EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3915-3931
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Impact of White Noise on Chaotic Behavior in a
Financial Fractional System with Constant and Variable

Order: A Comparative Study

Mohammed Berir1,2,∗

1 Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha,
Saudi Arabia
2 Department of Mathematics, Faculty of Science, Bakht Al-Ruda University, Duwaym,
Sudan

Abstract. In this study, we investigated the impact of white noise on the chaotic dynamics of a
financial system with Caputo fractional derivatives of constant- and variable-order. We solve these
fractional financial systems numerically. We analyze the chaotic behavior of these fractional-order
hyperchaotic systems through simulations, study the effects of white noise on chaotic dynamics,
and provide a comparison between fractional hyperchaotic systems of constant and variable orders.
The study reveals that white noise can either amplify or suppress chaos, with significant differences
observed between the constant- and variable-order cases. We obtained numerous intriguing graph-
ical findings for the model by evaluating various scenarios. The findings provide useful information
for better understanding financial market dynamics in the presence of noise.

2020 Mathematics Subject Classifications: 34C28, 34A08, 65P20

Key Words and Phrases: Constant and Variable-order fractional derivatives, White Noise,
hyperchaotic

1. Introduction

Fractional calculus has garnered significant interest in the modeling of intricate sys-
tems, particularly in the field of finance, owing to its adaptable nature and memory
characteristics[8, 30].Using the fractional concept, the authors investigated mathematical
models and obtained intriguing findings [4, 12]. Furthermore, it is evident that additional
research is required to render fractional calculus suitable for researchers worldwide. In re-
sponse, researchers have been assiduously engaged in the identification and formulation of
useful operators, with a particular emphasis on fractional operators. Stochastic non-linear
evolution equations play a vital role in a wide range of scientific and engineering fields,
allowing us to model and understand complex systems and phenomena. They analyze

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5462

Email address: midriss@bu.edu.sa (M. Berir)

https://www.ejpam.com 3915 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3916

the impact of white noise on chaotic financial systems, taking into account randomness
and nonlinearity[39]. This study investigates the effects of random disturbances on the
chaotic dynamics of systems with constant-order and variable-order fractions, employing
the principles of fractional calculus. It has become a popular area of research, attract-
ing a lot of attention. Hyperchaotic System (HS) exhibits the remarkable characteristic
of displaying multiple chaotic behaviors simultaneously[11, 14]. These types of systems
exhibit intricate chaotic dynamics that surpass those of conventional (HS). (HS) exhibit
the presence of multiple positive Lyapunov Exponents(LE), which signify a heightened
level of unpredictability and complexity in their behavior[10]. The distinct properties of
(HS) make them highly beneficial in a diverse range of fields that prioritize elements such
as randomness and complexity[42]. A deeper comprehension of processes that depend
on numerous variables is also brought about by the expansion of calculus into multivari-
able functions, opening the door for developments in disciplines like fluid dynamics and
thermodynamics[6, 40]. Calculus’s perennial relevance in the never-ending quest for knowl-
edge and creativity is shown by the theoretical breakthroughs and real-world applications
that scholars discover as they delve further into the subject[5, 13, 21].

Numerous scientific and technical domains have extensive uses for fractional stochastic
equations, which combine the ideas of stochastic processes with fractional calculus. These
equations are very helpful in domains like biology, physics, and finance since they are
used to simulate complicated systems with unpredictability and memory effects[17, 41].
For example, they may represent price dynamics in financial markets, mimic the behavior
of biological systems with long-range dependencies, and characterize anomalous diffusion
processes in porous surfaces. These equations offer more precise and adaptable models
for systems displaying both uncertainty and non-local behavior by fusing the tools of
fractional derivatives with stochastic dynamics[3].

Over time, (HS) have made steady progress since they were first introduced by Rossler
[37].Recently, researchers have made significant advances in the field of (HS), including
the development of various distinct systems. One notable example is the 2D (HS) that
has been proposed for image encryption [15].In addition, [45] developed a two-dimensional
(HS) using the optimization benchmark function [46]. developed a 2D hyperchaotic map
that has proven useful in generating pseudorandom numbers and encrypting color im-
ages. In their study, [16] introduced an image encryption algorithm that uses a 2D (HS).
An innovative system for secure communications was introduced by[2]. This system uti-
lizes a 2D cosine-sine interleaved chaotic approach. also studied a simple chaotic model
with complex chaotic behavior. The number is significant[19]. conducted a study on
the advancements in 3D hyperchaotic systems and their potential applications in secure
transmission[24]. An ultrawide parameter range was presented by[33] for a nondegenerate
hyperchaotic map. In their study, [43] introduced a hyperchaotic system that consists of
four dimensions. [26]. It performed a comprehensive investigation on the dynamics of
a multi-stable hyperchaotic Lorenz system and examined its several practical uses. [27].
The study conducted by [23].delved into the advancements in dynamical systems with five
state variables, specifically focusing on those with multiple positive (LE). The researchers
also explored the presence of coexisting hidden attractors [25].



