Analysis of the Structure of Chaotic Solutions of Differential Equations
MARYNA BELOVA, VOLODYMYR DENYSENKO, SVITLANA KARTASHOVA,
VALERIJ KOTLYAR, STANISLAV MIKHAILENKO
Department of Higher and Applied Mathematics,
Faculty of Information Technologies,
State University of Trade and Economics,
19 Kyoto Str., Kyiv, 02156
UKRAINE
Abstract: - This study deals with the relevant and important area of many fields of mathematics and physics -
chaotic systems. Three modified systems of Chua differential equations were considered, and the chaotic
structure of their solutions was compared with the structure of solutions of classical Lorentz and Rössler
chaotic systems. The following methods were used to achieve the set goal: the Runge-Kutta method, building a
phase portrait, determining Lyapunov exponents and noise level, and comparative analysis. A detailed analysis
of the structure of chaotic solutions of various differential equations was carried out. It was established that the
chaotic solution's structure depends on the differential equation's properties and the initial conditions.
According to the obtained results, one of the modifications of the Chua system is significantly superior to
classical chaotic systems and can be used as a chaos generator. Prospects for further research involve
expanding the scope of the study and the generalization of the obtained results for a wider class of systems of
differential equations.
Key-Words: - chaos theory, chaotic solutions, Chua system, differential equations, Lorentz attractor, Rössler
attractor.
Received: December 28, 2022. Revised: June 11, 2023. Accepted: June 29, 2023. Published: July 27, 2023.
1 Introduction
Analysis of the structure of chaotic solutions of
various differential equations is one of the key
topics of modern Mathematics and Physics. This
research is relevant as it is important for
understanding various complex systems that are the
subject of study in many sciences, such as Physics,
Chemistry, Biology, Economics, and others.
Chaos is the non-deterministic behaviour of
systems that have a complex structures. These
systems usually consist of many interacting
elements that can produce unpredictable results.
Differential equations are a key instrument for
studying complex systems. These equations describe
the change of physical quantities over time.
Analysis of the structure of chaotic solutions of
differential equations enables obtaining new
knowledge about the behaviour of systems that
usually cannot be described by simple rules.
So, studying the structure of chaotic solutions of
various differential equations is a relevant and
important area of modern Mathematics and Physics.
It can be applied in various fields of science and
technology: weather and climate forecasting, control
of chaotic systems, cryptography, forecasting the
development of pandemics and epidemics, creation
of artificial neural networks, etc.
The aim of the research: a study of the structure
of chaos in the system of Chua's equations and its
modifications, identification of factors affecting the
formation of chaotic solutions and their features.
Research objectives:
- Select a set of differential equations of different
complexity and perform their simulation;
- Study and compare the structure of chaotic
solutions for different parameters and initial
conditions;
- Assess the impact of internal and external factors
on the structure of chaotic solutions and their
behaviour.
