EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1991-2004
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Analysis of Novel 4D Rabinovich-Fabrikant Continuous
Dynamical System with Coexistence Attractors

Maysoon M. Aziz1,∗, Ghassan E. Arif2, Ahmad T. Ahmad2

1 Department of Mathematics, College of Computers Sciences and Mathematics,
University of Mosul, IRAQ
2 Department of Mathematics, College of Education for Pure Sciences,
University of Tikrit,Tikrit, IRAQ

Abstract. In this paper a new Rabinovitch-Fabrikant (R-F) four dimensional (4D) continuous
time dynamical system was generated from three dimensional (3D) Rabinovitch-Fabrikant dynam-
ical system using the state augmentation technique by adding new state variables u. The sys-
tem employs thirteen terms includes five cross-product terms and one irreversible function. The
dynamical behaviors of the system were investigated which include equilibrium points, stability
analysis, wave form analysis, phase space analysis, multistability, Hopf-bifurcation, the Lyapunov
exponent and Lyapunov dimension. The values of Lyapunov exponents are:L1 = 14.025946, L2 =
0.295151, L3 = −2.854401, L4 = −13.736833. and Lyapunov dimension is (3.83474) , so the system
is unstable and hyperchaotic with coexistence attractors. Chaos was handled in two ways: adap-
tive control and adaptive synchronization, it was found that the new system is stable and achieved
good results.

2020 Mathematics Subject Classifications: 65K10, 49M37, 90C06

Key Words and Phrases: Stability, Multistability, Lyapunov exponent, Adaptive Control,
Synchronization

1. Introduction

The origin of nonlinear dynamics goes back to French scientists (Henri poincar) [14], [2].
Dynamic systems are evolutionary process, which are generally characterized by differential
equation including mechanics and fluid flow [17], [16], [18], [1], [5]. Chaos theory is now
attractive to various fields such as physics, chemistry, medicine, biology, mathematics, and
engineering [13], [3], [6], [7]. Chaos control has received widespread attention of research
because contrability of chaotic attractors are index of utility in very totally different designs
like in communication and arificial intelligence [9], [12], [8], [4]. The synchronization of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4857

Email addresses: aziz maysoon@uomosul.edu.iq (M. M. Aziz),ghasanarif@tu.edu.iq,
ahm.taha10@gmail.com (A. T. Ahmad)

https://www.ejpam.com 1991 © 2023 EJPAM All rights reserved.



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1992

chaotic system is a very energetic subject in non-linear science, this phenomenon happens
where two non-linear system are combined, there are different types of synchronization
phenomena including complete synchronization, linear feedback synchronization, Hybrid
synchronization, Anti- synchronization, etc [10]. Due to the increasing importance in
many fields at the present time to generate a high dimensional system, a new 4D (R-F)
dynamical system was generated and it seen from the comparison table 1 that the new 4D
(R-F) is chaotic system with unstable real equilibrium points, several tools were used to
analyze the new system. It was found that the new 4D (R-F) coexistence of self-existing
chaotic attractors Table 1, comparison of the new 4D (R-F) dynamical system with results
of published papers.

Table 1: Comparision on new R-F system with published papers. [20], [11], [19] and [15]

2. New 4D Rabinovitch-Fabrikant(R-F) System

To construct a 4D system from 3D continuous time dynamical system, state augmen-
tation technique [15], was used by adding new state variable(u), with irreversible function
(trigonometric function) term, to the 3D dynamical system, that exhibits similar dynam-
ics.

The new thirteen terms system (1) with six non-linearities terms, given by four differ-
ential equations with three parameters a, b, k and x, y, z, u are state variables.



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1993

Expressed as:
ẋ = y(z − 1 + x2) + ax

ẏ = x(3z + 1− x2) + ay

ż = −2z(b+ xy)

u̇ = a(x− y) + ksin(u)

(1)

where a = 0.077, b = 0.1, k = 4.

3. System Analysis

To analyze the system we follow the steps:

3.1. Equilibria

We set the system equal to zero and find the equilibrium points.
So, the equilibrium points of the system are : E0 = (0, 0, 0, 0),

E1,2 = (±1.5378,∓0.065,−0.26909,∓1.768), E3,4 = (±0.0438,∓2.2831, 1.7635,∓2.5673).

3.2. Analysis of stability

3.2.1. Characteristic equation

The Jacobian matrix of new system (1), at equilibrium point E0(0, 0, 0, 0) is:
0.077 −1 0 0
1 0.077 0 0
0 0 −0.2 4
0 0 −0.2 4


It’s characteristic equation

λ4 − 3.954λ3 + 0.791129λ2 − 3.6993302λ − 0.8047432 = 0. The roots of characteristic
equation are:

λ1 = 4, λ2 = −0.2, λ3 = 0.077 + i, λ4 = 0.077 − i since λ1 is positive and λ3, λ4 are
complex numbers with positive real part, Therefor the new (R-F) system (1) is not stable.