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3917

In this paper, the concept of (HS), which is defined by fractional-order derivatives, is
introduced, and the impact of white noise is investigated[29]. In addition, we elaborate
on certain previous studies, including those detailed in[20]. Furthermore, we present a
variety of figures using the MATLAB 2024 software to investigate the influence of noise
on hyperchaotic systems (3, 4). Variable-order fractional calculus operators are a valuable
mathematical instrument for a more comprehensive examination of dynamical phenomena
due to their variable-order fractional order [31]. In reality, the mathematical systems
modeled by this innovative concept exhibit greater sensitivity and accuracy. It is important
to note that the process of obtaining an analytical solution for issues involving variable-
order fractional operators is frequently exceedingly intricate. Therefore, approximative
approaches are more likely to be employed to resolve such issues. In [9], the interested
reader can find several numerical methods for solving fractional problems of variable order.

Introducing white noise into chaotic systems can effectively decrease their complexity
and unpredictability. Studies suggest that noise has a substantial impact on chaotic dy-
namics, influencing the development of trajectories even when the starting conditions are
the same. The unpredictability in chaotic scattering problems is further complicated by
stochastic fluctuations, which can make it difficult to anticipate the ultimate states [32].
Nevertheless, research has demonstrated that low-intensity noise can decrease irregularity
in chaotic patterns, resulting in a regularization effect that amplifies organization within
the system [22]. Moreover, novel techniques for acquiring knowledge about chaotic systems
from noise-contaminated data reveal that noise may be effectively controlled, enabling pre-
cise forecasts despite intrinsic disorder [44]. Therefore, although noise adds complexity, it
might paradoxically enable more predictable and organized behavior in chaotic systems
under specific circumstances. This paper presents a novel approach to obtain the constant
and variable-order fractional estimates with stochasticity for the Financial Fractional Ca-
puto (HS). In order to derive these Hyperchaotic properties with stochastic properties,
we employ a novel numerical technique. Moreover, we build on previous studies, includ-
ing the findings reported in the reference[7]. The generated solutions play a crucial role
in elucidating compelling financial phenomena necessary for characterizing hyperchaotic
propagation. Moreover, we investigate the influence of noise on the precise solutions of
system (4,5) by displaying several visual representations.

Additionally, white noise is employed in the fields of chemistry and physics. For ex-
ample, these circuit systems based on memcapacitors are employed in the field of physics
to understand intricate systems such as turbulent flow or phase transitions. Scientists can
investigate the emergence of ordered structures from random initial conditions by tak-
ing into account the stochastic nature of these phenomena. The optimization of reaction
conditions and the design of efficient chemical reactors are facilitated by the use of white
noise in chemical kinetics to model reaction processes that involve random fluctuations.
The peculiarity of this article is the acquisition of the stochastic simulation of the systems
(4-5). We employ two distinct methodologies to generate these simulations: the extended
fractional constant and varibales orders of the mapping method. Additionally, we elabo-
rate on certain prior studies, including those detailed in [36, 38]. The solutions that were
generated are crucial in elucidating certain thrilling physical phenomena, as the Finan-



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3918

cial Fractional System is necessary to describe marketing propagation. Furthermore, we
present a variety of figures to investigate the influence of noise on the precise solutions of
the Financial Fractional Systems (4-5).

The article is organized in the following manner: This document presents definitions
and notations of variable-order fractional derivatives at the basic level. For section 2.
Analysis of the effects of white noise and simulations on the financial system using constant
and variable-order fractional methods. An analysis of Caputo hyperchaotic systems is
conducted in Section 3. Section 4 of the manuscript focuses on modeling the financial
system using white noise. The final conclusion is articulated in Section 5.

2. Preliminaries

This part provides essential definitions of Caputo variable-order fractional derivatives,
which will be applied in subsequent sections. These definitions are presented in this section.

Definition 1. ([35]). The beta function is given by β(t) is defined as

β(µ, λ) =

∫ 1

0
tµ−1(1− t)λ−1 dt, µ > 0, λ > 0 (1)

It is important to note that the beta function is symmetric, which means that β(µ, λ) =
β(λ, µ). In the interim, the beta function’s robust correlation with the gamma function is
a noteworthy attribute.