2 Literature Review
Henri Poincaré was the first researcher of chaos. In
the 1880s, he studied the behaviour of a system with
three gravitationally interacting bodies and found
that there could be non-periodic orbits that are
constantly neither moving away from nor
approaching a particular point. Later, many world
scientists made a great contribution to the study of
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Maryna Belova, Volodymyr Denysenko,
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chaotic structures, [1]. Many researchers focus on
this issue in their studies, given the wide range of
applications of chaotic systems. For example, the
work, [2], considered the nonlinear Schrödinger
equation with the Kerr Law Nonlinearity and
demonstrated the existence of various traveling
wave reactions. They also described the parametric
criteria for all these traveling solutions based on
physical factors. In, [3] the researchers also consider
the dynamic features of the dispersive extended
nonlinear Schrödinger equation (NLSE), where a
new method of expanding the F6−model is applied
to study solitary waves of the considered model. In,
[4] the authors obtained and studied a diverse range
of traveling wave structures in the perturbed Fokas–
Lenells model (p-FLM) using the extended (G/G2)-
expansion approach. In the paper, [5], studied the
dynamic characteristics of a meminductor and a
memcapacitor through the fractal-factional Caputo–
Fabrizio operator. The chaos scheme is modeled for
highly nonlinear and non-fractional meminductor
and meminductor governing differential equations
for studying chaos, hyperchaos, and coexisting
attractors. The use of Bernstein and Euler wavelets
was considered for solving a nonlinear fractional
biological model of two species predator–prey
model, [6]. This resulted in new chaotic models of
the population of predators and prey. Anees and
Iqtadar conducted an analysis based on the chaotic
sequences of the Lorentz system and the logistic
chaotic map, [7]. They found a serious problem of
hacking in symmetric security systems of chaotic
communications. In, [8] the authors derived
nonlinear governing equations for developing a
nonlinear dynamic model for nonlinear frequency
and chaotic responses of a doubly curved composite
panel reinforced with graphene nanoplates using
Hamilton’s principle and nonlinear von Kármán
theory. In the study, [9], the researchers carry out
simulations of several chaotic systems such as
multi-spin attractors, self-excited and hidden
attractors, period-doubling to chaos, periodic and
chaotic explosive oscillations, and various multiple
coexisting attractors using a new Atangana–Baleanu
time-fractional derivative. In work, [10], proposed a
self-hyperchaotic system-based perturbed
pseudorandom sequence generator to overcome this
problem. This hyperchaotic system is designed to
achieve complex dynamic behaviour. In work, [11],
also proposed a random number generator using a
fractional order Chua chaotic system.
Despite the existing research in this field, the
problem remains relevant due to the many unsolved
questions and the need for a more detailed
understanding of chaoticity. In this work, we try to
fill this gap by analyzing the structure of chaotic
solutions of differential equations, particularly the
system of Chua's equations and their modifications.
3 Methods
The following systems of differential equations are
considered in this work: the Lorentz equation, the
Rössler equation, and modifications of the Chua
equation.
1. Lorentz equation is a system of
inhomogeneous differential equations that describe
the dynamics of a system capable of transiting from
one state to another, where each state corresponds to
some parameters in this system. The Lorentz
equations were developed by Edward Lorentz in
1963 to describe turbulent fluid flow. These
equations were introduced to understand the
behaviour of atmospheric processes.
The Lorentz equation consists of three coupled
first-order differential equations:
(1)
where x(t) — convective movement intensity;
y(t) — the temperature difference of the ascending
and descending liquid flows; z(t) — deviation of the
vertical temperature distribution from the linear
regime; σ — the Prandtl number, a parameter that
affects the stretching of the system in the x
direction, where y is greater than x; α — a parameter
that affects the clustering or stretching of the system
in the z direction when the value of z is less than or
greater than a certain threshold (2); β — a parameter
that affects the interaction between x and z by
decreasing the value of z when x and y are greater
than a certain threshold (3).
(2)
(3)
where g — gravity acceleration; a — coefficient
of thermal expansion; H — the height of the liquid
layer; ΔT — the temperature difference between the
upper and lower levels; ν - kinematic viscosity of
the liquid; k — thermal conductivity of the liquid.
It is known that under condition (4) unstable
limit cycles assemble into stationary points, and
stationary points lose their stability, forming a
Lorentz attractor, which will be considered in this
study.
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2. Rössler attractor is an attractor found in the
Rössler system of differential equations (5)
discovered in 1976, which describes the dynamics
of chemical reactions occurring in some stirred
mixture.
(5)
where a, b, and c – are parameters that
determine the system behaviour. The classic
Rössler attractor occurs with the following values of
the parameters: a=0.2; b=0.2; c=5.7.
3. Chua circuit proposed in 1983 is the simplest
electrical circuit that demonstrates modes of chaotic
oscillations. The circuit (Figure 1) consists of two
capacitors, one inductor, a linear resistor, and a
nonlinear resistor with negative resistance — a
Chua diode.
Fig. 1: Chua circuit
The system of differential equations describing the
processes of this circuit looks as follows:
where
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
where h(x) — a nonlinear function involving
feedback through a resistor, a capacitor, and a Chua
diode; Ga і Gb – resistors; G — an active element
implemented through an operational amplifier,
transistor, or other element; C1 and C2 – containers;
L – inductance; iL – the current flowing through the
inductor; v, c1, c2 – the initial values of the voltage
on the capacitors C1, C2 and voltages on the Chua
diode; t – time; E – electromotive force.
