EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5576
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Stability Analysis of Fractional Chaotic and
Fractional-Order Hyperchain Systems Using Lyapunov

Functions

Thwiba A. Khalid1,2

1 Department of Mathematics, Faculty of Science, Al-Baha University, Albaha 65525,
Saudi Arabia
2 Department of Mathematics, Academy of Engineering and Medical Sciences,
Khartoum 12045, Sudan

Abstract. This study investigates the stability of nonlinear systems, particularly those charac-
terized by eigenvalues. We introduce dynamic Lyapunov functions as a mechanism for stability
analysis, especially when explicit solutions are not available. The authors provide stability criteria
at the equilibrium point, demonstrating exponential stability and ensuring a return to equilibrium
following disturbances. The results have a big effect on the design and analysis of control systems
because they provide a new way to achieve stability without using complicated calculations or
assumptions. The abstract delineates the Riemann–Liouville fractional integral, Caputo fractional
integrals and derivatives, and the Mittag–Leffler function. The research employs the root–Horwitz
criteria and introduces a novel formulation of the superchaotic Chen system. Fractional superchain
systems (FHCS) represent a sophisticated framework for investigation.

2020 Mathematics Subject Classifications: 34C28, 37C25, 34A08

Key Words and Phrases: Caputo Derivative, Mittag-Leffler Function, Hurwitz Matrices, Sta-
bility Analysis, Hyperchain Systems

1. Introduction

The study begins by emphasizing and reinforcing the importance of the basic ideas
and properties of fractional calculus, including derivatives and integrals of incorrect order.
It has become one of the most advanced fields in applied mathematics, physics, and many
other scientific and technical disciplines. This rise is evident from the many important
works that have contributed to the advancement and use of fractional calculus across differ-
ent settings and disciplines. Lai et al.’s study[43][3][17], which encompasses control theory
and dynamical systems, is one such example. The introduction highlights dynamical Lia-
punov functions as an innovative method for assessing the stability of nonlinear systems.

DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5576

Email address: tabdulrhman@bu.edu.sa (T. Khalid)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 2 of 16

This research looks at the differences between dynamical Liabanov functions and regular
Liabanov functions. It demonstrates that we can create dynamical Liabanov functions
without explicitly solving the differential equations they rely on. Fractional superchain
systems (FHCS) also serve as a framework for analyzing complex systems characterized by
fractal dynamics and strongly structured interactions. We use FHCS, the Adomian decom-
position method (ADM), and the variational iteration method (VIM) to solve fractional
differential equations. The Adomian decomposition method (ADM) provides a straight-
forward method for treating nonlinear differential equations, while the variational iteration
method (VIM) offers an adaptable strategy for managing nonlinearity. Alternative tech-
niques include the fractional Fourier transform, the Laplace transform, and numerical
methods. [19][38][20][18]. By combining these methodologies, researchers can improve the
modeling, analysis, and control of complex systems across many disciplines. The first sec-
tion classifies fractional derivatives into several types, such as the Riemann–Liouville (RL)
fractional integral, the Caputo (C) fractional derivative, the Caputo–Fabrizio (CF) frac-
tional derivative, and the Atangana–Pagliano–Cabuto (ABC) fractional derivative. This
is context for investigating fractional processes and their use in evaluating the stability
of nonlinear systems[24]. A comprehensive understanding of the mathematical framework
used in the research requires integrating each of these concepts. In Section 2, we talk
about how important it is to do stability analysis in fractional systems. We note that
for stability, the roots of a characteristic polynomial must have negative real parts, which
is based on the Routh-Hurwitz criterion [7] [33]. The introduction gives a strong base
for the discussion by looking at fractional operators, dynamical Lyapunov functions, and
how important they are for figuring out the stability of nonlinear systems[1]. This ade-
quately prepares the reader for further study. The text defines the fractional Caputo and
Riemann-Liouville derivatives, which are fundamental to the description and behavior of
fractIt is emphasized how important the Mittag-Leffler function is in fractional calculus,
especially when looking at fractional systems, which can give us new information about
how the system works[35][32]. This allows us to examine the synchronization between the
Lorenz and Liou systems in Section 3, implying that comprehending these connections
could provide valuable insights into chaotic dynamics and their control. This research
begins with the definitions and properties of the fractional processes used in the study.
The text also mentions fractional hyperchain systems (FHCS). The integration of frac-
tional calculus into fractal hyperchain systems leads to significant advances in the design
of control systems. By modeling system dynamics more accurately, enhancing stability
and performance, and offering greater flexibility, (FHCS) provides a powerful framework
for developing advanced control strategies suitable for complex, real-world applications.
This leads to more effective and efficient control systems across a variety of engineering
disciplines. This benefits diverse fields such as network theory, control systems, signal
processing, biology, finance, thermodynamics, and materials science. [23][40]. In their
study of stability, the authors look into how important it is to set criteria for the presence
of dynamic Lyapunov functions at stable exponential equilibrium points. The goal is to
determine the asymptotic stability of the systems under investigation.[25].