β(µ, λ) =
Γ(µ)Γ(λ)

Γ(µ+ λ)

where ( Γ(.) ) denotes the Gamma function.For the Mittag-Leffler function:

Definition 2. ([35]). The Mittag–Lefer function is given by

Eµ,λ(z) =

∞∑
j=0

zj

Γ(µj + λ)
, z ∈ C+, t ∈ R (2)

Note that for λ = 1, it is abbreviated as Eµ(t) = Eµ,1(t).

Definition 3. ([28]). The Caputo fractional derivative with a varying order β(t) is defined
as

V O
0 D

β(t)
t {X (t)} =

1

Γ(1− β(t))

∫ t

0
(t− τ)−β(t)X ′(τ) dτ, 0 < β(t) < 1. (3)

Definition 4. ([34]). The Caputo derivative with constant order β is given as follows:

C
0 D

β
t {X (t)} =

1

Γ(1− β)

∫ t

0

1

(t− w)β
d

dw
X (w)dw, 0 < β < 1. (4)



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3919

Definition 5. ([18]). Hyperchaotic Systems of Constant and Variable Order: White Noise
in Fractional CalculusWhite noise is a stochastic process in fractional calculus, especially
in constant and variable-order hyperchaotic systems. W (t),X[W (t)] = 0 characterized by
a mean of zero and a delta-correlated covariance function. Mathematically, it is expressed
as:

X[W (t)W (t′)] = σ2Γ(t− t′), (5)

where X denotes the expectation operator, σ2 represents the variance of the noise, δ(·) is
the Dirac delta function, and t and t′ are distinct time instances.White noise in fractional-
order systems can cause disturbances that shape the dynamic behavior, perhaps resulting
in hyperchaotic states, which are determined by the system’s order and the strength of the
noise.

3. A Mathematical Model of the Financial Fractional System

This section discusses the hyperchaos and chaos of a new fractional-order system,
touching on the effects of white noise on the predictability and stability of constant and
variable-order hyperchaotic systems. Through the use of white-noise driven fractional
differential equation models, this study examines the financial system[1]:

• fractional constant-order The hyperchaotic financial system of Caputo with σ is the
intensity of noise; W(t) is the white noise (Gaussian process).the interest rate X (t).
The investment demand is Y(t). The exponent Z(t) represents the price and the
rate of investment.

C
0 D

β
t {X (t)} = X (t)(Y(t)− a) + Z(t) + σXW (t),

C
0 D

β
t {Y(t)} = 1− (bZ(t) + Y2(t)) + σYW (t),

C
0 D

β
t {Z(t)} = −Y(t)− cZ(t) + σZW (t).

(6)

where a ≥ 0 is the saving amount, b ≥ 0 is the cost per investment, and c ≥ 0 is the
elasticity of demand of commercial market.

• fractional variable-order The hyperchaotic financial system of Caputo

V O
0 D

β(t)
t {X (t)} = X (t)(Y(t)− a) + Z(t) + σXW (t),

V O
0 D

β(t)
t {Y(t)} = 1− (bZ(t) + Y2(t)) + σYW (t),

V O
0 D

β(t)
t {Z(t)} = −Y(t)− cZ(t) + σZW (t).

(7)

4. A Novel Numerical Method for Stochastic Fractional Financial
Systems

Using the starting values of the variables that we will use to derive the phase portraits of
the fractional-order chaotic system (4)–(5), this section outlines the numerical scheme for



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3920

the numerical solution of model. We attempt to apply the standard discretizations found
in the literature to our model. Numerous numerical schemes and analytical techniques,
including the Chebyshev method, the domain decomposition method, and the homotopy
methods, can be applied in fractional calculus. However, because of the difficulties with
the stability and convergences of the approximate solutions, many of the drawbacks of the
approaches mentioned have yet to be resolved. MATLAB codes, which are essential in
chaotic and hyperchaotic systems, are one benefit of using the predictor-corrector approach
in our system. The solution of the fractional differential system (4)–(5) can be described
as follows in the remaining portion of this section Our study presents a new numerical
technique for modeling fractional differential operators of both constant and variable order
utilizing stochastic processes.

4.1. A numerical scheme for constant-order fractional financial system.

This study presents a numerical method for a fractional financial system that integrates
a stochastic Caputo fractional derivative with a constant-order fractional.