This study covers three modifications of Chua
equation systems:
3.1. Chua system of equations (16) with an
unsteady motion function h(x) (17) and additional
parameters:
(16)
(17)
3.2. Chua system of equations (18) with a non-
constant function h(x) (19) and additional
parameters:
(6)
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zcxcby
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(19)
3.3. Chua system of equations (20) containing an
additional equation that describes the current control
element
where A —is the current-controlling element and
c — is the system parameter.
A comparative analysis of the chaoticity of
modified Chua systems with classical systems of
Lorentz and Rössler differential equations (1, 5) was
carried out by creating a phase portrait and
calculating Lyapunov exponents (indicators) (21).
Various methods of analyzing chaotic systems, such
as fractal geometry and spectral analysis, were
proposed in the reviewed literature sources. In this
work, the analysis of phase portraits and the
Lyapunov exponent were chosen, since these
methods allow for a more detailed study of the
structure of chaotic solutions and to determine their
characteristics, such as the degree of chaos and
sensitivity to initial conditions. This approach
makes it possible to make a comparative analysis
with other chaotic systems and to determine the
peculiarities of the system of Chua's equations. For
this purpose, they were solved using the Runge-
Kutta fourth-order method.
where |dxFt| shows the distance between these
two trajectories in the state space of the system.
The values of the Lyapunov exponents indicate
the rate of increase or decrease of the distance
between two close trajectories over time, which is a
measure of the stability or chaos of the system.
When the system has all negative Lyapunov
exponents, all trajectories converge to a fixed point
and are stable. When at least one of the exponents is
positive, the system is unstable and has chaotic
behaviour.
A phase portrait is a geometric image of the
solutions of a dynamic system in the phase space,
which consists of the system coordinates, where
each coordinate corresponds to the state of the
system, and the geometric image reflects the nature
of changes in the system states over time.
The phase portrait in stable systems is usually a
set of nodes, foci, and centers. A node represents a
steady fixed state of a system, and a focus represents
periodic solutions. The center displays periodic
solutions that do not coincide but coincide in time
on average. As a rule, phase trajectories in stable
systems converge to fixed points or cycles, and
there are characteristic areas of convergence around
these points. This means the system has certain
regions of convergence, where the phase trajectories
deviate from these stable solutions and reflect
chaotic dynamics. In our study, we analyze these
convergence regions in detail and study the structure
of chaotic solutions of the system of Chua
equations.
The phase portrait in chaotic systems has a more
complex structure, reflecting system states' high-
frequency change. The difference between phase
portraits in chaotic systems is the curves that look
like randomly mixed bands, so-called Poincaré
bands. These bands show that phase trajectories can
be very sensitive to initial conditions. In other
words, small changes in initial conditions can result
in very different trajectories. This is called the
“butterfly effect”, where small changes in the initial
conditions can significantly affect the system's
behaviour in the future.
The effectiveness of the studied systems as
generators of chaos was compared through the noise
level assessment. The coefficient of variation, the
ratio of the root-mean-square deviation to the
absolute mean value of the sample were used as an
indicator for determining the noise level:
Modelling of equations and analysis of their
chaotic structure was carried out in the PyCharm
environment by building models using Python.
(20)
(21)
(22)
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Various approaches to the analysis of phase
portraits and the determination of Lyapunov
exponents are considered. Using the methods of
numerical modeling and differential geometry, a
detailed analysis of the phase portraits of the system
of Chua's equations was carried out. The calculation
of Lyapunov exponents for different sets of initial
conditions was performed, which made it possible to
estimate the degree of chaos of the system.
4 Results
1. The system of Lorentz equations was analyzed
according to its classical parameters: σ = 10, α = 28,
β = 8/3. With two sets of initial conditions:
1) x0=1, y0=1, z0=1
2) x0=-1, y0=-1, z0=-1
Consider the phase portraits of this system with
different sets of initial conditions (Figure 2).
