# Deep Chaos Synchronization

Majid Mobini, Georges Kaddoum This work is supported by the Tier2 Canada
research chair entitled ‘Towards a Novel and Intelligent Framework for the
Next generations of IoT Networks’.M. Mobini is with Department of Electrical,
Electronics and Communica- tion Engineering, Babol Noshirvani University of
Technology, Babol, Iran. (email: Mobini2002@gmail.com).G. Kaddoum is with
D´epartement de g´enie ´electrique, University of Qu´ebec, ´Ecole de
technologie sup´erieure, Montr´eal (QC), Canada.(email:
georges.kaddoum@etsmlt.ca).

###### Abstract

_Abstract-_ In this study, we address the problem of chaotic synchronization
over a noisy channel by introducing a novel Deep Chaos Synchronization (DCS)
system using a Convolutional Neural Network (CNN). Conventional Deep Learning
(DL) based communication strategies are extremely powerful but training on
large data sets is usually a difficult and time-consuming procedure. To tackle
this challenge, DCS does not require prior information or large data sets. In
addition, we provide a novel Recurrent Neural Network (RNN)-based chaotic
synchronization system for comparative analysis. The results show that the
proposed DCS architecture is competitive with RNN-based synchronization in
terms of robustness against noise, convergence, and training. Hence, with
these features, the DCS scheme will open the door for a new class of modulator
schemes and meet the robustness against noise, convergence, and training
requirements of the Ultra Reliable Low Latency Communications (URLLC) and
Industrial Internet of Things (IIoT).

###### Index Terms:

Deep learning, Chaotic synchronization, DCS, CNN, Lorenz system, RNN.

##  I INTRODUCTION

Synchronization is a fundamental requirement for a wide range of natural
phenomena and new industrial technologies [1]. This paper focuses on designing
a new family of chaos-based receivers that benefit from the advantages of Deep
Learning (DL) techniques. Historically, since the early 1980s, chaos-based
communication started with the implementation of electronic circuits
exhibiting chaotic behavior [2]. The second fundamental step towards chaos-
based communication systems was in 1990. Pecora and Carroll discovered that
synchronization can be achieved by coupling two chaotic systems with a common
signal [3]. After that, several innovative systems have been proposed
employing other outstanding features of chaotic waveforms. These waveforms are
wide band, nonlinear, noise-like, non-periodic, and hypersensitive to initial
conditions [4]. They also have excellent correlation properties and a simple
production process. Thus, chaotic waveforms could be used in a vast spectrum
of applications. They are major candidates for multi-user spread-spectrum
schemes because of their wideband characteristics [5]- [8]. In addition to the
spread-spectrum applications, numerous chaos based modulations have been
proposed for use in digital wireless communications because of their
robustness against the destructive effects of jamming and fading channels [9].
Moreover, many studies have been done on Low Probability of Interception (LPI)
features [10] and chaos-based secure communication schemes [11].

In [12], the authors classified secure chaos-based techniques into four
generations. The first three generations were based on continuous chaotic
synchronization in which the bandwidth efficiency was very low. The fourth
generation, called impulsive methods, employed impulsive chaotic
synchronization to solve the low bandwidth efficiency drawback. The main
design challenge of chaos-based secure schemes is how to deliver a secret
message over a public channel using a transceiver structure that shows strong
noise rejection capabilities. Several attempts made to design robust digital
and analog chaos-based secure communication systems [13], [14]. In general,
traditional studies of chaos synchronization rely on the fact that the
equations of chaotic systems are known beforehand. This assumption is not
realistic for practical chaotic systems in which only one or some
observational signals are available [15].

Many recent studies have demonstrated the capability of ML-based approaches
for modeling chaotic systems whose equations of motion are unknown. A special
approach known as “reservoir computing” caused considerable progress in model-
free prediction of chaotic systems [16], [17]. It is currently an active
research area that uses natural potentials of Recurrent Neural Networks
(RNNs). Reservoir computing has been widely used in the modeling of temporal
processes. Also, reservoir computing has employed successfully in several
practical applications, e.g. speech recognition, handwriting recognition,
robot motor control, and financial forecasting [18]. More importantly, by
transmitting just one scalar signal, synchronization can be obtained between
well-trained reservoir computers. This technique can also extract Lyapunov
exponents and captures the dynamics of a chaotic system [19].

On the other hand, labeling is a difficult and time-consuming process in some
practical scenarios, such as chaos-based communications. To tackle this
challenge, in our paper we proposed a self-supervised (untrained) structure
for a chaos-based receiver. This new design is based on the DIP (Deep Image
Prior [20]) approach to achieve a self-supervised configuration. To this aim,
we employ a modern version of Generative Adversarial Networks (GANs), known as
Deep Convolutional Generative Adversarial Networks (DCGANs), which have a
feed-forward structure [21]. Such a design benefits from advantages of
conventional DCGANs and self-supervised nature of the DIP approach. Because of
these features, the proposed design leads to a high-performance self-
supervised structure that relies less on the input data.

