}$ | 2.29 $\times$ $10^{-3}$ | 1.6 $\times$ $10^{-3}$ | 1.19 $\times$ $10^{-3}$ | 1.03 $\times$ $10^{-3}$ | 0.89 $\times$ $10^{-3}$
1 | 3.69 $\times$ $10^{-3}$ | 2.28 $\times$ $10^{-3}$ | 1.52 $\times$ $10^{-3}$ | 1.09 $\times$ $10^{-3}$ | 0.96 $\times$ $10^{-3}$ | 0.88 $\times$ $10^{-3}$
3 | 3.19 $\times$ $10^{-3}$ | 1.89 $\times$ $10^{-3}$ | 1.39 $\times$ $10^{-3}$ | 0.9 $\times$ $10^{-3}$ | 0.84 $\times$ $10^{-3}$ | 0.79 $\times$ $10^{-3}$
5 | 2.89 $\times$ $10^{-3}$ | 1.59 $\times$ $10^{-3}$ | 1.29 $\times$ $10^{-3}$ | 0.89 $\times$ $10^{-3}$ | 0.71 $\times$ $10^{-3}$ | 0.59 $\times$ $10^{-3}$
7 | 2.59 $\times$ $10^{-3}$ | 1.54 $\times$ $10^{-3}$ | 1.19 $\times$ $10^{-3}$ | 0.79 $\times$ $10^{-3}$ | 0.59 $\times$ $10^{-3}$ | 0.39 $\times$ $10^{-3}$
Table 1: The critical intensity $D_{c}$ of noise as a function of light
intensity $I_{0}$ and of stiffness $\mu$.

Table 1 collects the results on the critical value of the noise $D_{c}$ as a
function of the noise intensity $I_{0}$ and $\mu$.

Evolution of the largest Lyapunov exponent as a function of the noise for
different values of the daylight duration $D_{L}$ (i) of the external light
and of the phase difference $\phi$ (ii). The duration of the daylight and the
phase have no relevant impact on the chaotic state in the wake/sleep system.
Thus the only parameters which affect the chaotic state of our system are the
stiffness and the intensity of the external stimulus. Indeed we know that the
condition to maintain the self-sustaining oscillations in a van der Pol type
oscillator is that the dissipation coefficient (here, the stiffness $\mu$)
must be relatively small tsaf ; which is in agreement with the results
obtained since the increase in stiffness decreases the interval $]0;D_{c}]$
system stability.

The critical value of the noise $D_{c}$ indicates the maximum of the noise for
which the cycle is stable; Beyond this value the disorder intervenes in the
system and it becomes impossible to predict not only the amplitude of the
sleep (the desire to sleep) but also its period. Since sleep plays a big role
in the development of the human body and many health problems are linked to a
bad wake sleep cycle lumiere , noise could be one of the reasons for the human
body to malfunction. Quantitatively, this occurs that the critical
illumination noise correlation $D_{c}$ is pretty constant for a wide interval
of the illumination intensity $I_{0}$, and weakly decreases above a certain
threshold, $I_{0}\simeq 3$ for these parameters.

##  IV Coherence resonance

In this Section it is highlighted the phenomenon of coherent resonance in the
sleep awakening system through numerical investigations. Indeed, coherence
resonance Nagumo ; cham  entails that the stochastic system recovers some sort
of regularity of the trajectories in spite of the fact that noise is
increased. In other words, one finds that the appropriated noise added to the
wake-sleep cycle can improve the system behavior. How to measure these
improvements calls for some measures of the coherence, as will be discussed
below in Sect. IV.1 prior to the results on the coherence discussed in Sect.
IV.2.

###  IV.1 Tools to quantify coherent resonance

In order to study the effects of coherent resonance, the system is subjected
to the sole influence of noise, Eq. (4) with $I_{0}=0$, that is to say in
absence of excitatory signal. To characterize the coherent resonance effect,
it is necessary to introduce a measuring tool such as the normalized
autocorrelation $C(\tau)$ Ctau1  defined by:

$C(\tau)=\dfrac{\langle\tilde{x}(t)\tilde{x}(t-\tau)\rangle}{\langle\tilde{x}(t)^{2}\rangle},\qquad
with\qquad\tilde{x}(t)=x(t)-\langle x(t)\rangle.$ (8)