Fig. 2: Lorentz attractor: a) under initial conditions x0=1, y0=1, z0=1; б) under initial conditions x0=-1, y0=-
1, z0=-1
The Lorentz attractor, which occurs in the case
of the parameters: σ = 10, α = 28, β = 8/3, is
characterized by chaotic behaviour, which manifests
itself as complex, non-periodic oscillations. Phase
portraits of the Lorentz attractor (Figure 2) have a
characteristic shape resembling two twisted scales
that are unevenly stretched into space. Moreover, as
Figure 2a and Figure 2b demonstrate, a “butterfly
effect” — sensitivity to initial conditions — is
observed for this equation system.
The respective Lyapunov exponents for two sets
of parameters are:
1) λ1 = 0.9293; λ2 = -14.572; λ3 = -8.022;
2) λ1 = -0.9293; λ2 = -14.572; λ3 = -8.022.
It is worth noting that the negative sign of λ1 for
the second initial conditions indicates that these
conditions are an attractor particle that is symmetric
with respect to the origin of coordinates.
2. The system of Rössler’s equations was
analyzed according to two sets of parameters:
1) a = 0.2; b = 0.2; c = 5.7;
2) a = 0.1; b = 0.1; c = 14,
and initial conditions: x0 = 1, y0 = 0.2, z0 = 3.
The phase portraits of these two experiments are
shown in Figure 3.
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Fig. 3: Rössler attractor: а) for the parameters a = 0.2; b = 0.2; c = 5.7; б) for the parameters a = 0.1; b = 0.1;
c = 14
Studying the phase portraits of the Rössler
dynamic system, we also observe the occurrence of
chaotic attractors. For the considered parameters,
the Lyapunov exponents are as follows:
1) λ1 = -1.09868411; λ2 = 0.4384828245735064;
λ3 = -0.4384828245735064;
2) λ1 = 0.11727913; λ2 = -1.57868094; λ3 = -
9.29501389.
The equation λ2 is positive for the first set of
parameters and the second λ1, confirming that the
considered system is chaotic.
The initial conditions were randomly generated
in the range [-1;1] to study three modified Chua
systems. A total of 1,000 experiments were
conducted for each option of the system of
equations.
3. Chua system of equations (16) was studied
with the following parameters: a = 1.1428, b =
0.7142, c = 0.4285, d = -1, e = -0.7. Figure 4 shows
phase portraits for two sets of initial conditions from
the variety of results obtained. One positive
Lyapunov exponent was obtained for one of them,
in the second case all are negative (more details in
Table 1).
Sets of initial conditions for solving the system
of Chua equations (16):
1) x0 = 0.21; y0 = -0.54; z0 = 0.85;
2) x0 = 0.74; y0 = -0.95; z0 = 0.73.
Fig. 4: Phase portraits of the Chua 1 system: a) under initial conditions: x0 = 0.21; y0 = -0.54; z0 = 0.85; b)
under initial conditions: an x0 = 0.74; y0 = -0.95; z0 = 0.73
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Both phase portraits shown in Figure 4 have a
complex and non-periodic structure. We observe a
high sensitivity to the initial conditions: the
“butterfly effect”, which indicates that the
hypothesis of the chaotic nature of the system under
study is not rejected at this research stage. However,
the values of the Lyapunov exponents listed in
Table 1 should be considered.
Table 1. The value of the Lyapunov exponent for
the Chua 1 system
Initial conditions
Lyapunov exponents
x0
y0
z0
λ1
λ2
λ3
0.21
-0.54
0.85
0.0039
-2.05
-0.42
0.74
-0.95
0.73
-1.47
-0.37
-0.14
0.40
-0.33
0.35
-0.23
-1.11
-0.24
0.76
-0.16
0.12
-0.38
-0.86
-1.08
0.99
-0.94
0.99
-0.95
-0.39
-0.17
According to Table 2, only one of the considered
sets of initial conditions has a positive Lyapunov
exponent. Therefore, it can be argued that the
system shows high sensitivity to initial conditions
and manifests chaotic properties.