The results of this paper can be extended to a wide range of applications,
from health monitoring, and chaos-based security, to human behavior analysis
and pattern studies in dynamic social networks [22]- [25]. The proposed model
is flexible for use in a variety of applications with different requirements.
It can be used to achieve real-time synchronization in applications that are
vulnerable to timing and delay such as control of heart fibrillation. Chaotic
conditions of the human heart cause arrhythmia and it can lead to death if
left uncontrolled [26]- [28]. To retain a normal heart rhythm, accurate timing
can be obtained using the proposed synchronization scheme. On the other hand,
there is a lot of evidence that chaotic synchronization provides valuable
information and a deeper understanding of physiological mechanisms underlying
the brain [29]. When the timing is not the main challenge, de-noising features
of the proposed approach can be exploited. Thus, it can be used for noise
reduction in practical chaotic signals such as Electroencephalography (EEG),
Electromyography (EMG), and other chaotic physiological signals [30]- [32].

###  I-A Basic Definitions

Synchronisation gives a conception of high correlations between connected
systems. In its elementary definition, synchronisation points out to the
tendency to have the same dynamical behaviour. In the context of
communications, a local copy of chaotic signals can be recreated in the
receiver side using appropriate synchronization circuitry. This is the most
popular approach in coherent receivers, for recovering the original chaotic
samples from the received noisy signals [33]. These recovered samples enters a
correlator as a reference signal for data detection. The Bit Error Rate (BER)
performance of these receivers depends on the “closeness” of the reference to
the original chaotic samples. Using these receivers have some significant
advantages over non-coherent receivers in terms of data rate, noise
performance, and bandwidth efﬁciency. However, all the above advantages can be
attained provided that the synchronization is preserved, Otherwise, all these
advantages will be lost [34].

To gain a robust chaotic synchronization organization and excellent noise
performance, several synchronization policies and various designs for the
receivers are reported in the literature. Essential concepts of chaotic
synchronization are explained in [3] by finding some effective applications to
secure chaos-based communications. In a coupled Lorenz system as a remarkable
synchronization scheme [35], [36], two identical chaotic generators are
synchronized using a shared drive signal. This type of synchronization is
stable and robust to perturbations in the drive signal [37], [38].

In this paper, chaotic map selection and other important design decisions are
based on the intended applications. These applications are associated with
several important issues such as latency, power consumption, and security
[39]. The Lorenz map, as a high dimensional chaotic map, has a more
complicated mathematical structure and high chaotic complexity. One-
dimensional chaotic maps are more suitable for applications with low latency
requirements because they need fewer computational operations. On the other
hand, simpler chaotic maps suffer from security limitations. This drawback is
due to the limited chaotic range, low chaotic complexity, and higher dynamic
behavior degradation rate [40]. In a chaotic masking communication scheme, the
robustness to channel noise is a determining measure [41]. In this case, the
original message is added to a chaotic signal. At the receiver side, the
chaotic signal is subtracted from the received signal to retrieve the original
data. In order to mask the original message, this signal should be generally
much weaker than the state variable of the transmitter’s chaotic system, and
thus the message signal is vulnerable to channel noise. In this situation with
high-security requirements, the Lorenz map can be a good choice. In [42] the
authors have compared the Lorenz system with a Rössler system and it is shown
that in the Lorenz case, even for rather strong noise, the complete
synchronization is preserved.

In this study, we assume that the received signal is corrupted by Additive
White Gaussian Noise (AWGN). This assumption has the following advantages:

1) Tractability: Noise effects are very important in chaotic synchronization
because of the sensitivity of chaotic systems. In fact, a small noise may lead
to instability and synchronization error [43]. This assumption assists in
detailed problem tracking by avoiding calculation complexity.

2) Necessity: Noise must be considered because it exists always before any
other effects of the communication channel. Other destructive effects could be
modeled as serial filter blocks after the noise block [44].

3) Generality: The relative performance of different communication schemes
determined using the AWGN channel model remains valid under real channel
models, i.e., systems under fading conditions generally follow the same trend
as in the AWGN channels [44], [45], and [46].

###  I-B Background and Motivations

To cope with the destructive effects of the noise and distortion, several
noise reduction methods have been recommended for chaotic signals, such as the
local projection approach and lifting wavelet transform [47], [48]. This issue
is even more challenging when the chaotic parameters or initial conditions are
unknown. In this case, there are efforts to approximate the noise-free
trajectory by estimation of the parameters and initial conditions [49]- [51].
However, in most of the reported works, the chaotic signal cannot be recovered
precisely. On the other hand, since noise and chaos have similar spectral
behaviour, regular frequency domain filtering degrades the signal of interest
[52].

Recently, DL-based techniques have attracted great attention because of
effectiveness in data-driven analysis of nonlinear dynamics [53]-[56]. Among
these, Recurrent Neural Network (RNN) is frequently used to learn complex
dynamics from the input data [57]-[60]. Unfortunately, classical RNN methods
suffer from vanishing gradients and tendency to take into account only short-
term dependencies [61]. In [62], the authors compared the prediction
performance of the RNN and Long-Short Term Memory (LSTM) in the presence of
noise. They showed that when LSTM was trained on noisy data, it reduced the
noise contribution in the prediction process. This feature can be used for
noise reduction goals. However, when LSTM was trained on clean data it became
susceptible to perturbations in the input data.

In a surprising new article [20], Deep Image Prior (DIP) method is suggested
for image de-noising and inpainting using DCGANs [21]. In DIP, a deep
convolutional neural network generator such as DCGAN is initialized with some
random weights and these weights should be optimized in such a way that force
the network to generate an image that resembles the target image as much as
possible. This procedure does not use any prior information, or training on
large data sets. Motivated by the potentials of the DIP method [20], this
article presents some approaches to take advantages of the DL in order to
promote the capabilities of the existing chaos-based communication systems.