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 3 of 16

2. Preliminaries

This section delineates the definitions and characteristics of fractional operators that
will be employed in our research.

Definition 1. [34] Consider a set L1 (r0,+∞) given that 0α is less than or equal to 1.
The formal definition of the Riemann-Liouville (RL) fractional integral of function f is
specified as:

RLIαr0g(r) =
1

Γ(α)

∫ r

r0

(r − x)α−1g(x)dx (1)

Let Γ(·) be the Gamma function.

Definition 2. [45] Consider a function g in the closed set H1 (r0,+∞) with 0α ≤ 1. The
Caputo (C) fractional derivative of the function f is defined as follows[29]:

CDα
r f(t) =

1

Γ(1− α)

∫ r

r0

f ′(x)

(r − x)α
dx (2)

Definition 3. [39] Let g ∈ H1 (r0,+∞) and 0 < α ≤ 1. An expression for the
Caputo-Fabrizio (CF) fractional derivative of function g is given as[12]:

CFDα
r g(r) =

1

2

B(α)(2− α)

1− α

∫ t

r0

f ′(x) exp

[
− α

1− α
(r − x)

]
dx (3)

Consider a normalizing function B(α) defined by the equation B(0) = B(1) = 1.
The definition of the fractional integral associated with the fractional derivative of CF is
provided under

CF
r Iαr0g(r) =

2(1− α)

B(α)(2− α)
g(r) +

2α

B(α)(2− α)
RL
r I1r0g(r) (4)

Definition 4. [37]Consider a function g in the closed set H1 (t0,+∞) with 0α ≤ 1.The
fractional derivative of function f according to the Atangana-Baleanu-Caputo (ABC)
method is expressed as:

ABC
r0 Dα

r g(r) =
B(α)

1− α

∫ r

r0

g′(x)Eα

[
− α

1− α
(r − x)α

]
dx (5)

A precise specification of the fractional integral that regulates the ABC fractional
derivative is presented.

AB
r Iαr0g(r) =

1− α

B(α)
f(r) +

α

B(α)
RL
t Iαr0g(r) (6)

Definition 5. [28] Let α be more than or equal to zero. A Mittag-Leffler function is
defined using two parameters, denoted as α and β[13].

Eα,β(z) =

∞∑
j=0

zj

Γ(αj + β)
, z ∈ C



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 4 of 16

Remark 1. [14] Assuming that β = 1, we may assert Assuming that β = 1, we may
assert

Eα,1(z) = Eα(z) =

∞∑
j=0

zj

Γ(αj + 1)

In this analysis, we examine the Mittag-Leffler function with a single parameter α.
Provided that α = β = 1, one may derive

E1,1(z) =
∞∑
j=0

zj

j!
= exp(z)

Theorem 1. [41] A characterization of The derivative of the Mittag-Leffler function is as
follows:

dEα,β

dz
(z) = E2

α,α+β(z)

3. Stability and the Routh-Hurwitz criteria of fractional-order

This section presents a nonlinear fractional-order system. This study employs an
enhanced direct Lyapunov approach using contemporary fractional derivative components
to analyzes the stability of fractional-order systems and the Rath-Hurtwitz criterion for
chaotic fractional-order systems. The stability of the hyperbolic equilibrium point in any
dynamical system with an integer derivative is ascertained by the signs of the real parts of
the eigenvalues of its Jacobian matrix[30]. If all eigenvalues of the Jacobian matrix possess
negative real portions, the hyperbolic equilibrium point is asymptotically stable[11]. This
is analogous to the Routh-Hurwitz criteria for algebraic methods. The Routh-Hurwitz
criteria is a recognized method for assessing the stability of linear systems represented as
ẋ = Λx(r), where x(r) ∈ Rn and Λ is a n × n real matrix, without resolving roots. This
criterion also assesses stability by examining the system’s characteristic equation, which
may be expressed as p(θ) = θn + a1θ

n−1 + a2θ
n−2 + ... + an = 0 where all coefficients ai

are real constants. The n Hurwitz matrices are defined as

M2 =

(
a1 1
a3 a2

)
,M3 =

 a1 1 0
a3 a2 a1
a5 a4 a3

 ,



a1 1 0
a3 a2 a1
a5 a4 a3

· · ·

...
. . .