CDα
a+S(t) = ψ1(t,S(t)) + σy(t)dB(t) (8)

S(t) = S(0) + 1

Γ(β)

∫ t

0
(t− w)β−1ψ1(S, w) dw +

∫ t

0
σy(t)dB(t) (9)

At time t = tn+1, Equation 9 is as follows:

S(tn+1) = S(0) + 1

Γ(β)

∫ tn+1

0
(t− w)β−1ψ1(S, w) dw +

∫ tn+1

0
σy(t)dB(t) (10)

At time t = tn

Sn = S(tn) = S(0) + 1

Γ(β)

∫ tn

0
(t− w)β−1ψ1(S, w) dw +

∫ tn

0
σy(t)dB(t) (11)

S(tn+1)− S(tn) =
1

Γ(α)

[∫ tn+1

0
(tn+1 − w)α−1g(S,w) dw −

∫ tn

0
(tn − w)α−1g(S,w) dw

]
.

(12)
Numerical solution of Equation (10) obtained via Lagrange polynomial interpolation

is as follows:

Sn+1 = S0 +
βhβ

Γ(β + 2)

r∑
m=0

ψ1(tm,Sm)
[
(n−m+ 1)β(n−m+ 2 + 2β)− (n−m)β(n−m+ 2 + 2β)

]
− hβ

Γ(β + 2)

r∑
m=0

ψ1(tm−1,Sm−1)
[
(n−m+ 1)β+1 − (n−m)β(n−m+ 1 + β)

]
+ ασy(t)(cn)(B(tn+1)−B(tn)).

(13)



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3921

We therefore present the numerical solution that we have developed for the constant-order
fractional(Hs), which is as follows:

Xn+1 = X0 +
βhβ

Γ(β + 2)

r∑
m=0

ψ1(tm,Xm)
[
(n−m+ 1)β(n−m+ 2 + 2β)− (n−m)β(n−m+ 2 + 2β)

]
− hβ

Γ(β + 2)

r∑
m=0

ψ1(tm−1,Xm−1)
[
(n−m+ 1)β+1 − (n−m)β(n−m+ 1 + β)

]
+ ασy(t)(cn)(B(tn+1)−B(tn)).

(14)

Yn+1 = Y0 +
βhβ

Γ(β + 2)

r∑
m=0

ψ1(tm,Ym)
[
(n−m+ 1)β(n−m+ 2 + 2β)− (n−m)β(n−m+ 2 + 2β)

]
− hβ

Γ(β + 2)

r∑
m=0

ψ1(tm−1,Ym−1)
[
(n−m+ 1)β+1 − (n−m)β(n−m+ 1 + β)

]
+ ασy(t)(cn)(B(tn+1)−B(tn)).

(15)

Zn+1 = Z0 +
βhβ

Γ(β + 2)

r∑
m=0

ψ1(tm,Zm)
[
(n−m+ 1)β(n−m+ 2 + 2β)− (n−m)β(n−m+ 2 + 2β)

]
− hβ

Γ(β + 2)

r∑
m=0

ψ1(tm−1,Zm−1)
[
(n−m+ 1)β+1 − (n−m)β(n−m+ 1 + β)

]
+ ασy(t)(cn)(B(tn+1)−B(tn)).

(16)

4.2. A numerical scheme for Variable-order fractional financial system.

This study presents a numerical method for a fractional financial system that inte-
grates a stochastic Utilizing a variable-order fractional, the Caputo fractional derivative
is employed.

CD
β(t)
a+

S(t) = ψ1(t,S(t)) + σy(t)dB(t) (17)

S(t) = S(0) + 1

Γ(β)

∫ t

0
(t− w)β(t)−1ψ1(S, w) dw +

∫ t

0
σy(t)dB(t) (18)

At time t = tn+1, 18 takes the following form:

S(tn+1) = S(0) + 1

Γ(β(t))

∫ tn+1

0
(t− w)β(t)−1ψ1(S, w) dw +

∫ tn+1

0
σy(t)dB(t) (19)



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3922

At t = tn

Sn = S(tn) = S(0) + 1

Γ(β(t))

∫ tn

0
(t− w)β(t)−1ψ1(S, w) dw +

∫ tn

0
σy(t)dB(t) (20)

S(tn + 1) = S(0) + 1

Γ(β(t))

∫ tn

0
(t− w)β(t)−1ψ1(S, w) dw +

∫ tn

0
σy(t)dB(t) (21)

Numerical solution of Equation (9) obtained via Lagrange polynomial interpolation is
as follows:

Sn+1 = S0 +
β(t)hβ(t)

Γ(β(t) + 2)

r∑
m=0

ψ1(tm,Sm)
[
(n−m+ 1)β(t)(n−m+ 2 + 2β)− (n−m)β(t))

]
− hβ

Γ(β(t) + 2)

r∑
m=0

ψ1(tm−1,Sm−1)
[
(n−m+ 1)β(t)+1 − (n−m)β(t)(n−m+ 1 + β)

]
+ β(t)σy(t)(cn)(B(tn+1)−B(tn)).