3.2.2. Criterion of Routh stability

Routh criteria states that new R-F system (1) is steady that each term in Routh Array
table’s first column are positive values, is necessary and sufficient condition.

a4 = 1, a3 = −3.95, a2 = 0.791129, a1 = −3.6993302, a0 = −0.8047432

b1 =
a3a2 − a4a1

a3
= 1.72672, c1 =

b1a1 − a3b2
b1

= −5.5421

Since the characteristic equation contains different signs and the first column contains
negative values. So, the new system (1) is unstable.



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1994

Table 2: Routh Arrays for new R-F system.

3.2.3. Hurwitz stability criteria

Using the coefficients of the characteristic equation, we form a square matrix in which the
number of rows and columns equals the degree of the equation:

∆1 = a3 = −3.954 < 0,∆2 =

∣∣∣∣ a3 a1
a4 a2

∣∣∣∣ = −6.827454266 < 0,∆3 =

∣∣∣∣∣∣
a3 a1 0
a4 a2 a0
0 a3 a1

∣∣∣∣∣∣ =

10.4683797,∆4 =

∣∣∣∣∣∣∣∣
a3 a1 0 0
a4 a2 a0 0
0 a3 a1 0
0 a4 a2 a0

∣∣∣∣∣∣∣∣ = 30.4502404. Since (∆1,∆2) < 0, the new system

(1) is unstable.

3.2.4. Lyapunov function

Let the Lyapunov function of new R-F system (1) is

V (x, y, z, u) =
1

2
(x2 + y2 + z2 + u2) (2)

V̇ (x, y, z, u) = xẋ+ yẏ + zż + uu̇ (3)

Substitute the new system (1) in equation (3) we get:

V̇ (x, y, z.u) = ax2 + ay2 − 2bz2 + 4xyz − 2xyz2 + axu− ayu+ ku sin(u) (4)

substitute the initial conditions in (4) we get: V̇ (x, y, z, u) > 0. Hence the system is
unstable.

3.3. Graphical numerical analysis

New system (1) has been solved by Runge-Kutta technique of fourth order with initial
values [x0, y0, z0, u0] = [4,−11, 18, 3].

3.3.1. Wave-form

The wave form x(t), z(t) for system (1) is characterized with a non-aperiodic shape, shown
in figure (1), which is one of the basic characteristic of chaotic dynamical system :



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1995

(a) (b)

Figure 1: The wave-form of new system (1): (1a)x− t, (1b)z − t.

3.3.2. Phase space of new system (1)

I this paragraph figure (2)(a,b) shows the chaotic attractors in (x−y), (x−z), (y−z), (z−u)
plane, and figure (3) in (x, y, z), (y, z, u) space:

(a) (b)

(c) (d)

Figure 2: 2-D for the chaotic attractor for new system (1) (2a)x − y(2b)x − z.(2c)y −
z.(2d)z − u.

3.4. Multistability

Multistability (coexisting attractors) of nonlinear dynamical system mean , by chang-
ing the initial condition for same parameters of the system , and changing the initial
condition with changing parameters is achieve the coexistence of many aspects according
to the table (2), as in figure (4).



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1996

(a) (b)

Figure 3: 3-D chaotic attractor for new system (1).

The presence of multistability does not necessarily indicate chaos, but it can provide
a favorable environment for the emergence of chaos in dynamical system. Multistability
can increase the complexity and richness of the system’s behavior and provide a basis for
emergence of more complex dynamics, such as chaos.

Table 3: Coexistence with different parameters and initial conditions.

(a) (b)

Figure 4: Multistability of three attractors with three initial conditions corresponding to
table (3).

3.5. Hopf-bifurcation

The Jacobian matrix(J) have pair of complex eigenvalues with modulus approximately
equal one, so the new system (1) has Hopf-Bifurcation. Analyzing new system (1) using
bifurcation diagrams. The bifurcation parameter (a) is changed interval [-0.5-0.05] with
the rest of the parameters fixed, shown in figure (5a), also the bifurcation parameter (b)



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1997

is changed with the rest of parameters fixed show in figure(5a). Which shows the chaotic
behavior of the system (1).

(a) (b)

Figure 5: Bifurcation diagram for the system (1).