###  I-C Contributions

This paper is the first to introduce DL-based chaotic synchronization along
with some approaches to improve trainable and untrained systems. In chaotic
synchronization and other communication systems, large data sets and labeled
data are not available. In addition, for a reasonable performance in regular
DNN-based structures, learning and testing steps must be performed for the
same channel realization results in a lot of overhead due to learning [63].
Innovative aspects of this paper are briefly itemized below:

$\bullet$ As the main achievement, we introduce a Deep Chaos Synchronization
system (DCS) that needs no data set. The proposed DCS benefits from the
inherent security of the chaotic signals and advantages of DNNs. Moreover, it
can offer a low synchronization error, especially for longer sequences. The
above-mentioned features make DCS an appealing candidate for use in chaos-
based Code Division Multiple Access (CDMA) systems [64], encrypted data
transmission [65], [66], Ultra-Reliable Low Latency Communications (URLLC)
[67], Industrial Internet of Things (IIoT) [67], and Wireless Sensor Networks
(WSNs) [68], [69].

$\bullet$ In addition, we suggest a trainable RNN-based synchronization
structure gains de-noising properties of the sequence to sequence RNN, as a
special arrangement of the RNNs family for comparison with our proposed DCS
approach. We accepted the traditional RNN as a well-known trainable approach
since it can give a measure for the de-noising strength of the proposed
method. Depending on the intended application, each has its own benefits and
can be used.

In order to initial condition estimation, we offer a Genetic Algorithm
(GA)-based approach, which gives a suitable reconstruction of the transmitted
signals. A correct estimation is vital in chaotic synchronization because of
their sensitivity to the initial condition. With accurate initial conditions,
the slave system can be synchronized with the master system almost
instantaneously, thereby eliminating the problem of transients. This technique
can still keep synchronization where the largest conditional Lyapunov exponent
is positive and other strategies cannot be synchronized [35].

In order to select a proper map among various chaotic maps, we tested several
chaotic maps as the DCS system input. We found a tractable trade-off between
processing time and noise robustness. In other words, we show that using the
Rössler map [40] and discrete-time Henon map [72], the processing time
reduced. The average amplitude error between the recovered drive signal and
the original signal shows that the Lorenz map is superior to other maps from a
noise robustness point of view.

The rest of this paper is organized as follows: In Section II, the basics of
the Lorenz map and some other chaotic maps are investigated. Also, the
employed initial condition estimation method is described in this Section.
Moreover, some basic information about DCGANs, the structure of the proposed
transmitter/receiver, and other information related to the DCS system is
presented in Section II. In Section III, the RNN-based synchronization system
is introduced for comparison with our DCS method. In Section IV, simulation
results and discussions are presented. The conclusions are explained in
Section V.

Figure 1: DCS model based on DCGAN as the generative model.

##  II DCS MODEL

In this section, the basics of the Lorenz chaotic synchronization system are
revisited and then the offered Deep Chaos Synchronization model is presented.

###  II-A Conventional Lorenz System

Consider the following conventional Lorenz system of Equations [1]:

$\centering\begin{split}\dot{x}=\rho(y-x),\\\ \dot{y}=rx-y-xz,\\\
\dot{z}=xy-\beta z,\\\ \end{split}\@add@centering$ (1)

where $x$, $y$, and $z$ are the state variables at the drive (Master)
subsystem and $\rho$, $r$, and $\beta$ are control parameters. When these
parameters are chosen as $\rho=10$, $r=28$, and $\beta=8/3$ the equations lead
to chaotic dynamics while the initial conditions shape the chaotic behavior
[3]. At the receiver side, the received noisy signal can be written as:

$\centering\mu(t)=x(t)+\zeta(t),\@add@centering$ (2)

where $\mu(t)$ is a given noisy observation, $\zeta(t)$ is Additive White
Gaussian Noise with zero-mean and power spectral density $\sigma_{n}^{2}$. The
response (Slave) uses the $\mu(t)$ as driver, in order to reconstruct Lorenz
state variables $y_{r}$ and $z_{r}$:

$\centering\begin{split}\dot{y}_{r}=r\mu-y_{r}-\mu z_{r},\\\ \dot{z}_{r}=\mu
y_{r}-\beta z_{r}.\\\ \end{split}\@add@centering$ (3)

We presume that all control parameters $\rho$, $r$, and $\beta$ of the master
sub-system are known at the receiver side. The initial values of the master
subsystem are $(x_{0},y_{0},z_{0})$ and the slave subsystem is initiated
similarly, except that we do not know one of the initial values $(x_{0})$. In
a Lorenz system starting from two slightly different initial states
$(x_{0},y_{0},z_{0})$ and $(x_{r}{}_{0},y_{r}{}_{0},z_{r}{}_{0})$, the
trajectories diverge from each other. At the end of this section, a GA-based
solution to this challenge is presented.

###  II-B Comparison with Other Chaotic Maps

In addition to the Lorenz system, there are many other chaotic systems that
generate chaos and can be utilized in communication systems. For example,
Rössler systems can be decomposed into stable drive-response sub-systems to
produce a robust synchronization [40]. Moreover, several discrete-time maps,
such as the Henon map, are also decomposable into synchronizing sub-systems
without the burden of solving differential equations [72]. In Section IV, we
discuss the merits and demerits of using these systems. To investigate the
effect of different maps on de-noising performance of the DCS system, we first
adopt a signal from the Rössler system as input [40]:

$\centering\begin{split}\dot{x}=\omega y-z,\\\ \dot{y}=\omega x+0.15y,\\\
\dot{z}=0.4+z(x-8.5),\\\ \end{split}\@add@centering$ (4)

with $\omega=0.95$, and another chaotic signal from discrete-time Henon system
[72]:

$\centering\begin{split}x[n+1]=1+y[n]-z[n]x^{2}[n],\\\ y[n+1]=bx[n],\\\
z[n+1]=z[n]-0.5+\beta x^{2}[n],\\\ \end{split}\@add@centering$ (5)

with $b=0.25$, and $\beta=0.279$.