(22)
We therefore present the numerical solution that we have developed for the variable-order
fractional hyperchaotic system, which is as follows:

Xn+1 = X0 +
β(t)hβ(t)

Γ(β(t) + 2)

r∑
m=0

ψ1(tm,Xm)
[
(n−m+ 1)β(t)(n−m+ 2 + 2β)− (n−m)β(t))

]
− hβ

Γ(β(t) + 2)

r∑
m=0

ψ1(tm−1,Xm−1)
[
(n−m+ 1)β(t)+1 − (n−m)β(t)(n−m+ 1 + β(t))

]
+ ασy(t)(cn)(B(tn+1)−B(tn)).

(23)

Yn+1 = Y0 +
β(t)hβ(t)

Γ(β(t) + 2)

r∑
m=0

ψ1(tm,Ym)
[
(n−m+ 1)β(t)(n−m+ 2 + 2β(t))− (n−m)β(t))

]
− hβ(t)

Γ(β(t) + 2)

r∑
m=0

ψ1(tm−1,Ym−1)
[
(n−m+ 1)β(t)+1 − (n−m)β(t)(n−m+ 1 + β(t))

]
+ ασy(t)(cn)(B(tn+1)−B(tn)).

(24)

Zn+1 = Z0 +
β(t)hβ(t)

Γ(β(t) + 2)

r∑
m=0

ψ1(tm,Zm)
[
(n−m+ 1)β(t)(n−m+ 2 + 2β(t))− (n−m)β(t))

]
− hβ(t)

Γ(β(t) + 2)

r∑
m=0

ψ1(tm−1,Zm−1)
[
(n−m+ 1)β(t)+1 − (n−m)β(t)(n−m+ 1 + β(t))

]
+ β(t)σy(t)(cn)(B(tn+1)−B(tn)).

(25)



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3923

5. Modeling the Financial System with White Noise

In this section, we define the specific financial system under study, introducing the
governing equations and parameters. The (HS) is examined both in the absence and
presence of white noise. Numerical simulations are conducted to explore how white noise
influences the chaotic dynamics for both constant and variable orders. From the previous
Figure(1), it is evident that in the absence of noise (i.e., when σ = 0), there are no dis-
tinct types of white noise in systems (4-5). Figure(2) shows that when the noise is absent
(i.e., when σ = 0.2), there are several types of white noise. Figures(3-5) demonstrates
that when the noise is absent (i.e., when σ = 0.5), there are more efficient forms of white
noise. Nevertheless, the presence of noise leads to the degradation of all these solutions,
causing the hyperchaotic behavior to become flat as the magnitude of the noise increments.

0 1 2 3 4 5 6 7 8 9 10

t

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

0 1 2 3 4 5 6 7 8 9 10

t

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

Figure 1: shows a 2D for constant and variable fractional order values for different σ = 0, 0.2, 0.5.



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3924

-4 -3 -2 -1 0 1 2 3 4

x

-2

-1

0

1

2

3

4

y

Constant Fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4

x

-2

-1

0

1

2

3

4

y

Variable fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4

x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Constant Fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4

x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2
z

Variable fractional-order with White Noise

-2 -1 0 1 2 3 4

y

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Constant Fractional-order with White Noise

-2 -1 0 1 2 3 4

y

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Variable fractional-order with White Noise

Figure 2: present Comparison between Constant and Variable Fractional Order at σ = 0.



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3925

-3 -2 -1 0 1 2 3 4

x

-2

-1

0

1

2

3

4

5

y

Constant Fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4

x

-3

-2

-1

0

1

2

3

4

5

y

Variable fractional-order with White Noise

-2 -1 0 1 2 3 4

x

-1.5

-1

-0.5

0

0.5

1

1.5

z

Constant Fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4

x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2
z

Variable fractional-order with White Noise

-1 0 1 2 3 4 5

y

-1.5

-1

-0.5

0

0.5

1

1.5

z

Constant Fractional-order with White Noise

-3 -2 -1 0 1 2 3 4 5

y

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Variable fractional-order with White Noise

Figure 3: present Comparison between Constant and Variable Fractional Order at σ = 0.2.