Autocorrelation is calculated on the system variable $x$ and the variable
$\tilde{x}(t)$ corresponds to the oscillations of the variable $x(t)$ around
its mean value $\langle x(t)\rangle$. Coherent resonance occurs when
autocorrelation senses the most the noisy evolution. In other words, one
speaks of coherent (anti-coherent) resonance when the amplitude of the
autocorrelation decreases the least (more) quickly. Coherent resonance can
also be detected through the correlation time cham ; Ctau1 :

$\tau_{cor}=\frac{1}{Var(x(t))}\int_{0}^{+\infty}C(\tau)d\tau,\qquad
with\qquad Var(x(t))=\langle x(t)^{2}\rangle-\langle x(t)\rangle^{2}.$ (9)

if this quantity exhibits a resonance for a particular value of the noise.

###  IV.2 Results on coherence resonance

Figures 6 show, for the nonlinear damping $\mu=0.23$, the autocorrelation for
three values of the noise: $8.5$ $\times$$10^{-5}$ (i), $9.9$
$\times$$10^{-5}$ (ii) and $15.5$ $\times$$10^{-5}$ (iii). The autocorrelation
amplitude decreases the fastest in Fig. 4 (ii), _i.e._ for the intermediate
value of noise $9.9$ $\times$$10^{-5}$. This determines that there is a value
of the noise, close to $9.9$ $\times$$10^{-5}$, for which the noise depresses
the most the autocorrelation of the wake-sleep system. To support this result,
we have plotted the evolution of the correlation time $\tau_{cor}$ for some
values of $\mu$ as a function of the intensity of the noise in Fig. 7. The
following observations emerge:

  * •

Assuming that the optimal desired effect of noise is to enhance the coherent
resonance, or to achieve the highest correlation time, it is clear that the
lowest noise level, ideally approaching zero, ($D=0$) would be the most
desirable. In fact, any increase of the noise initially reduces the
correlation time, which deteriorates the coherence in the temporal evolution
of the system. The interpretation in physiological consequences is not so
obvious, it might be beneficial to shorten the correlation time, and thus to
not “propagate” the disturbances in the subsequent cycles. In any case what we
observe is that the system becomes more and more incoherent in the presence of
noise cham ; mbac , which creates the deterioration not only of the amplitude
of the system (the desire to sleep) but also of the duration of sleep which
can subsequently lead to sleep-related illnesses;

  * •

There is an intensity of the noise for which the system is the least coherent
corresponding to the correlation time is the lowest (Fig 7). In other words,
for an exact value of noise, the regularity of the response of the wake-sleep
system is most degraded; indeed it is clear that in the system an anti-
coherent resonance phenomenon has occurred, inasmuch there is an optimal noise
above which noise degrades the regularity of the cycle.

  * •

Finally we can see in this figure that the stiffness $\mu$ of our system have
a considerable impact on the phenomenon of coherent resonance. Indeed,
whatever the value of the stiffness, the noise reflecting the anti-coherent
resonance remains the same. Indeed, when $\mu$ increases, the value of the
noise reflecting the anti-coherent resonance decreases. Table 2 gives us more
information.

Stiffness $\mu$ | 0.023 | 0.13 | 0.23 | 0.39 | 0.46
---|---|---|---|---|---
Noise D |  $19.9$ $\times$$10^{-5}$ |  $11.9$ $\times$ $10^{-5}$ |  $9.9$ $\times$$10^{-5}$ |  $9.8$ $\times$$10^{-5}$ |  $9.4$ $\times$$10^{-5}$
Table 2:  Evolution of anti-coherence noise $D$ as function of the stiffness
$\mu$.

Contrary to the phenomenon of coherent resonance in the system of FithzHugh-
Nagumo Nagumo  where there is a value of the noise which improves the
regularity of the system, there emerges with regard to the wake-sleep cycle an
optimal value of the noise which deteriorates the regularity of the cycle. In
other words, noise contributes to the degradation of the evolution of the
wake/sleep system in humans in the absence of daylight, as measured by the
correlation time (9). This result is in agreement with those presented on the
van der Pol oscillators with limit multi-cycles cham ; the study of coherent
resonance has been done on birhythmic systems and it has been shown that there
is a lower value of noise which gives the best intensity necessary to obtain
(anti) coherence resonance.