###  II-C Proposed DCGAN-based Receiver

In this study, motivated by the appealing factors of the DIP model [20], the
one-dimensional arrangement of DIP is implemented using a DCGAN. In
particular, we adapt the input and output of the DIP model and use it as a
baseline which we term the Deep Chaos Synchronization. The receiver of the DCS
is composed of three stages, as shown in Fig. 1. The implementation steps of
the DCS are illustrated in Algorithm 1.

In the first stage, a less noisy signal is achieved from the DIP method. When
the objective is to generate data identical to a target distribution, a
Generative Adversarial Network (GAN) provides data from a similar distribution
efficiently [70]. However, GANs are unstable to train and this regularly
generates nonsensical outputs.

DCGAN is one of the attempts to scale up GANs using CNNs. Compared to
conventional GANs, spatial pooling functions are replaced with convolutional
layers, allowing the DCGAN to learn its own spatial downsampling. Also, DCGAN
adopts Batch Normalization (BN) which stabilizes learning by normalizing the
input to each unit to have zero mean and unit variance. Moreover, ReLU
activation functions are considered for all layers. The RMSProp optimizer is
used as a variant of the traditional gradient descent method that adopts a
momentum parameter for faster convergence of the training process.

In the second stage, we offer a GA-based approach in order to estimate the
initial condition using the achieved clear signal from the previous stage. An
accurate initial state estimation results in a proper reconstruction of the
original signal.

In the final stage, the estimated initial value enters the response sub-system
in order to reconstruct Lorenz attractors. By doing the above mentioned three
stages the inherent advantages of a chaotic system are combined with the noise
reduction capability of the DIP so as to have remarkable performance. However,
the complexity increases in comparison with the conventional Lorenz
synchronization system, but compared to RNN-based designs, DCS needs no
labeled data or large data set. Moreover, no need to send train signals result
in a lot of overhead reduction.

As shown in Fig. 1, the DCS receiver is implemented using a DCGAN located at
the input of the response subsystem. We aim to tackle the inverse problem of
de-noising a drive signal $x(t)\in\mathbb{R}^{N}$ which has been transmitted
on a communication channel with Gaussian noise $\zeta(t)\in\mathbb{R}^{N}$.
Therefore, given a noisy observation $\mu(t)$, the basic idea is to reproduce
a signal $\chi(t)$ that is similar to $x(t)$. To implement the DCGAN, We need
to configure and organize the three blocks: 1) the generator, 2) the
discriminator, and 3) the training process. The discriminator can be
implemented by successive convolutional layers, BN layers, and Relu activation
functions. In each of the convolutional layers, we downsample the spatial
dimension of the input. The generator of the DCGAN consists of a sequence of
transpose convolutional layers that upsample the input random sample to
generate an artificial signal. A DCGAN can be trained in the same way as a
regular GAN. Different from the conventional DCGAN models, in the DIP
approach, we optimize the ”$random~{}weights$” to force the network to produce
an output similar to the target. Therefore, G (k, w)is the output of DCGAN
with weights $w$ and latent vector $k$. In other words, instead of minimizing
$\|\chi-\mu\|^{2}$, the network optimization problem of the DCS can be
formulated as:

$\centering
w^{*}=\underset{w}{\mathrm{argmin}}\|G(k,w)-\mu\|^{2},\@add@centering$ (6)

where $\mu\in\mathbb{R}^{N}$ is the observed noisy signal and
$G(k,w^{*})=\chi$ is the output clean sequence. This non-convex optimization
problem can be solved easily using Gradient Descend (GD) method. As an
alternative optimization strategy, we can regularize the network using a two
stages optimization, which has shown converges better than simple Mean Squared
Error (MSE) optimization [71], [73]. Therefore, we first train the adversarial
network using a noisy sequence $\mu$, an RMSProp optimizer, and the loss
function in (6). Then, this problem can be solved through a simple GD
algorithm to obtain the final solution.

1: Initialize:

Observed Noisy Sequence $\mu$, Random vector $w$;

2: DCGAN pseudo-code:

2-1: Generator;

\- 1D Convolution Transpose

\- Batch Normalization

\- Relu

\- Tanh (At the end of Layers)

2-2: Discriminator (Input random vector k)

\- 1D Convolution transpose

\- Batch Normalization

\- IRelu

2-3: DCGAN training loop

\- Draw training example from the $\mu$ and $w$

\- Generate a network $G(w,k)$

\- Network training using (6)

$w=RMSprop(G(w,k),LR,MOM)$

\- Update loss

$Output=Network(w)$

3: DIP Output

\- Solving (6) using GD method

$\chi=G^{*}(w,k)$

4: Initial condition estimation $(\chi,\hat{x})$

\- GA based calculation of the $x_{0}$

5: Reconstruction of $x_{r},y_{r},z_{r}$

6: End

Algorithm 1 DCS Receiver Pseudo-code

###  II-D Initial Condition Estimation using GA

The chaotic trajectory reproduction is extremely susceptible to the initial
conditions. On the other hand, in most practical scenarios, the initial
conditions are hard to measure precisely. So, as a reasonable assumption, we
assume that we have only the upper and lower bounds for initial conditions.
After an effectively initial condition estimation, we can get a noise-free
signal, and the slave system can follow the dynamics of the master system
almost instantaneously, thereby eliminating the issue of transients. With
these advantages, initial condition estimation is one of the hot topics in
chaos-based studies. Some articles have emphasized that when alternative
synchronization techniques fail, this approach can achieve synchronization
[74-78]. For example, in [48], [75] some heuristic algorithms are employed to
initial state estimation and robustness of the chaotic synchronization to
noise.