M. Berir / Eur. J. Pure Appl. Math, 17 (4) (2024), 3915-3931 3926

-4 -3 -2 -1 0 1 2 3 4 5

x

-2

-1

0

1

2

3

4

5

y

Constant Fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4

x

-3

-2

-1

0

1

2

3

4

5

y

Variable fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4 5

x

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Constant Fractional-order with White Noise

-4 -3 -2 -1 0 1 2 3 4

x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2
z

Variable fractional-order with White Noise

Figure 4:

-2 -1 0 1 2 3 4 5

y

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Constant Fractional-order with White Noise

-3 -2 -1 0 1 2 3 4 5

y

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Variable fractional-order with White Noise

Figure 5: present Comparison between Constant and Variable Fractional Order at σ = 0.5.



REFERENCES 3927

6. Conclusion

In this study, a comparative examination of how white noise affects chaotic behavior
in a financial fractional system considering both constant and changing orders. The study
demonstrates that systems with changeable orders have intricate dynamics and are highly
responsive to stochastic disturbances. This work investigated the control of both constant-
order and variable-order fractional (HS) by the use of Caputo derivatives. Figures(1–5)
depict the chaotic dynamics of fractional order Caputo fractional (HS) with constant and
changing orders. From our analysis of the images, we deduce that the simulations empha-
size the distinctions between constant- and variable-order Caputo derivatives. Changes
in the fractional order provide intricacy to the chaotic behavior. The Caputo (HS) was
examined using fractional derivatives. Our findings indicate that obtaining a comprehen-
sive understanding of the chaotic regions in (HS) with variable-order fractional deriva-
tives is more straightforward compared to (HS) with constant-order fractional derivatives.
Therefore, the comprehension of the dynamics of a (HS) enhances when it incorporates
a variable-order fractional derivative.We found that the stochastic component stabilizes
the chaos of the financial fractional system at zero. In the future, we may investigate the
chaotic attractors linked to fractional derivatives or additive noise.In the future, we may
conduct an investigation into the financial fractal fractional system that includes additive
noise.

References

[1] Mohammed Salah Abd-Elouahab, Nasr-Eddine Hamri, and Junwei Wang. Chaos
control of a fractional-order financial system. Mathematical Problems in Engineering,
2010, 2010.

[2] Mohamed A Abdoon and Abdulrahman BM Alzahrani. Comparative analysis of
influenza modeling using novel fractional operators with real data. Symmetry,
16(9):1126, 2024.

[3] Mohamed A Abdoon, Mohamed Elbadri, Abdulrahman BM Alzahrani, Mohammed
Berir, and Abubker Ahmed. Analyzing the inverted fractional rössler system through
two approaches: numerical scheme and lham analysis. Physica Scripta, 2024.

[4] Himani Agarwal, Manvendra Narayan Mishra, and Ravi Shanker Dubey. On frac-
tional caputo operator for the generalized glucose supply model via incomplete aleph
function. International Journal of Mathematics for Industry, page 2450003, 2024.

[5] Ibtehal Alazman, Manvendra Narayan Mishra, Badr Saad Alkahtani, and
Ravi Shanker Dubey. Analysis of infection and diffusion coefficient in an sir model
by using generalized fractional derivative. Fractal and Fractional, 8(9):537, 2024.

[6] Mawada Ali, Fathelrhman EL Guma, Ahmad Qazza, Rania Saadeh, Nahaa E Al-
subaie, Mohammed Althubyani, and Mohamed A Abdoon. Stochastic modeling of



REFERENCES 3928

influenza transmission: Insights into disease dynamics and epidemic management.
Partial Differential Equations in Applied Mathematics, 11:100886, 2024.

[7] Awatif Muflih Alqahtani, Arun Chaudhary, Ravi Shanker Dubey, and Shivani
Sharma. Comparative analysis of the chaotic behavior of a five-dimensional frac-
tional hyperchaotic system with constant and variable order. Fractal & Fractional,
8(7), 2024.

[8] Saad Althobaiti, Ravi Shanker Dubey, and Jyoti Geetesh Prasad. Solution of local
fractional generalized fokker–planck equation using local fractional mohand adomian
decomposition method. Fractals, 30(01):2240028, 2022.

[9] Seda IGRET Araz, Mehmet Akif Cetin, and Abdon Atangana. Existence, uniqueness
and numerical solution of stochastic fractional differential equations with integer and
non-integer orders. Electronic Research Archive, 32(2):733–761, 2024.

[10] Mohammed Berir. A fractional study for solving the smoking model and the chaotic
engineering model. In 2023 2nd International Engineering Conference on Electrical,
Energy, and Artificial Intelligence (EICEEAI), pages 1–6. IEEE, 2023.

