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Research ArticleDynamics, Chaos Control, and Synchronization in
aFractional-Order Samardzija-Greller Population System withOrder
Lying in (0, 2)
A. Al-khedhairi,1 S. S. Askar,1,2 A. E. Matouk ,3 A. Elsadany,4
and M. Ghazel5
1Department of Statistics and Operations Researches, College of
Science, King Saud University, P.O. Box 2455,Riyadh 11451, Saudi
Arabia2Department of Mathematics, Faculty of Science, Mansoura
University, Mansoura 35516, Egypt3Department of Basic Engineering
Sciences, College of Engineering, Majmaah University, Al-Majmaah
11952, Saudi Arabia4Mathematics Department, College of Sciences and
Humanities Studies Al-Kharj, Prince Sattam Bin Abdulaziz
University,Al-Kharj, Saudi Arabia5Mathematics Department, Faculty
of Science, Hail University, Hail 2440, Saudi Arabia
Correspondence should be addressed to A. E. Matouk;
[email protected]
Received 28 January 2018; Revised 3 July 2018; Accepted 16 July
2018; Published 10 September 2018
Academic Editor: Matilde Santos
Copyright © 2018 A. Al-khedhairi et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
This paper demonstrates dynamics, chaos control, and
synchronization in Samardzija-Greller population model with
fractionalorder between zero and two. The fractional-order case is
shown to exhibit rich variety of nonlinear dynamics.
Lyapunovexponents are calculated to confirm the existence of wide
range of chaotic dynamics in this system. Chaos control in this
modelis achieved via a novel linear control technique with the
fractional order lying in (1, 2). Moreover, a linear feedback
controlmethod is used to control the fractional-order model to its
steady states when 0 < α < 2. In addition, the obtained
resultsillustrate the role of fractional parameter on controlling
chaos in this model. Furthermore, nonlinear feedback
synchronizationscheme is also employed to illustrate that the
fractional parameter has a stabilizing role on the synchronization
process in thissystem. The analytical results are confirmed by
numerical simulations.
1. Introduction
Dynamic analysis of engineering and biological models hasbecome
an important issue for research [1–10]. One of themost fascinating
dynamical phenomena is the existence ofchaotic attractors. Due to
the importance of chaos, it has beeninvestigated in various
academic disciplines [11–15]. Thesensitivity to initial conditions
which characterizes the exis-tence of chaotic attractors was first
noticed by Poincaré [16].
According to their potential applications in a wide varietyof
settings, fractional-order differential equations (FODEs)have
received increasing attention in engineering [17–23],physics [24],
mathematical biology [25-26], and encryptionalgorithms [27].
Moreover, FODEs play an important rolein the description of memory
which is essential in mostbiological models.
Chaos synchronization and control in dynamical systemsare
essential applications of chaos theory. They have becomefocal
topics for research since the elegant work of Ott et al. inchaos
control [28] and the pioneering work of Pecora andCarroll in chaos
synchronization [29]. Chaos control issometimes needed to refine
the behavior of a chaotic modeland to remove unexpected performance
of power electronics.Synchronization of chaos has also useful
applications tobiological, chemical, physical systems and secure
communi-cations. Furthermore, synchronization and control in
chaoticfractional-order dynamical systems have been investigatedby
authors [30–32].
The integer-order Samardzija-Greller population modelis a system
of ODEs that generalizes the Lotka-Volterra equa-tions and
expresses the behaviors of two species predator-prey population
dynamical system. This model was proposed
HindawiComplexityVolume 2018, Article ID 6719341, 14
pageshttps://doi.org/10.1155/2018/6719341
http://orcid.org/0000-0001-5834-4234https://doi.org/10.1155/2018/6719341
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by Samardzija and Greller in 1988 [33]. They had proved
theexistence of complex oscillatory behavior in this model. In1999,
chaos synchronization had firstly been investigated inthis model by
Costello [34]. Oancea et al. utilized this systemto achieve the
pest control in agricultural systems [35]. In2018, Elsadany et al.
proved the existence of generalizedHopf (Bautin) bifurcation in
this model [36]. Recently,some works investigating synchronization
in the fractional-order Samardzija-Greller model with order less
than onehave appeared [37-38]. Up to the present day,
dynamics,chaos control, and synchronization have not been
investi-gated in the fractional-order Samardzija-Greller system
withorder lying in (0, 2).