3.6. Lyapunov Exponent

The Lyapunov exponent is simple way of describing the dynamics of a chaotic sys-
tem. The system is chaotic if at least one Lyapunov exponent greater than zero. The
values of Lyapunoc exponents are L1 = 14.025946, L2 = 0.295151, L3 = −2.854401, L4 =
−13.736833. and

∑4
i=1 Li = −2.270137therefore the Lyapunov dimension is DL = 3 +

L1+L2+L3
|L4| = 3.83474 . So, the system (1) is Hyper chaotic system since the necessary

condition are satisfied: Two Lyapunov exponent are positive from the four Lyapunov
exponents, as shown in figure (6).

Figure 6: Lyapunov Exponent of new system(1).



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1998

4. ADAPTIVE CONTROL TECHNIQUE

4.1. Theoretical results

To achieve the stability of the new hyperchaotic R-F system (1) by adaptive control
technology, with the undefined parameter (b).

ẋ = y(z − 1 + x2) + 0.077x+ v1(t)

ẏ = x(3z + 1− x2) + 0.077y + v2(t)

ż = −2z(b+ xy) + v3(t)

u̇ = ax− ay + k sin(u) + v4(t)

(5)

where (v1(t), v2(t), v3(t), v4(t) are feedback controllers and (x, y, z, u) are state variable.
The adaptive control functions are:

v1(t) = −yz + y − x2y − 0.077x− β1x

v2(t) = −3xz − x+ x3 − 0.077y − β2y

v3(t) = 2b̂z + 2xyz − β3z

v4(t) = −ax+ ay − k sin(u)− β4u

where the constants (β1, β2, β3, β4) > 0, and b̂ is the estimate parameter of (b): Substi-
tuting the adaptive control functions into (5), we get:

ẋ = −β1x

ẏ = −β2y

ż = 2(b̂− b)z − β3z

u̇ = −β4u

(6)

Let the parameter estimation error:

eb = b− b̂ (7)

Substituting (7) in (6) we obtain:

ẋ = −β1x

ẏ = −β2y

ż = −2ebz − β3z

u̇ = −β4u

(8)

Consider the Lyapunov function: V (x, y, z, u, e) = 1
2(x

2 + y2 + z2 + u2, e2b)
V is positive-definite on R5.
Differentiating V we get:

V̇ = xẋ+ yẏ + zż + uu̇+ ebėb (9)



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 1999

Also,

ėb = − ˙̂
b (10)

We substituting the equation (10) and the system (8) in equation (9) we get:

V̇ = −β1x− β2y − β3z − β4u− eb(2z
2 +

˙̂
b) (11)

Not that:
˙̂
b = −2z2 + β5eb (12)

Substitute (12) in (11) we obtain:

V̇ = −β1x
2 − β2y

2 − β3z
2 − β4u

2 − β5e
2
b (13)

where (β5) is positive.
We substitute (β1, β2, β3, β4.β5) and the initial value in equation (13) we get V̇ (x, y, z, u, eb) <

0. So, the following proposition 1 is proved.
Proposition 1: By adaptive control design, the chaotic system (5) with unknown pa-
rameter is stabilized for every initial value, where the estimated parameter is obtaind by
(12) and (β1, β2, β3, β4.β5) are greater than zero.

4.2. Numerical results

Simulation for controlled hyperchaotic system (8) with initial values x0 = 3, y0 =
−2, z0 = −5, u0 = 7, β1 = 10, β2 = 15, β3 = 20, β4 = 5, β5 = 10 and unknown parameter
(b̂ = 0.9). Figure (7) shown the controlled state path of new system:

Figure 7: Paths of state variable x, y, z, u for controlled system.

5. Strategy for adaptive synchronization

5.1. Theoretical results

This section adaptive synchronization of hyperchaotic system, when the parameter (b)
is unknown parameter.



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 2000

The drive system, we consider the chaotic new (R-F) dynamical system described by:

ẋ1 = y1z1 − y1 + x21y1 + 0.077x1

ẏ1 = 3x1z1 + x1 − x31 + 0.077y1

ż1 = −0.2z1 − 2x1y1z1

u̇1 = 0.077x1 − 0.077y1 + 4sinu1

(14)

where (x1, y1, z1, u1) are state variables. The response system, is given by:

ẋ2 = y2z2 − y2 + x22y2 + 0.077x2 + v1(t)

ẏ2 = 3x2z2 + x2 − x32 + 0.077y2 + v2(t)

ż2 = −2bz2 − 2x2y2z2 + v3(t)

u̇2 = 0.077x2 − 0.077y2 + 4sinu2 + v4(t)

(15)

where (v1(t), v2(t), v3(t), v4(t)) are the feedback control and (x2, y2, z2, u2).
Are state variables.
adaptive synchronization error given by e1 = x2 − x1, e2 = y2 − y1, e3 = z2 − z1, e4 =

u2 − u1.
Hence, the dynamical system of synchronization error:

ė1 = e2e3 + z1e2 + y1e3 − e2 + e21e2 + y1e1 + x21e2 + 0.077e1 + v1(t)

ė2 = 3(e1e3 + z1e1 + x1e3) + e1 − e31 + 0.077e2 + v2(t)

ė3 = −2be3 − 2(e1e2e3 + z1e1e2 + y1e1e3 + x1e2e3 + z1e1e2 + y1e1e3 + x1e2e3) + v3(t)

ė4 = 0.077e1 − 0.077e2 + 4sine4 + v4(t)
(16)

Defined adaptive control function:

v1(t) = −e2e3 − z1e2 − y1e3 + e2 − e21e2 − y1e1 − x21e2 − 0.077e1 − β1e1

v2(t) = −3(e1e3 + z1e1 + x1e3)− e1 − e31 − 0.077e2 − β2e2

v3(t) = 2b̂e3 + 2(e1e2e3 + z1e1e2 + y1e1e3 + x1e2e3 + z1e1e2 + y1e1e3 + x1e2e3)− β3e3

v4(t) = −0.077e1 + 0.077e2 − 4sine4 − β4e4
(17)

where (β1, β2, β3, β4) are positive real values, and (b̂) is the estimated value of the param-
eter (b).

Substituting (17) into (16) we get the dynamical system of synchronization error:

ė1 = −β1e1

ė2 = −β2e2

ė3 = −2(b− b̂)e3 − β3e3

ė4 = −β4e4

(18)



M. M. Aziz, G. E. Arif, A. T. Ahmad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1991-2004 2001

The parameter estimation error is:

eb = b− b̂ (19)

Substitute (19) into (18) we obtain synchronization error system is:

ė1 = −β1e1

ė2 = −β2e2

ė3 = −2ebe3 − β3e3

ė4 = −β4e4

(20)

To prove the stability of the system (20), Consider the quadratic Lyapunov function is
considered as:

V (e1, e2, e3, e4, ek) =
1

2
(e21 + e22 + e23 + e24 + e2b) (21)

which is a positive definite on R5. Not that:

ėb = − ˙̂
b (22)

Differentiating equation (21) and substituting the system (20) and (22) we get:

V̇ = −β1e
2
1 − β2e

2
2 − β3e

2
3 − β4e

2
4 − eb(

˙̂
b+ 2e3) (23)

Hence, the estimated parameter is:

˙̂
b = −2e3 + β5eb (24)

Where (β5) is a positive.
Substitute (24) into (23) we obtion:

V̇ = −β1e
2
1 − β2e

2
2 − β3e

2
3 − β4e

2
4 − β5e

2
b (25)

Which is a negative on R5. Hence, by Lyapunov stability its immediately that the param-
eter error and error synchronization decay exponentially to zero. Thus the proposition 2
is proved.
Proposition 2: The identical chaotic of the drive and the response systems, with the
unknown parameter (b) are synchronized for all initial value by adaptive control (17),
where the estimated given by (24) and constant βi, (i = 1, 2, 3, 4, 5) are positive, since
limi→∞ ei → 0 , Therefor the system (20) is stable.

5.2. Numerical results

By the 4th-order Runge-kutta method to solve the master system (14), slave sys-
tem (15), and synchronization error system. We take initial value of the system (14)
(x1(0), y1(0), z1(0), u1(0)) = [3, 6, 5, 12] and initial value of the system (15). (x2(0), y2(0), z2(0), u2(0)) =

[19, 10,−4, 5] ,the parameter unknown b̂ = 2.4 and βi = 6 for i = 1, 2, 3, 4, 5. Dynamic
synchronization error system shown in figure (8).



REFERENCES 2002

Figure 8: The convergent for system (16) with adaptive control (17).

6. Conclusion

This paper present the successful creation of a new 4D Rabinovich-Fabrikant dy-
namical system from a 3D version using the state augmentation technique. A compre-
hensive investigation of the system’s dynamical behaviors was conducted, revealing an
unstable hyperchaotic with coexistence self-excited attractors, the system multistability,
Hopf-bifurcation, wave form analysis and phase space analysis. Were meticulously ana-
lyzed. Importantly, the Lyapunov exponents are: L1 = 14.025946, L2 = 0.295151, L3 =
−2.854401, L4 = −13.736833. confirmed the system’s unstability. While the Lyapunov
dimension DL = 3.83474. underscord. It’s hyperchaotic state. Despite the inherent chaos,
the system’s stability was effectively attained through the implementation of adaptive con-
trol and adaptive synchronization techniques, illustration the potential of these methods
for controlling and managing complex dynamical systems.

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