[11] Mohammed Berir, Rania Saadeh, Mohamed A Abdoon, Ahmad Qazza, and Dalal
Almutairi. A fractional study for solving the sir model and chaotic system. IAENG
International Journal of Applied Mathematics, 54(2), 2024.

[12] Nishant Bhatnagar, Kanak Modi, Lokesh Kumar Yadav, and Ravi Shanker Dubey.
An efficient technique to study of time fractional whitham–broer–kaup equations.
International Journal of Mathematics for Industry, page 2450024, 2024.

[13] Ravi Shankar Dubey, Manvendra Narayan Mishra, and Pranay Goswami. Effect
of covid-19 in india-a prediction through mathematical modeling using atangana
baleanu fractional derivative. Journal of Interdisciplinary Mathematics, 25(8):2431–
2444, 2022.

[14] M Elbadri, MA Abdoon, M Berir, and DK Almutairi. A symmetry chaotic model
with fractional derivative order via two different methods. symmetry 2023, 15, 1151.
htps. doi. org/10.3390/sym15061151# Academic Editors: Christos Volos and Jorge
E. Maćıas Dı́az Received, 14, 2023.

[15] Uğur Erkan, Abdurrahim Toktas, and Qiang Lai. 2d hyperchaotic system based on
schaffer function for image encryption. Expert Systems with Applications, 213:119076,
2023.

[16] Xiaohong Gao. Image encryption algorithm based on 2d hyperchaotic map. Optics
& Laser Technology, 142:107252, 2021.

[17] Faeza Lafta Hasan, Mohamed A Abdoon, Rania Saadeh, Ahmad Qazza, and
Dalal Khalid Almutairi. Exploring analytical results for (2+ 1) dimensional breaking



REFERENCES 3929

soliton equation and stochastic fractional broer-kaup system. AIMS Mathematics,
9(5):11622–11643, 2024.

[18] Dongliang Hu, Xiaochen Mao, and Lin Han. Stochastic stability analysis of a frac-
tional viscoelastic plate excited by gaussian white noise. Mechanical Systems and
Signal Processing, 177:109181, 2022.

[19] Lilian Huang, Chuang Li, Jin Liu, Yu Zhong, and Hao Zhang. A novel 3d non-
degenerate hyperchaotic map with ultra-wide parameter range and coexisting attrac-
tors periodic switching. Nonlinear Dynamics, 112(3):2289–2304, 2024.

[20] Hossein Jafari, Muslim Yusif Zair, and Hassan Kamil Jassim. Analysis of fractional
navier–stokes equations. Heat Transfer, 52(3):2859–2877, 2023.

[21] Rinku Jain, Kirti Arekar, and Ravi Shanker Dubey. Study of bergman’s minimal
blood glucose-insulin model by adomian decomposition method. Journal of Informa-
tion and Optimization Sciences, 38(1):133–149, 2017.

[22] HK Jassim, H Ahmad, A Shamaoon, and C Cesarano. An efficient hybrid technique
for the solution of fractional-order partial differential equations. Carpathian Mathe-
matical Publications, 13(3):790–804, 2021.

[23] Omer Kocak, Uğur Erkan, Abdurrahim Toktas, and Suo Gao. Pso-based image
encryption scheme using modular integrated logistic exponential map. Expert Systems
with Applications, 237:121452, 2024.

[24] Gervais Dolvis Leutcho, Huihai Wang, Theophile Fonzin Fozin, Kehui Sun,
Zeric Tabekoueng Njitacke, and Jacques Kengne. Dynamics of a new multistable
4d hyperchaotic lorenz system and its applications. International Journal of Bifur-
cation and Chaos, 32(01):2250001, 2022.

[25] Hongmin Li, Shuqi Yu, Wei Feng, Yao Chen, Jing Zhang, Zhentao Qin, Zhengguo Zhu,
and Marcin Wozniak. Exploiting dynamic vector-level operations and a 2d-enhanced
logistic modular map for efficient chaotic image encryption. Entropy, 25(8):1147,
2023.

[26] Junhong Li and Ning Cui. Dynamical analysis of a new 5d hyperchaotic system.
Physica Scripta, 98(10):105205, 2023.

[27] Xilin Liu, Xiaojun Tong, Miao Zhang, and Zhu Wang. A highly secure image encryp-
tion algorithm based on conservative hyperchaotic system and dynamic biogenetic
gene algorithms. Chaos, Solitons & Fractals, 171:113450, 2023.

[28] B Parsa Moghaddam, Sh Yaghoobi, and JA Tenreiro Machado. An extended
predictor–corrector algorithm for variable-order fractional delay differential equations.
Journal of Computational and Nonlinear Dynamics, 11(6), 2016.