In this paper, in order to initial condition estimation, we offer a simple GA-
based method that provides an excellent reconstruction of the transmitted
signal. The GA-based optimization is a convenient tool for non-linear problems
that are not appropriate for analytical optimization methods. In addition, GA
approach is fit for solving complex multi-variable problems, for example,
where we would like to estimate several chaotic parameters and initial
conditions. In the literature different methods have been proposed for initial
condition estimation, e.g. Differential evolution algorithm-based methods
[58], or Hybrid Mehta-heuristic methods [59]. Since the main idea behind this
study is to implement an untrained structure for chaos-based receivers and our
main focus is not on the precise parameters and initial value estimation, we
select the GA to simplify the implementation.

We focus on the simultaneous start property and the security aspects of the
initial condition estimation method [73]. The simultaneous start property is
important in the synchronization of two coupled discrete-time systems, because
it enables us to obtain real-time synchronization without transient time. The
receiver can start de-noising process after receiving the first sample. In
other words, because the receiver gains deep learning structures, it is
inherently predictive, and after receiving the first sample, it can predict
the next samples. In the synchronization model presented, if the response sub-
system knows the initial conditions of the transmitter, or estimates them
accurately, it can make a copy of the transmitted signal without the need for
additional signaling. From a security point of view, when the adversary
network has access to the wireless medium and public samples $x_{0}$, we can
secure the DCS system by sharing $y_{0}$ and $z_{0}$ with legal users as
private keys. We can use any of the transmitters outputs $x_{0},$ $y_{0}$, and
$z_{0}$ for signal transmission or security purposes. Estimating other
variables imposes time and computational load and without loss of generality,
we take a single variable case $x_{0}$ to decrease the computational
complexity.

In our setting, equal parameters $\rho$, $r$, and $\beta$ are considered for
the drive and response systems because the synchronization between them is
stable provided that the parameters are identical. Here, $(x_{0},y_{0},z_{0})$
and $(x_{r}{}_{0},y_{r}{}_{0},z_{r}{}_{0})$ are the initial states of the
drive and response systems, respectively. We have adopted the $x$ component to
operate as a drive signal and it is also assumed that the receiver does not
know the initial state of the $x$ component (I. e. $x_{0}$ ) correctly and
only knows the upper and lower bounds of it. To cope with the above-mentioned
trajectory divergence phenomenon, in [75], the authors recommended a piece-
wise estimation for each segment of the observed signal. Mathematically,
initial state estimation in such a problem reduces to the optimization of the
following objective function:

$\centering\underset{x_{0}}{\mathrm{argmin}}{[f_{o}{}_{b}{}_{j}(x_{0})=\frac{1}{T_{s}}\sum_{t=1}^{T_{s}}{(\hat{x}(t)-\chi(t))}^{2}],}\@add@centering$
(7)

where $x_{0}$ is initial condition of the $x$ component, for the Lorenz map.
The idea is to minimize the objective function $f{}_{o}{}_{b}{}_{j}(x_{0})$
given by the mean-squared error between a candidate chaotic signal solution
$\hat{x}(t)$ and the output of the above mentioned neural network $\chi(t)$.
If an initial population is determined for $(x_{0})$, the solution will be
found by an iterative approach such as GA. The initial population can be
modified based on some rules such as selection, mutation, and crossover to
achieve the optimal solution [79]. In particular, we use a decimal
representation of genes in the range [0 0.1], uniform mutation, and one-point
crossover. It is necessary to notice that here we determine the accuracy of
our estimates by the initial population definition. In other words, the larger
the initial population, the higher the accuracy of estimation, and accordingly
the reconstructed signal $\hat{x}$ will be closer to the primary signal $x$.
Iterative solving of the system of equations in (1) can give the best solution
$\hat{x}(t)$. This can be performed in Python with the help of
scipy.integrate.odeint function. Ultimately, we can reconstruct Lorenz
attractors $y_{r}$ and $z_{r}$ at the receiver side using parameters and the
estimated initial condition.

##  III TRAINABLE RNN-BASED SYNCHRONIZATION MODEL

In this section, we introduce a trainable RNN-based synchronization scheme for
comparative investigation. Fig. 2 illustrates the suggested model.

Figure 2: Trainable RNN-based synchronization System.

It is noted that since the DCS is the ﬁrst self-supervised chaotic
synchronization system, the performance comparison cannot be provided with
other self- supervised systems. Therefore, the comparison with the RNN-based
synchronization system can provide a general measure of de-noising power of
them, although, they belong to the two different training classes.