...
...

...
0 . . .

0 0


when aj = 0 exceeds n. Many terms P (θ) possess negative real portions if and only if the
determinant of all Horowitz matrices is positive. Consequently, employing integer arith-
metic, a theoretical analysis is a crucial endeavor in fractional differential computation.



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 5 of 16

Matignon understands the subsequent fractional differential equations and formulae when
the order of the derivative lies between 0 and 1[31].
Theorem 2. [25] The fractional-order nonlinear system is characterised as

0D
α
r x(r) = g(r, x(r)) (7)

here g = (g1, g2, ..., gn)
T Let x(r) = (x1(r), x2(r), . . . , xn(r))

T ∈ Rn with initial value
x(0) = (b1, b2, . . . , bn)

T .The function gi(i = 1, 2, . . . , n) denotes continuous differential
nonlinear functions, whereas α indicates the fractional order of the derivative. The oper-
ator 0Dr denotes the Caputo (or Riemann-Liouville) fractional-order derivative operator.
Denotes the Caputo (or Riemann-Liouville) derivative. Assuming no loss of generality, let
x∗ = 0 be the equilibrium point, and let x(r) ∈ Rn. There exists a positive definite matrix
P that fulfils

xn(r)
T pg(x(r)) ≤ 0 (8)

then the origin of the system 8is asymptotically stable.
Lemma 1. [27]Let us consider a real continuous and differentiable function denoted as
u(r). Taking into account any t ≥ t0 and 0α ≤ 1, It is possible to infer the following:

ABC
r0 Dα

r

(
u2(r)

)
≤ 2u(r)ABC

r0 Dα
r u(r), (9)

CF
r0 Dα

r

(
u2(r)

)
≤ 2u(r)CF

r0 Dα
t u(r), (10)

C
r0D

α
r

(
u2(r)

)
≤ 2u(r)Cr0D

α
t u(r). (11)

Proposition 1.[8]

• If m1,m2,m3 are Routh–Hurwitz determinants which are defined as follows:

m1 = n1, m2 = n1n2n3, m3 = n1n2n− 3n2
1n4n

2
3

• If D(P ) > 0, n1 > 0, n2 < 0 and α > 2/3 then the equilibrium point Eis unstable.

• If D(P ) < 0, n1 > 0, n2 > 0, n3 > 0, n4 > 0, and α < 1/3, then the equilibrium point
E is locally asymptotically stable. Also, if D(P ) < 0, n1 < 0, n2 > 0, n3 < 0, n4 > 0,
then the equilibrium point Ēis unstable.

• IfD(P ) < 0, n1 > 0, n2 > 0, n3 > 0, n4 > 0andn2 = n1n4n3 + n3n1 , then the
equilibrium point Ēis locally asymptotically stable, for allα ∈ (0, 1).

• n4 > 0is the necessary condition for the equilibrium point Eto be locally asymptot-
ically stable.

then forα = 1, the equilibrium pointĒ of system 11 is locally asymptotically stable
if and only if m1 > 0,m2 > 0,m3 = 0, n4 > 0. (7) Moreover, the conditions of (7) are
sufficient conditions for the equilibrium pointĒ to be locally asymptotically stable for all
α ∈ [0, 1).



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4. Simulation and Analysis

This section analyzes the synchronization between the Lorenz and Liu systems, identi-
fying the Lorenz system as the main system and the Liu system as the responsive system.
It incorporates the proposed unified system by Lu et al. into the Lorenz, Chen, and Liu
systems, offering a thorough analytical framework. By examining the extensive equations
of the unified fractional degree system,[5].