Table 1. The value of the Lyapunov exponent for
the Chua 2 system
Initial conditions
Lyapunov exponents
x0
y0
z0
λ1
λ2
λ3
-0.27
0.18
-0.05
0.0511
0
-10.506
0.48
-0.35
0.47
0.2723
-1.2711
-17.201
-0.9
0.45
0.95
0.5434
-1.2303
-25.912
-0.19
0.97
-0.65
1.5629
-2.0178
-29.977
0.65
-0.81
-0.8
-0.4492
-0.4305
-27.765
The analysis of the phase portraits and the
Lyapunov exponents of the system of Chua
equations (16) gives grounds to summarize: as the
phase portrait shows chaotic properties, it can be
stated that the system is chaotic, regardless of the
sign of the Lyapunov exponent. It can also be noted
that the studied system shows complex and
unpredictable dynamics, which are inherent in
chaotic systems.
4. The studied modification of the Chua system
of equations (18) considered the parameters: a =
15.6, b = 28, c1 = -1, c2 = -0.7, r = 1.2, m0 = -8/7, m1
= -5/7. For example, Figure 5 shows phase portraits
for two sets of initial conditions:
1) x0 = -0.272; y0 = 0.177; z0 = -0.05;
2) x0 = -0.189; y0 = 0.969; z0 = -0.653.
Fig. 5: Phase portraits of the Chua 2 system: a) under initial conditions: x0 = -0.272; y0 = 0.177; z0 = -0.05; b)
under initial conditions: x0 = -0.189; y0 = 0.969; z0 = -0.653
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As the phase portraits in Figure 5 show,
trajectories of the system are non-repetitive and
show unpredictable dynamics, being also
significantly dependent on the initial conditions.
The value of the Lyapunov exponent for some
randomly selected sets of initial conditions listed in
Table 2 was considered to confirm the hypothesis
about the chaotic nature of the system.
As Table 2 shows, the array of Lyapunov
exponent for each experiment has one positive
value. Therefore, the obtained results indicate the
chaotic nature of the Chua system (18).
3.3 The studied third modification of the Chua
system of equations (18) with the current control
equation is considered for the parameter values: α =
15.6, β = 28, γ = -1, σ = 0.1. For example, Figure 6
and Figure 7 demonstrate phase portraits and dot
plots for two sets of initial conditions:
1) x0 = 0.048; y0 = 0.0319; z0 = 0.6017; A0 =
0.83374
2) x0 = 0.1; y0 = 0.08; z0 = 1; A0 = 1.5.
The phase portrait illustrated in Figure 6a shows
more complex dynamics with many starting points,
the boundaries on the phase portrait are more
blurred. However, there are several shutters, which
may indicate invariant areas or symmetries in the
system.These images (Figure 6 and Figure 7) may
indicate complex chaotic dynamics in the system. It
should also be noted that there is a striking
difference between the phase portraits in Figures 6
and Figure 7 with a slight change in the initial
conditions, which confirms the hypothesis of high
chaotic system. This may indicate an invariant plane
or other symmetry in the system. The image has
sharp boundaries, indicating certain areas in the
phase space with different dynamics. Table 3 shows
the values of Lyapunov exponents for the studied
system of equations with several sets of initial
conditions.
Fig. 6: a) Phase portrait of the Chua 3 system under initial conditions: x0 = 0.1; y0 = 0.08; z0 = 0.1; А0 = 1.5; b)
Phase portrait in the form of a scatter plot.
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Fig. 7: а) a shutter centered at the point (0,0) dividing the phase portrait into two symmetrical parts; b) a chaotic
placement of points, without any patterns.