A standard RNN includes of a hidden layer $\bf h$ and an output $\bf\chi$
works on an input sequence $\bf x$= ${x(1),...,x(T)}$. The hidden state $h(t)$
is updated at each time step as follows:

$\centering\ h(t+1)=\psi(x(t),h(t)),\@add@centering$ (8)

where $\psi$ is a non-linear activation function, such as a logistic sigmoid
function. An RNN learns a probability distribution over a sequence by being
trained to predict the next symbol in a sequence. The output at each time step
$t$ is the conditional distribution $p(x(t+1)|x(t),...,x(1))$. From this
learned distribution, it can sample a different sequence by iteratively
sampling. The RNN is a remarkable tool for learning complex dynamics and, at
the same time, has a lot of noise reduction capabilities. It is also proper
for Natural Language Processing (NLP) and signal prediction problems. Among
these capabilities, we focus on the de-noising potentials of the Encoder-
Decoder sequence to sequence RNNs as a special structure of the RNNs family.
It consists of two RNNs, one of them encodes a sequence into a fixed-length
vector and the second RNN decodes the vector into another sequence [80]. In
the RNN-based signal prediction problem, the model predicts the next sequence
given a previous sequence of vectors. This will enable us to generate a new
sequence, one vector at a time. The architecture of the de-noising problem is
similar to the prediction process, except that the signal given to the input
of encoder is noisy, and the desired output of the decoder should be a clear
signal. We adopt a newer version of dynamic RNN functions, known as Gated
Recurrent Unit (GRU), which realizes the RNN for us [81]. GRU has no output
gate and incorporates the input and previous gates into an update gate. This
new gate manages how much the internal state is combined with a candidate
activation.

We select GRU because it requires less computation while provides comparable
results with LSTM. In addition, GRU can be easily implemented by GRUCell in
TensorFlow. Different GRU structures are supported by TensorFlow in the
tf.nn.rnn_cell module [82], [83]. Fig.3 illustrates the arrangement of the
stacked GRU sequence to sequence RNN.

Figure 3: The stacked GRU sequence to sequence RNN structure.

The $G$ is the value of the hidden output of the encoder in the last time
step. We should repeat this value as an input to the further RNN to make it
remember the context of the present at all moments when predicting the future.
We have now defined the RNN and should train the model using an optimizer. The
input and output neuron’s dimensions should be $m=d=1$ because the number of
input train and test sequences are one
$n{}_{t}{}_{r}{}_{a}{}_{i}{}_{n}=n{}_{t}{}_{e}{}_{s}{}_{t}=1$. We consider a
supervised framework and generate two sequences of the noisy chaotic samples
with length $T$ as $train$ and $test$ signals. In the meantime, we keep their
noise-free version (or $target$) for final evaluation. In this framework, we
train the model showing the $train$ sequence and reading a prediction from the
output neurons in the test phase. Even with a corrupted $test$ signal, this
prediction is close to the ground truth [59]. The RNN is trained to suppress
the contributions from the noisy observations and minimize the following
objective function using an ADAM optimizer [84]:

$\centering L=\sum_{t=1}^{N}\frac{1}{2}(\mu(t)-\chi(t))^{2}.\@add@centering$
(9)

Here,

$\centering\chi(t+1)=\psi_{\sigma}(\mu(t+1),G(t)),\@add@centering$ (10)

where $h(t)\in\mathbb{R}^{N}$ are the internal states, $N$ is the number of
memory units, and $\chi(t)$ is the predicted/de-noised signal. $\psi_{\sigma}$
denotes the RNN trained against noise with a standard deviation $\sigma$. In
order to reconstruct Lorenz attractors, $\chi$ enters the initial condition
estimation block. After initial state estimation, the desired attractors can
easily re-produced by the Lorenz circuit.

##  IV SIMULATION RESULTS

In this section, a number of simulations are considered for a comparative
analysis of the DCS, RNN-based synchronization, and traditional Lorenz coupled
system in (3). The considered parameters are listed in Table I.

TABLE I: List of the Parameters. $~{}Variable~{}$ | $~{}~{}~{}Description~{}~{}~{}$ | $~{}RNN~{}$ | $~{}DCS~{}$
---|---|---|---
N | Size of the RNN | 64 | -
$n{}_{t}{}_{e}{}_{s}{}_{t}$ | Number of test signals | 1 | 1
$n{}_{t}{}_{r}{}_{a}{}_{i}{}_{n}$ | Number of training signals | 1 | -
T | Length of train / test signals | 1024 | 1024
$N{}_{G}{}_{F}$ | Convolutional filter per layer | $-$ | 64
Iter | Iterations during the training | 800 | 800
Mom | Momentum of RMSProp | $-$ | 0.9
m | Output dimension | 1 | 1
d | Input dimension | 1 | 1
LR | Learning Rate | $10^{-}{}4$ | $10^{-}{}4$

Many methods have been proposed to implement chaotic trajectories in python.
We solve the Lorenz equations, using scipy.integrate.odeint function according
to the method presented in [85]. This function integrates the system of
Ordinary Differential Equations (ODE) and returns the solution. We produce
1024 data points with a time step= 0.1. It is presumed that all control
parameters of the master sub-system are known at the receiver side. To produce
chaos dynamics, the control parameters are selected as $\rho=10$, $r=28$, and
$\beta=8/3$. The initial values of the master subsystem are
$(x_{0},y_{0},z_{0})=(0.1,0.1,0.1)$ and the slave subsystem is the same,
except for one of the values $(x_{0})$, which is assumed to be unknown and
must be estimated. To this aim, we offer the GA-based initial condition
estimation method. The parameters of the GA-based approach are listed in Table
II.