So, our objective in this paper is to investigate the
richdynamics and achieve chaos synchronization and control inthe
fractional-order Samardzija-Greller model with orderlying in (0,
2). Moreover, applying the fractional Routh-Hurwitz criteria [23,
39] enables us to illustrate the role ofthe fractional parameter on
synchronizing and controllingchaos in this model. Motivated by the
previous discussion,numerical verifications are performed to show
the existenceof wide range of chaotic dynamics in this model using
theaids of phase portraits and Lyapunov exponents.
2. Mathematical Preliminaries
Here, Caputo definition [17] is adopted as
Dα f t = Jl−α f l t , 1
where α ∈ℜ+, f l denotes the ordinary lth derivative of f t ,l
is an integer that satisfies l − 1 < α ≤ l, and the operator Jβ
isdefined by
Jβg t = 1Γ βt
0t − τ β−1g τ dτ, β > 0, 2
where Γ is defined as
Γ ς =∞
0tς−1e−tdt 3
Therefore, fractional modeling provides more accuracyin both
theoretical and experimental results of the ecologicalmodel which
are naturally related with long range memorybehavior which is very
important in modeling ecologicalsystems. Thus, increasing the range
of the fractional parame-ter α from the interval (0, 1) that is
commonly used in mostliteratures to the interval (0, 2) provides
greater degrees offreedom in modeling the population system. In
addition,increasing the interval of fractional parameter increases
thecomplexity in the system since the fractional parameter α isin a
close relationship with fractals which are abundant inecosystems.
Furthermore, as α is increased, more adequatedescription of the
whole time domain of the process isachieved and the system may show
rich variety of complexbehaviors such as sensitive dependence on
initial conditions.
Consider the following fractional-order system:
DαX = JX, 4
which represents a linearized form of the nonlinear system:
DαX =G X , 5
where 0 < α < 2, X ∈ℜn, G ∈ C Ψ,ℜn . Moreover, the
follow-ing inequalities [40]
arg λi >απ
2 , i = 1,… , n, 6
determine the local stability of a steady state X ∈ℜn of
thefractional-order system (4), where λi is any eigenvalue ofthe
matrix J. The stability region for α ∈ 0, 1 and α ∈ 1, 2is shown in
Figures 1(a) and 1(b), respectively. Moreover,stability conditions
and their applications to nonlinear sys-tems of FODEs were
investigated in [23, 41].
3. A Three-Dimensional Samardzija-Greller Model
A three-dimensional Samardzija-Greller model [33] isdescribed
as
dxdt
= 1 − y x + c − az x2,
dydt
= −1 + x y,
dzdt
= −b + ax2 z,
7
where the prey of the population is denoted by x and
thepredators of the population are denoted by y and z. Thepredators
y, z do not interact directly with one anotherbut compete for prey
x, and a, b, c are nonnegative param-eters used to discuss the
bifurcation phenomena in themodel as stated by Samardzija and
Greller in 1988 [33].Furthermore, the positive value of the
parameter a indi-cates that two different predators y and z consume
theprey x. However, if parameter a is vanished, a classicalversion
of Lotka-Volterra model is obtained. By replacingfirst-order
derivatives in system (7) with fractional deriva-tives of order 0
< α < 2, we obtain
Dαx = 1 − y x + c − az x2,Dαy = −1 + x y,Dαz = −b + ax2 z
8
So, a greater degree of accuracy can be obtained in
ourpopulation model by using the property of evolution (8)due to
the existence of fractional derivatives which isessential to the
proposed Samardzija-Greller model
2 Complexity
-
because of its ability to provide a realistic description ofthe
population model which involves processes with mem-ory and
hereditary properties. Furthermore, fractionalderivatives provide
greater degrees of freedom in modelingpredator-prey ecosystems and
closely related to fractalswhich are abundant in population models
as well as itsability to describe the whole time domain for the
process.
To obtain the steady states of system (8), we set Dαx = 0,Dαy =
0,Dαz = 0, then the system has three nonnegativeequilibria given as
E0 = 0,0,0 , E1 = 1, 1 + c, 0 , and E2 =m, 0, 1 +mc /ma , where m =
b/a.