Table 3. Values of Lyapunov exponents for the Chua 3 system
Initial conditions
Lyapunov exponents
x0
y0
z0
A0
λ1
λ2
λ3
λ4
0.048
0.0319
0.6017
0.8337
0.0537
-0.0675
-2.4923
-0.1005
0.4
0.02
0.59
0.82
0.0492
-0.0621
-2.4929
-0.1092
0.03
0.01
0.58
0.81
0.0455
-0.0569
-2.4935
-0.1167
0.02
0.001
0.4
0.7
0.0035
-0.0044
-0.3063
-0.0049
-0.01
-0.02
0.2
0.5
-0.0239
0.0301
-2.4944
-0.0671
0.1
0.08
1
1.5
0.0645
-0.0826
-2.4915
-0.1367
There is one positive exponent in each set,
therefore, it can be concluded, based on the analysis
of the phase portraits above that the system is
chaotic.
5 Discussion
Table 4 compares the noise level (%) of the studied
systems based on the Chua equation system with the
classical Lorentz and Rössler chaotic systems.
Table 2. The value of the coefficient of variation, %
for the solutions of the considered systems
System of
equations
x
y
z
A
Lorentz
424,22
450,67
31,36
Rössler
2424,35
1638,74
578,50
Chua 1
264,50
1816,90
315,99
Chua 2
22,38
602,55
45,23
Chua 3
19437,73
15329,24
46345,01
690,42
So, Chua systems of equations 1 and 2 have a
noise level similar to the system of Lorenz equations
and are inferior to the system of Rössler equations.
Instead, the modified Chua 3 system of equations
far surpasses the results of other systems and can be
used as a chaos generator.
Unfortunately, a recent study that considered a
chaos generator based on the classical Chua
equation, does not explain how they estimated the
noise level, so a comparison cannot be made, [11].
Many authors studied chaos generators. For
example, the paper, [12], studied a new system of
chaotic generation: a circulating chaotic system. The
researchers studied the dynamics of the system and
showed that it has some important characteristics of
chaotic systems, such as sensitivity to the initial
conditions and randomness, and also showed that
the system has many stable attractors and can have a
wide variety of behaviours depending on the
parameters of the system. In, [13] the authors
proposed an electro-optical source of chaos based
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DOI: 10.37394/23201.2023.22.10
Maryna Belova, Volodymyr Denysenko,
Svitlana Kartashova, Valerij Kotlyar,
Stanislav Mikhailenko
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on the theory of converting phase modulation into
intensity modulation and an analog-digital hybrid
time-delay feedback loop.
However, the results presented in this work were
obtained in a simpler way and can be practically
implemented.
6 Conclusions
The growing importance of chaotic systems in
various fields of science and technology tasks
scientists to analyze the structure of chaotic
solutions. The research carried out in this work is
focused on the analysis of the structure of chaotic
solutions of several differential equations.
The results of the study give grounds to draw
the following conclusions. All the studied
differential equations have chaotic solutions with a
highly complex structure. It was found that the
structure of chaos in these solutions depends on the
initial conditions and parameters of the equation. It
was shown that analysing the structure of chaotic
solutions is an important step in understanding the
behaviour of chaotic systems.
The significance of the research is that the
analysis of chaotic solutions of differential
equations allows a deeper understanding of the
structure of complex systems and their behaviour in
various conditions. This can find practical
applications in various sciences and fields, from
theoretical physics and mathematics to biology and
engineering.
Our main contribution is developing a modified
Chua equation system that includes new parameters
and current control equations. This modification
revealed high chaoticity and complex dynamics,
distinguishing it from the classical system of Chua
and others. The modified system of Chua's
equations can be applied to solve various applied
problems, such as cryptography, pseudorandom
number generation, steganography, and
communication systems. Due to its chaos and
complex dynamics properties, this modified system
opens up new possibilities for creating reliable and
efficient algorithms in these areas.
An important direction of future research is
extending our analysis to other classes of chaotic
systems and determining their specific features. It is
recommended to discuss the advantages and
limitations of using our analysis, particularly in the
context of various applied problems.
References:
[1] Aubin, D., Dalmedico, A. D.: Writing the
History of Dynamical Systems and
Chaos: Longue Durée and Revolution,
Disciplines and Cultures. Historia
Mathematica, Vol. 29, No 3, pp. 273-339,
2002. https://doi.org/10.1006/hmat.2002.2351
[2] Adil J., Hassan A., Zamir H.: Bifurcation
study and pattern formation analysis of a
nonlinear dynamical system for chaotic
behavior in traveling wave solution. Results in
Physics, Vol. 37, 2022.
https://doi.org/10.1016/j.rinp.2022.105492.