REFERENCES 3930

[29] Wael WMohammed, Clemente Cesarano, Doaa Rizk, Elkhateeb S Aly, and Mahmoud
El-Morshedy. Impact of white noise on the exact solutions of the stochastic riemann
wave equation in quantum mechanics. Symmetry, 15(11):2070, 2023.

[30] Mounia Mouy, Hamid Boulares, Saleh Alshammari, Mohammad Alshammari, Yam-
ina Laskri, and Wael W Mohammed. On averaging principle for caputo–hadamard
fractional stochastic differential pantograph equation. Fractal and Fractional, 7(1):31,
2022.

[31] Tsafack Nestor, Akram Belazi, Bassem Abd-El-Atty, Md Nazish Aslam, Christos
Volos, Nkapkop Jean De Dieu, and Ahmed A Abd El-Latif. A new 4d hyperchaotic
system with dynamics analysis, synchronization, and application to image encryption.
Symmetry, 14(2):424, 2022.

[32] Alexandre R. Nieto, Jesús M. Seoane, Miguel A. F. Sanjuán, and Miguel A. F.
Sanjuán. Final state sensitivity in noisy chaotic scattering. Chaos Solitons Fractals,
2021.

[33] Olurotimi S Ojoniyi and Abdulahi N Njah. A 5d hyperchaotic sprott b system with
coexisting hidden attractors. Chaos, Solitons & Fractals, 87:172–181, 2016.

[34] KB Oldham. The Fractional Calculus: Theory and Applications of Differentiation
and Integration of Arbitrary Order. Academic Press, 1974.

[35] Igor Podlubny, Aleksei Chechkin, Tomas Skovranek, YangQuan Chen, and Blas
M Vinagre Jara. Matrix approach to discrete fractional calculus ii: partial frac-
tional differential equations. Journal of Computational Physics, 228(8):3137–3153,
2009.

[36] Ahmad Qazza, Mohamed Abdoon, Rania Saadeh, and Mohammed Berir. A new
scheme for solving a fractional differential equation and a chaotic system. European
Journal of Pure and Applied Mathematics, 16(2):1128–1139, 2023.

[37] OE5889510996 Rossler. An equation for hyperchaos. Physics Letters A, 71(2-3):155–
157, 1979.

[38] Rania Saadeh, Mohamed A. Abdoon, Ahmad Qazza, and Mohammed Berir. A nu-
merical solution of generalized caputo fractional initial value problems. Fractal And
Fractional, 7(4):332, 2023.

[39] Stefan G Samko. Fractional integration and differentiation of variable order. Analysis
Mathematica, 21(3):213–236, 1995.

[40] Shivani Sharma, Ravi Shanker Dubey, and Arun Chaudhary. Caputo fractional model
for the predator-prey relation with sickness in prey and refuge to susceptible prey.
International Journal of Mathematics for Industry, 2024.



REFERENCES 3931

[41] Jagdev Singh, Hassan Kamil Jassim, Devendra Kumar, and Ved Prakash Dubey.
Fractal dynamics and computational analysis of local fractional poisson equations
arising in electrostatics. Communications in Theoretical Physics, 75(12):125002, 2023.

[42] JE Soĺıs-Pérez, JF Gómez-Aguilar, and A Atangana. Novel numerical method for
solving variable-order fractional differential equations with power, exponential and
mittag-leffler laws. Chaos, Solitons & Fractals, 114:175–185, 2018.

[43] Fei Yu, Li Liu, Shuai Qian, Lixiang Li, Yuanyuan Huang, Changqiong Shi, Shuo Cai,
Xianming Wu, Sichun Du, and Qiuzhen Wan. Chaos-based application of a novel
multistable 5d memristive hyperchaotic system with coexisting multiple attractors.
Complexity, 2020(1):8034196, 2020.

[44] Lei Zhang, Shaoqiang Tang, and G. He. (2) learning chaotic systems from noisy data
via multi-step optimization and adaptive training. Chaos, 2022.

[45] Shenli Zhu, Xiaoheng Deng, Wendong Zhang, and Congxu Zhu. Construction of a new
2d hyperchaotic map with application in efficient pseudo-random number generator
design and color image encryption. Mathematics, 11(14):3171, 2023.

[46] Shenli Zhu, Xiaoheng Deng, Wendong Zhang, and Congxu Zhu. Construction of a new
2d hyperchaotic map with application in efficient pseudo-random number generator
design and color image encryption. Mathematics, 11(14):3171, 2023.