TABLE II: GA Parameters.
$Variable~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}Description$
---
population size 10000
Number of variables 1
Maximum iterations 10
Crossover fraction 0.1
Mutation fraction 0.1

###  IV-A De-noising Performance

We can easily make the RNN model using NumPy and TensorFlow packages to
produce a clear signal from a noisy observation. This ability comes from that
the RNN tends to weaken the contribution of the noise and captures its own
learned dynamics. Fig. 4 shows the convergence curve of the RNN, in which the
loss is plotted versus the number of iterations during the training process.
In this simulation, the RNN is trained at $\sigma^{2}_{n}=0.3$ while is tested
at $\sigma^{2}_{n}=0.5$. If $\sigma^{2}_{n}$ is set at a big value, the RNN
might learn nothing but the noise. On the other hand, if the training
$\sigma^{2}_{n}$ is low, the RNN can only learn the clean signal. For a proper
training $\sigma^{2}_{n}$ value must help the RNN to learn both types of
samples. The proposed RNN-based system can be trained at a speciﬁc and proper
noise condition, while operating throughout the range. during the training
stage, we employ ADAM optimizer to minimize the loss function presented in
(9). One can see that the RNN converges to the best case after several
iterations. This demonstrates that the training phase not only imposes
computational complexity, but also can be time consuming for the receiver and
results in a lot of overhead.

Figure 4: Optimizing iterations during the training of RNN.

Fig. 5 (a) compares the target $(x)$ and de-noised $(\chi)$ signals. In order
to facilitate visual comparison, the first 250 samples of these signals are
plotted. The RNN model follows the target dynamics, but the synchronization is
degraded after about 200 time steps. In other words, the RNN is not able to
capture the dynamics of the drive signal and its error signal has a large
domain. Therefore, it is necessary to re-train the receiver after the above
time. Re-training process consumes time and other resources, and results in
high computational complexity. In contrast, as shown in Fig. 5 (b), the DCS
model, which is implemented using Torch and TensorFlow libraries, follows the
driving dynamics perfectly.

t

(a)

t

(b)

Figure 5: (a) Target and de-noised signal ($\chi$) by RNN (b) Target and de-
noised signal ($\chi$) by DCS.

Fig. 6 shows de-noising error, i.e., $(x(t)-\chi(t))$ of the DCS compared with
the RNN system in the same scale. We observe that, although the DCS does not
use pre-training, it follows dynamics of the target signal and the error
amplitude remains fixed over time.

Figure 6:  Comparison between de-nosing error $x(t)-\chi(t)$ of the RNN and
DCS.

Fig. 7 (a) plots the average amplitude error of the DCS for different number
of iterations. The average amplitude error converges to the best case rapidly
after about 600 iterations. Fig. 7 (b) shows the average amplitude error
between the recovered drive signal and the original signal for different
values of $\sigma_{n}^{2}$ . As illustrated in this figure, for large noise
values, when the number of iterations increases to 800, the error becomes very
small. Increase the number of iterations from 800 to 2000 has a negligible
impact on the error performance. In Fig. 7 (c), the robustness of the DCS
system is measured for different chaotic maps. Execution time of the DCS
algorithm is also measured for each of the maps. The average amplitude error
between the recovered signal and the original signal shows that even for a
rather strong noise, the Lorenz map is more robust. However, when we use the
Rössler and Henon maps presented in (4) and (5), the processing time is
reduced by $20\%$ and $25\%$, respectively. The Lorenz map has a more
complicated structure and high chaotic complexity, while the Rössler and Henon
maps are faster and more suitable for applications with low latency
requirements such as URLLC and IIoT. On the other hand, simpler chaotic maps
suffer from security limitations, and the Lorenz map is more suitable for
applications such as synchronization in extremely noisy channels and secure
communications. According to the above discussion, future systems should make
different compromises to meet their requirements.

Number of Iterations
(a)

$\sigma^{2}_{n}$

(b)

$\sigma^{2}_{n}$

(c)

Figure 7: (a) DCS convergence. (b) Effect of increasing the number of
iterations. (c) A comparison among the noise performance of the Lorenz,
Rössler, and Henon maps.

###  IV-B Synchronization Error Comparison

The following simulations deal with the output signals of the response system.
Fig. 8 show a comparison between the output of the traditional Lorenz coupled
system in (3) and the proposed DCS. Fig. 8 (a) shows the master signal $z(t)$
and reconstructed attractor $z_{r}(t)$ using the traditional system. The
parameters in the transmitter and receiver of the conventional system are the
same. We assume that in the receiver, the initial state of $(x_{0})$ is a
random number in the range [0, 0.1]. Fig. 8 (b) shows the reconstructed
attractor $z_{r}(t)$ by the DCS. The amplitude of the synchronization error is
very small because of the noise reduction and initialal state estimation in
DCS, and the synchronization is more persistent over time.

(a)

(b)

Figure 8: Master signal z(t) and reconstructed signal in (a) conventional
system (b) DCS.

Fig. 9 presents $z(t)-z_{r}(t)$ as the synchronization error criterion. To
make the visual comparison easier, error signals are plotted over a smaller
time span. In comparison with the traditional coupled Lorenz system and the
RNN-based synchronization system, the DCS model conserves the dynamics of the
master signal while it has a very small error amplitude.

Figure 9:  Comparison between synchronization error of the different systems
for $\sigma_{n}^{2}=0.5$.