4.1. Stabilization of Unified Chaotic System

In 2002, Lü et al. introduced the unified system 10, which integrates the Chen, Lü,
and Lorenz systems. The fractional representation is specified as follows. Chen and
Ueta discovered a similar yet separate chaotic attractor, identified as the counterpart of
the Lorenz system. Recently, the L¨u system was formulated, establishing a connection
between the Lorenz system and the Chen system. Lü et al. have recently created a
unified chaotic system that includes the Lorenz and Chen systems as two dual systems
at the boundaries of its parameter range. The written workThe system illustrates the
continuous shift from the Lorenz system to the Chen system and demonstrates chaotic
behavior throughout the spectrum of the crucial system parameter[26]. The fractional-
order unified chaotic system is a member of the fractional-order Lorenz chaotic system
family.The administration of various chaotic attractors within Lorenz system families is
enhanced by the use of a fractional order unified chaotic system. The fractional order
unified chaotic system is described as follows:

0D
α
r x1 = (25ϑ+ 10) (x2 − x1) ,

0D
α
r x2 = (28− 35ϑ)x1 + (29ϑ− 1)x2 − x1x3,

0D
α
r x3 = x1x2 − (ϑ+ 8)/3x3,

(12)

In this context, ϑ belongs to the set of real numbers, and 0 < α < 1. The system
is referred to as the Lorenz system when ϑ = 0 and as the Chen system when ϑ =



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 7 of 16

1. It denotes a continuous chaotic system as ϑ ranges from 0 to 1. We discern three
equilibrium positions inside the system: (0, 0, 0), (

√
72− 7ϑ2,

√
72− 7ϑ− 2ϑ2, 6ϑ), and

(−
√
72− 7ϑ2,−

√
72− 7ϑ− 2ϑ2,−6ϑ). Employing a fractional order of α = 0.9 and a

starting condition of (2,2,1), the unified system 13 exhibits chaotic behavior, characterized
by various equilibria as ϑ varies from 0 to 1. Figure 1 depicts an unstable equilibrium[6].

The controlled stabilization of a Unified Chaotic System 12 can be reformulated into
the form 13.

0D
α
r x(r) = Λx+ f(x(r))x(r)−Bx(r) (13)

where

Λ =

 −(25 + ϑ) 25 + ϑ 0
28− 35ϑ 29ϑ− 1 0

0 0 −8+ϑ
3

 ,

is a linear matrix

f(x(t)) =

 0
x1x3
x1x2


is a nonlinear term,

B =

 0
1
0


is the control matrix

4.2. Regulating The Erratic Dynamics of the Liu system

The Liu system is an obscure chaotic system that has intricate and chaotic dynamics
akin to the more renowned Lorenz and Chen systems. examined the fractional-order
chaotic Liu system, characterized as[4][36]:

c
0D

α
r x1(r) = −a(x2x1),

c
0D

α
r x2(r) = bx1 − cx1x3,

c
0D

α
r x3(r) = −dx3 + hx21

(14)

When the parameters are set to a = 10, b = 40, c = 10, d = 2.8, h = 4, andα = 1.05, system
13 demonstrates chaotic behavior, as illustrated in Figure 2. The controlled fractional-
order chaotic system14 may be reformulated into the structure of 14.

c
0D

α
r x(r) = Ax(r) + f(x(r))x(r)−Kx(r) (15)



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 8 of 16

where

Λ =

 −a 0 0
0 b 0
0 0 −d

 , f(x(r)) =

 0
−cx1x3
hx21

 x(r) =

 x1(r)
x2(r)
x3(r)


Considering the system’s chaotic characteristics and the constrained state variables,

we can easily determine that |x1(r)| < 2 using numerical simulation. Consequently, the
expression x⊤(r)g(x(r))x(t) = −ex1x

2
2 < 2x22 necessitates that c = 2. asserts that if

the matrix S = Λ⊤+Λ
2 − hI is negative definite, then the system 14 exhibits asymptotic

stability. The maximum eigenvalue of the matrix Λ⊤+Λ
2 is λmax = 2.5. If the feedback

gain matrix H = hI satisfies the condition h > 4.5, the system attains asymptotic stability
at the origin[26].