##  V Stochastic resonance

Stochastic resonance has been proposed Gammaitoni98  as the peculiar effect
for which noise – in the appropriate proportion, enhance the response of a
nonlinear system to a periodic drive. The concept of stochastic resonance has
subsequently been expanded to describe any phenomenon where the presence of
the right amount of noise in a nonlinear system is beneficial – in a wide
sense – for the system response. The purpose of this Section is therefore
first to briefly introduce some tools that quantify the “benefits”, and then
to apply the concepts to the wake-sleep system to ascertain whether noise can
have some positive influence.

###  V.1 Tools to quantify stochastic resonance

The word resonance in the term stochastic resonance was originally used
because the change in output Signal-to-Noise Ratio (SNR) over noise intensity
has a maximum (or minimum). The SNR is introduced to quantify the contribution
of the signal with respect to the noise; to do so one first defines the Root
Mean Square (RMS) cham ; Rms  by:

$RMS=\int x(t)x(t-t^{\prime})dt^{\prime},$ (10)

which allows to determine the SNR cham ; Rms1

$SNR=\dfrac{RMS(E\neq 0,D\neq 0)}{RMS(D\neq 0)},$ (11)

as the ratio of the RMS of noise and signal divided by the RMS in the presence
of noise only. This method was used in ref.cham  on Van Der Pol which was
inspired by the different mathematical methods of the book Rms . When
interpreted as an analysis of the sleep-awake cycle, one could surmise that it
is a measure of the regularity of the sleep cycle Mendonca19 . The noise value
that maximizes the SNR is the one that improves the signal and creates
resonance. The simulations that follow aim to ascertain if noise can play such
a constructive role.

###  V.2 Results of stochastic resonance

Figure 8 displays the evolution of the SNR as a function of the noise
intensity $D$ for several values of the light intensity $I_{0}$. There is a
value of the noise, $D=3.4$ $\times$$10^{-5}$ for which the SNR is maximum,
which highlights the phenomenon of stochastic resonance. When the light
intensity $I_{0}$ is increased, the curve is shifted upwards. Thus it is clear
that the increase or decrease in the intensity of light applied to the system
does not change the value of the noise which favors the wake-sleep cycle
evolution in humans. To confirm this hypothesis, we plot in Fig. 9 the SNR for
different values of the duration of the light $D_{L}$ (i) and of the phase
$\phi$ (ii). It appears that the duration and the phase of the light do not
change the value of the noise for which there is resonance, namely $D=3.4$
$\times$$10^{-5}$ .
Therefore it appears that the parameters of the external light do not change
the resonance peak which is in agreement with the results obtained on the van
der Pol type oscillators mbac  where the noise value corresponding to the peak
resonance does not change when the correlation time is changed. Nevertheless
let us note that the eigen parameters of the system can considerably change
this resonance peak with regard to the bi-rhythmic oscillators cham .

##  VI Conclusions

Sleep is essential and helps maintain a healthy body. It is during sleep that
the regeneration of the immune, nervous, skelatal and muscular systemtakes
place lumiere , restoring the reserve of adenosine triphosphate lit02  (ATP:
molecule that acts as an energy current in the cell), and so forth; sleep must
therefore follow a well-defined cycle. The wake/sleep cycle studied here is
modeled by a rhythmic van der Pol oscillator having higher order nonlinearity
in the presence of an external excitation shaped like natural light. As noise
is inevitable in biological models, and illumination noise is nowadays a
further source of disturbance, it is interesting to investigate the effects of
an additive noise blanc  on the circadian pacemaker. The aim is to find out in
which case noise might or might not promote the regularity of to sleep in this
model.

To conduct this type of study, it is necessary to highlight several phenomena,
namely: the bifurcation for the study of the presence of chaos, coherent and
stochastic resonance for the beneficial effects of noise, both in the absence
and in the presence of the external stimulus of light. From the simulations it
is possible to draw three main conclusions.

First, there is a critical noise $D_{c}$ beyond which the system becomes
chaotic; so above this noise level an utter disorder occurs in the wake/sleep
cycle and it becomes impossible to predict the amplitude and period of sleep.
This critical noise varies according to the stiffness $\mu$ of the system and
the light intensity $I_{0}$ but remains constant for the other parameters
($D_{L}$, $\phi$) of the external stimulus.