Because the visual comparison may not be accurate, in another experiment we
measured the average synchronization error for the above mentioned span
$(t=250)$ and for different values of the noise power. The results are shown
in the Table III. Observe that the DCS method is always superior to the other
two methods.

TABLE III: Average synchronization error of different methods for different noise values. | RNN-based Method | Conventional Method | DCS
---|---|---|---
$\sigma_{n}^{2}=0.1$ | 0.05101 | 0.07180 | 0.02791
$\sigma_{n}^{2}=0.2$ | 0.05111 | 0.07181 | 0.02851
$\sigma_{n}^{2}=0.3$ | 0.04428 | 0.07171 | 0.02779
$\sigma_{n}^{2}=0.4$ | 0.05193 | 0.07192 | 0.02780
$\sigma_{n}^{2}=0.5$ | 0.05195 | 0.07195 | 0.02781
$\sigma_{n}^{2}=0.6$ | 0.05501 | 0.07196 | 0.02781
$\sigma_{n}^{2}=0.7$ | 0.06333 | 0.07197 | 0.02782

More exactly, consider a scenario, where a chaotic signal with 1024 samples
generated, and portions of the signal with specified lengths are randomly
selected for transmission and performance evaluation. We repeated this
experiment ten times for each determined length and calculated the average
synchronization error.

As shown in Fig. 10, the DCS system has a competitive performance with the
RNN-based system and even works better for signals with $T>250$. Furthermore,
in this figure, both proposed systems have been compared with the evolutionary
de-noising method presented in [75]. To make a fair comparison, GA-based
initial condition estimation is used for all of them, according to the values
in table 2. because of the benefits of neural networks, both of the proposed
systems are more robust than the evolutionary de-noising method that only uses
initial condition estimation, specifically when $T>250$. It is important to
note that we have deliberately selected large noise and high perturbation of
the initial values in this experiment so that the performance of the various
methods is visually distinguishable. Adjusting parameters like the initial
population and mutation rate in GA can counteract these damaging effects and
eliminate synchronization errors.

Figure 10: Comparison between synchronization error of the different systems
for $\sigma_{n}^{2}=0.5$.

##  V Conclusions

This paper presents the ﬁrst attempt of using DL in chaos-based communication
systems with providing an excellent synchronization between two coupled Lorenz
systems. The results showed that the DCS reduces the synchronization error
compared to the traditional systems and RNN-based approach. Moreover, the
proposed DCS is easy to implement since it does not require training on large
data sets. Thus, it can be used in coherent wireless communication scenarios
where users or mobile units are generally hard to train. Practical
implementations of DCS-based applications e.g., chaos-based CDMA systems and
low latency communications is an interesting topic for future researches.

## Acknowledgment

We are grateful to the Tier2 Canada research chair entitled  ‘Towardsa Novel
and Intelligent Framework for the Next generations of IoT Networks’ for
support this project.

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|  MAJID MOBINI received his M.Sc. degree in communications engineering with
the first rank from the Department of Electrical and Computer Engineering,
Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in
2013, and the Ph.D. degree in communications engineering from the Babol
Noshirvani University of Technology, in 2019. He was a Visiting Scholar with
Cyberspace Research Institute, Shahid Beheshti University, Tehran, Iran, from
2013 to 2014. He is currently a Research Associate with the Department of
Electrical and Computer Engineering, Babol Noshirvani University of
Technology. His current research interests include chaos-based wireless
Communication, optimization, artiﬁcial intelligence, Internet of Things,
cyber-physical systems, machine vision, biomedical instruments, and
interdisciplinary researches. He is a Reviewer of the IEEE ACCESS and the
Journal of Soft Computing and Information Theory.
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|  GEORGES KADDOUM received the Bachelor’s degree in electrical engineering
from the École Nationale Supérieure de Techniques Avancées (ENSTA Bretagne),
Brest, France, and the M.S. degree in telecommunications and signal
processing(circuits, systems, and signal processing) from the Université de
Bretagne Occidentale and Telecom Bretagne (ENSTB), Brest, in 2005 and the
Ph.D. degree (with honors) in signal processing and telecommunications from
the National Institute of Applied Sciences (INSA), University of Toulouse,
Toulouse, France, in 2009. He is currently an Associate Professor and Tier 2
Canada Research Chair with the École de Technologie Supérieure (ÉTS),
Université du Québec, Montréal, Canada. In 2014, he was awarded the ÉTS
Research Chair in physical-layer security for wireless networks. Since 2010,
he has been a Scientific Consultant in the field of space and wireless
telecommunications for several US and Canadian companies. He has published
over 200+ journal and conference papers and has two pending patents. His
recent research activities cover mobile communication systems, modulations,
security, and space communications and navigation. Dr. Kaddoum received the
Best Papers Awards at the 2014 IEEE International Conference on Wireless and
Mobile Computing, Networking, Communications (WIMOB), with three coauthors,
and at the 2017 IEEE International Symposium on Personal Indoor and Mobile
Radio Communications (PIMRC), with four coauthors. Moreover, he received IEEE
Transactions on Communications Exemplary Reviewer Award for the year 2015,
2017, 2019. In addition, he received the research excellence award of the
Université du Québec in the year 2018. In the year 2019, he received the
research excellence award from the ÉTS in recognition of his outstanding
research outcomes. Prof. Kaddoum is currently serving as an Associate Editor
for IEEE Transactions on Information Forensics and Security, and IEEE
Communications Letters.
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