4. Stability of System (8) with α Lying in (0, 2)
Assume that system (8) is written in the following form:
Dαx1 t = f1 x1, x2, x3 ,Dαx2 t = f2 x1, x2, x3 ,Dαx3 t = f3 x1,
x2, x3 ,
9
α ∈ 0, 2 . The characteristic polynomial of the steady stateX∗
of system (9) is given as
P λ = λ3 + r1λ2 + r2λ + r3 = 0 10
The discriminant of P λ is given by
D P = 18r1r2r3 + r21r22 − 4r3r31 − 4r32 − 27r23 11
If all roots of (10) satisfy the inequalities (6), then
thesteady state of the linearized form of system (9) is
locallyasymptotically stable (LAS). To discuss the local stability
ofX∗, we prove the following theorem:
Theorem 1. The steady state X∗ of system (9) is LAS if argλi J
X∗ > απ/2, i = 1,2,3, α ∈ 0, 2 , where J is the Jaco-bian matrix
computed at the steady state X∗.
Proof 1. Let si be a very small perturbation defined as si =xi −
x∗i , where i = 1,2,3. Therefore, (9), with derivatives inthe
Caputo sense, can be written as
Dαsi t = f i si + x∗i , 12
with initial values si 0 = xi 0 − x∗i . A linearization of
theabove equation, based on the Taylor expansion, can takethe
form
Dαs = Js, 13
where α ∈ 0, 2 , J is computed at the steady state X∗ = x∗1 ,x∗2
, x∗3 , and s = s1 s2 s3
T . Hence,
Dαs = PQP−1 s,
Dα P−1s =Q P−1s ,14
where Q = diag λ1, λ2, λ3 , λi is an eigenvalue of J, and P
isthe corresponding eigenvector. Suppose that γ = P−1s =γ1 γ2
γ3
T , then it follows that
Dαγ =Qγ,Dαγi = λiγi
15
The above system can easily be solved by the aid
ofMittag-Leffler functions as follows
γi t = 〠∞
k=0
tαkλkiΓ 1 + αk γi 0 , i = 1,2,3 16
If the eigenvalue λi satisfies the condition argλi J X∗ >
απ/2, then the perturbations si t are decreas-ing which implies
that X∗ is LAS.
Moreover, the fractional Routh-Hurwitz (FRH) stabilitycriterion
is provided by the following lemma:
�훼�휋/2 Re (�휆)
Im (�휆)
−�훼�휋/2
Unstable
Unstable
Stable
Stable
Stable
(a)
Stable
Unstable
Unstable Unstable
Unstable
StableRe (�휆)
�훼�휋/2
−�훼�휋/2
Im (�휆)
(b)
Figure 1: Stability and instability regions of system (4): (a)
the case 0 < α < 1; (b) the case 1 ≤ α < 2.
3Complexity
-
Lemma 1 (see [39]).
(i) The steady state X∗ is LAS iff
r1 > 0,r3 > 0,
r1r2 > r3,17
provided that the discriminant of (10) is positive.
(ii) If the discriminant of (10) is negative, and r1 ≥ 0,r2 ≥ 0,
r3 > 0 , then the steady state X∗ is LAS whenthe fractional
parameter α is less than 2/3, whileif fractional parameter α is
greater than 2/3, andr1 < 0, r2 < 0, then X∗ is not LAS.
(iii) If the discriminant of (10) is negative, and r1 > 0,r2
> 0, r1r2 = r3, then X∗is LAS when 0 < α < 1.
(iv) Point X∗ is LAS only if r3 > 0.
Hence, the following lemmas are provided.
Lemma 2 (see [23, 39]). If the discriminant of (10) is
negativeand r1r2 = r3, r1 > 0, r2 > 0, then the steady state
X∗ is LASjust when 0 < α < 1.
Lemma 3 (see [42]). Assume that system (5) is written in
theform
DαX t =CX t + h X t , 18
where α ∈ 0, 2 , C ∈ℜn×n is a constant matrix, and h X t isa
nonlinear vector function such that
limX t →0
h X tX t
= 0, 19
where represents the l2-norm, then the zero steady stateof
system (5) is LAS provided that arg λi C > απ/2,i = 1,… , n.
4.1. Stability Conditions for the Steady State E0. The
charac-teristic equation of the steady state E0 is described as
P λ = λ3 + bλ2 − λ − b = 0 20
It is easy to check that r3 = −b < 0, so by applying
thestability condition (iv), we conclude that the steady state E0of
system (8) is unstable.