[3] Ali F., Jhangeer A., Muddassar M., Almusawa
H.: Solitonic, quasi-periodic, super nonlinear
and chaotic behaviors of a dispersive extended
nonlinear Schrödinger equation in an optical
fiber. Results in Physics, Vol. 31, 2021.
https://doi.org/10.1016/j.rinp.2021.104921
[4] Jhangeer, A., Rezazadeh, H., Seadawy, A.: A
study of travelling, periodic, quasiperiodic
and chaotic structures of perturbed Fokas–
Lenells model. Pramana, Vol. 95, pp. 1-11,
2021. https://doi.org/10.1007/s12043-020-
02067-9
[5] Abro, K.A., Atangana, A.: Numerical Study
and Chaotic Analysis of Meminductor and
Memcapacitor Through Fractal–Fractional
Differential Operator. Arabian Journal for
Science and Engineering, Vol. 46, pp. 857–
871, 2021. https://doi.org/10.1007/s13369-
020-04780-4
[6] Kumar, S., Kumar, R., Cattani, C., Samet, B.:
Chaotic behaviour of fractional predator-prey
dynamical system. Chaos, Solitons &
Fractals, Vol. 135, 2020
https://doi.org/10.1016/j.chaos.2020.109811
[7] Anees, A., Iqtadar H.: "A novel method to
identify initial values of chaotic maps in
cybersecurity." Symmetry, Vol. 11, No. 2,
2019. https://doi.org/10.3390/sym11020140
[8] Al-Furjan, M.S.H., Habibi, M., won Jung, D.,
Chen, G.: Mehran Safarpour, Hamed
Safarpour, Chaotic responses and nonlinear
dynamics of the graphene nanoplatelets
reinforced doubly-curved panel. European
Journal of Mechanics - A/Solids, Vol. 85,
2021.
https://doi.org/10.1016/j.euromechsol.2020.10
4091
[9] Kolade, M. O., Gómez-Aguilar J.F., Karaagac
B.: Modelling, analysis and simulations of
some chaotic systems using derivative with
Mittag–Leffler kernel. Chaos, Solitons &
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2023.22.10
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Fractals, Vol. 125, pp. 54-63, 2019.
https://doi.org/10.1016/j.chaos.2019.05.019
[10] Yi, Z., Changyuan, G., Jie, L., Shaozeng D.:
A self-perturbed pseudo-random sequence
generator based on hyperchaos. Chaos,
Solitons & Fractals: X, Vol. 4, 2019
https://doi.org/10.1016/j.csfx.2020.100023
[11] Ozkaynak, F. A Novel Random Number
Generator Based on Fractional Order Chaotic
Chua System. Elektronika Ir Elektrotechnika,
Vol. 26, No. 1, pp. 52-57, 2020
https://doi.org/10.5755/j01.eie.26.1.25310
[12] Rajagopal, K., Akgul, A., Pham, V. T.,
Alsaadi, F. E., Nazarimehr, F., Alsaadi, F. E.,
Jafari, S.: Multistability and coexisting
attractors in a new circulant chaotic system.
International Journal of Bifurcation and
Chaos, Vol. 29, No. 13, 2019.
https://doi.org/10.1142/S0218127419501748
[13] Cheng, M., Luo, C., Jiang, X., Deng, L.,
Zhang, M., Ke, C., Fu, S., Tang, M., Shum,
P., Liu, D.: An Electro-Optic Chaotic System
Based on a Hybrid Feedback Loop. Journal of
Lightwave Technology, Vol. 36, No. 19, 2018.
https://doi.org/10.1109/JLT.2018.2814080
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WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2023.22.10
Maryna Belova, Volodymyr Denysenko,
Svitlana Kartashova, Valerij Kotlyar,
Stanislav Mikhailenko
E-ISSN: 2224-266X
85
Volume 22, 2023