4.3. Challenges in Stabilising the Fractional-Order Hyperchaotic Chen
System (FHCS)

Fractional Hyper Chain Systems (FHCS) provide an innovative methodology for sta-
bility analysis in fractional differential systems, providing numerous benefits compared
to conventional techniques such as the Adomian Decomposition Method (ADM) and the
Variational Iteration Method (VIM). FHCS offers advanced modelling capabilities that
accurately represent the intricate behaviours of real-world systems characterised by non-
integer order dynamics. Its highly organised structure facilitates a more thorough ex-
amination of systems with interconnected components, especially beneficial in networked
systems or multi-agent situations. FHCS additionally provides comprehensive stability
analysis, using novel stability criteria that surpass conventional Lyapunov methods in
efficacy. This facilitates improved design and control tactics, augmenting versatility in
controller design. FHCS can streamline computing demands and offer analytical insights,
rendering it appropriate for various applications in engineering and applied sciences. The
study illustrates the practical applications of FHCS in addressing intricate issues in engi-
neering and applied sciences, connecting theory with practice.[22][21][16][10] [6]:

ẋ1 = a1 (x2 − x1) + x4,

ẋ2 = d1x1 − x1x3 + c1x2,

ẋ3 = x1x2 − b1x3,

ẋ4 = x2x3 + rx4

(16)

where a1, b1, c1, d1, k ∈ R. When a1 = 35, b1 = 3, c1 = 12, d1 = 7, and 0.085 < r ⩽ 0.798,
the system 16 is hyperchaotic[15]. Subsequently, we examine the fractional variant of this
system, which is represented by

0D
α
t x1 = a (x2 − x1) + x4,

0D
α
t x2 = dx1 − x1x3 + cx2,

0D
α
t x3 = x1x2 − bx3, 0

Dα
t x4 = x2x3 + rx4

(17)



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 9 of 16

where α denotes the fractional order. For the values a = 35, b = 3, c = 12, d = 7,
and r = 0.5, this system possesses a singular equilibrium point at (0, 0, 0, 0). In the
commensurate system, computer simulations indicate that the minimum order required to
produce hyperchaos is 0.946. The regulated fractional-order chaotic system (4.6) can be
reformulated as (3.4).

0D
α
t x(t) = Ax(t) + g(x(t))x(t)−Kx(t)

where

A =


−a a 0 1
d c 0 0
0 0 −b
0 0 0 r

 , g(x(t)) =


0 0 0 0
0 0 −x1 0
0 x1 0 0
0 0 x2 0


Given the chaotic nature of the system and the bounded state variables, we can readily
derive

x⊤(t)g(x(t))x(t) = x2x3x4 =
x2x3
x4

x24 ⩽ 43x24

through numerical simulation, namely, c = 43. Therefore, if the matrix S = Λ⊤+Λ
2 −hI

is negative definite, the system (4.7) is asymptotically stable. The maximal eigenvalue of

the matrix Λ⊤+Λ
2 is approximately λmax ≈ 20. If the feedback gain matrix H = hI meets

the condition > 63, the system is asymptotically stable at the origin.

max width0.85 max width

Figure 1: Chaotic attractors in the Lorenz system with α = 1 and chaotic attractors in
the Lorenz system with α = 0.94, respectively.



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 10 of 16

Figure 2:Chaotic attractors in Lü system with α = 1, Chaotic attractors in Lü system
with α = 0.94.



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 11 of 16

Figure 3: Chaotic attractors in hyperchaotic Chensystem with α = 1, Chaotic attractors
in hyperchaotic Chen system with α = 0.94. respectively.



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 12 of 16

Figure 4:Time series considering a Caputo Lorenz system withα = 1 and Time series
considering a Lü system system with α = 0.94 respectively

Figure 5: Time series considering a Lü system system withα = 1 Time series considering
a Lü system system with α = 0.94.

Figure 6: Time series considering a hyperchaotic Chen system system withα = 1 Time
series considering a hyperchaotic Chen system system with α = 0.94.



T. Khalid / Eur. J. Pure Appl. Math, 18 (1) (2025), 5576 13 of 16

5. Conclusion

The paper comprehensively investigates the synchronization of Lorenz and Liu systems,
and establishes a framework for understanding their dynamics using a unified chaotic sys-
tem. Fractal superchain systems (FHCS) are used as a framework for analyzing complex
systems characterized by fractal dynamics and highly ordered interactions[9]. It is shown
that the lowest fractal order is necessary to achieve superchaos, highlighting the impor-
tance of initial conditions in influencing the behavior of the system. The results improve
the understanding and stability of chaotic systems. This indicates the chaotic character of
the system and its use in chaos control[42][44]. The results demonstrate the potential for
better analysis of system models and identification of attractor chaos and hybrid systems
that integrate fractal chaotic dynamics and neural networks for control and prediction pur-
poses. Thus, the proposed methods can be used for other systems to obtain more accurate
results. These methods can provide numerical simulations and model solutions, which
enhance the control and reliability for future research and engineering applications[28][2].

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