The measurement of the correlation rate in the absence of the external
stimulus makes it possible to observe that there is an appropriate dose of the
noise for which the regularity of the response of the wake-sleep system is the
most degraded; hence the demonstration of the phenomenon of anti-coherent
resonance. Indeed, there exists a value of the noise $D$ depending of the
value of stiffness $\mu$ for which the regularity of the cycle is rather
degraded. Table 2 gives us more information.
Finally, about stochastic resonance, the evaluation of the SNR allows to
determine that there is a value of the noise, $D=3.4$ $\times$ $10^{-5}$,
which favors the evolution of our system. This implies that noise can be
beneficial for the wake-sleep cycle model in the presence of natural light. In
fact, this noise value does not vary according to the stiffness and the
parameters of natural light.

In short, in this van der Pol model noise can cause a dysfunction in the
wake/sleep cycle in humans by making the cycle completely unpredictable
(amplitude and period of the cycle), which can lead to certain sleep-related
diseases over time. Nevertheless there is an optimal noise which can improve
the evolution in the presence of the illumination stimulus. We hope that this
work might prove of importance in biology, for it allows us to better
understand the behavior of circadian rhythms and gives results that can be
exploited to model the sleep/awake cycle in different illumination conditions,
including the effects of artificial light oscillations.

## Conflict of Interest

The authors declare that they have no conflict of interest.

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Figure 1: Phase portrait in a noisy van der pol-type circadian pacemaker
driven by light for $D=10^{-6}$ (i), $D=5$ $\times$$10^{-5}$ (ii) and
$D=10^{-4}$ (iii). ( $I_{0}=0$, $\mu=0.23$ ) . The other parameter is:
$\tau_{x}=24.2$.

Figure 2: (i) Effects of the stiffness $\mu$ on the variation of the largest
Lyapunov exponent versus the intensity of noise $D$ in the absence of the
light intensity $I_{0}=0$. (ii) Critical noise boundary in the ($\mu$,$D_{c}$)
plane. The other parameter is: $\tau_{x}=24.2$.

Figure 3: Effects of the noise intensity $I_{0}$ on the variation of the
largest Lyapunov exponent versus the noise intensity $D$ for several different
values of the stiffness coefficient; (i): $\mu=0.13$;$(ii):\mu=0.23$; and
(iii):$\mu=0.48$. The other parameters are : $D_{L}=12h$, $\phi=0$,
$\tau_{x}=24.2$. Figure 4: Effects of the stiffness on the critical noise
boundary in the $(I_{0},D_{c})$ plane. The other parameters are : $D_{L}=12h$
,$\phi=0$ and $\tau_{x}=24.2$ .

Figure 5: (i) Effects of the duration of the daylight $D_{L}$ on the variation
of the largest Lyapunov exponent versus the noise intensity $D$ with $\phi=0$.
(ii) Effects of $\phi$ on the variation of the largest Lyapunov exponent
versus the noise intensity $D$. The other parameters used are : $I_{0}=2.5$,
$\tau_{x}=24.2$ .

Figure 6: Effects of the noise intensity $D$ of the variation of the
autocorrelation function $C$ versus $\tau$; (i): $D=8.5$ $\times$$10^{-5}$;
(ii): $D=9.9$ $\times$$10^{-5}$ and (iii): $D=15.5$ $\times$$10^{-5}$. The
other parameters used are: $\mu=0.23$, $I_{0}=0$, $\tau_{x}=24.2$.  Figure 7:
Effects of the stiffness $\mu$ on the variation of the correlation time
$\tau_{cor}$ versus the noise intensity $D$. The other parameters used are
defined in Fig. 6.

Figure 8: (i) Effects of the stiffness $\mu$ on the variation of the signal to
noise ratio $SNR$ versus the noise intensity $D$. (ii) Effects of the light
intensity $I_{0}$ on the variation of the signal to noise ratio $SNR$ versus
the noise intensity $D$. The other parameters used are: $\tau_{x}=24.2$
,$D_{L}=12h$ and $\phi=0$.

Figure 9: (i) Effects of the daylight duration $D_{L}$ on the variation of the
signal to noise ratio SNR versus the noise intensity $D$ for
$I_{0}=2.5$,$\phi=0$). (ii) Effects of $\phi$ on the variation of the signal
to noise ratio SNR versus the noise intensity $D$ for $I_{0}=2.5$,
$D_{L}=12h$. The other parameters used are defined in Fig. 1.