4.2. Stability Conditions for the Steady State E1. The
charac-teristic equation of the steady state E1 is described as
P λ = λ3 + b − a − c λ2 + 1 + ac − bc + c λ+ c + 1 b − a = 0
21
So by utilizing the FRH conditions (i)–(iv), we obtain
thefollowing results:
(1) If the discriminant of (21) is negative; the steadystate E1
is LAS provided that a + c < b, b ≤ a + 1 +1/c, α < 2/3.
However, if a + c > b, b > a + 1 + 1/c, α > 2/3, the
steady stateE1 is unstable.
(2) The steady state E1 of system (8) is LAS only if b >
a.
Remark 1. The stability conditions (i) and (iii) are not
satis-fied for the steady state E1.
4.3. Stability Conditions for the Steady State E2. The
charac-teristic equation of the steady state E2 is described as
P λ = λ3 + 2 −m λ2 + 2b + 1 + 2mbc −m λ+ 2b mc + 1 1 −m = 0,
22
where m = b/a. So by utilizing the FRH conditions (i)–(iv),we
obtain the following results:
(1) When the discriminant of (22) is positive, the steadystate
E2 of system (8) is LAS iff
b < a,
c > 3ma − 2a − b − 2ab2mab23
(2) When the discriminant of (22) is negative, the steadystate
E2 is LAS provided that b < a, c ≥m − 2b − 1/2mb, α < 2/3.
However, the steady state E2 is unstableif b > 4a, c < m − 2b
− 1 /2mb, α > 2/3. Moreover, ifthe discriminant of (22) is
negative, b < 3a, then E2is LAS for all 0 < α < 1.
(3) The steady state E2 of system (8) is LAS only if b <
a.
5. Hopf Bifurcation in Samardzija-GrellerModel (8)
Although exact periodic solutions do not exist in auton-omous
fractional systems [43], some asymptotically peri-odic signals
converge to limit cycles have been observed bynumerical simulations
in many fractional systems. Theselimit cycles attract the nearby
solutions of such systems.Therefore, Hopf bifurcation (HB) in
autonomous fractionalsystems is expected to occur around an
equilibrium pointof such systems if it has a pair of complex
conjugate eigen-values and at least one negative eigenvalue.
However, deter-mining the precise bifurcation type is
difficult.
In Samardzija-Greller system (8), the occurrence of HB
isexpected around E1 = 1, 1 + c, 0 at the critical parameter
c∗ = 2 + 2 2 + tan2 απ/2
1 + tan2 απ/2 ,24
4 Complexity
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at which the transversality condition holds and 2 1 − 2< c
< 2 1 + 2 , a < b. For α = 0 98, a = 0 8, b = 3, and c =0 07,
system (8) converges to a limit cycle as shown inFigure 2.
6. Chaos in Samardzija-Greller Model (8)
Based on the algorithm given in [44] and using the parameterset
a, b, c = 8/10, 3, 3 , the maximum values of Lyapunovexponents
(MLEs) of system (8) are computed. The calcula-tions of the MLEs
give the values 0.21315, 0.038468, 0.0089,and 0.1562 when α = 1 1,
α = 1 05, α = 1 00, and α = 0 99,respectively. In Figure 3, it is
shown that the chaoticattractor of system (8) related to the
fractional-order caseis more complicated than the attractor related
to theinteger-order case.
Another parameter set a, b, c = 3, 4, 10 is used toexplore a
variety of complex dynamics and new chaoticregions in the
Samardzija-Greller model (7) and its corre-sponding
fractional-order form. The results are depicted inFigure 4 which
shows that the complex dynamics of thesystem exist in a wide range
of the fractional parameter α.These foundations are confirmed by
the calculation of maxi-mal Lyapunov exponents which are depicted
in Figure 5.
7. Chaos Control in System (8)
In the following, we will control chaos in system (8) when1 <
α < 2 using linear control technique and also we willdiscuss the
role of fractional parameter α on controllingchaos in
Samardzija-Greller model using a linear feedbackcontrol
technique.
7.1. Chaos Control of Samardzija-Greller System (8) with αLying
in (1, 2) via Linear Control. Assume that the
controlledSamardzija-Greller system is described by
Dαx = 1 − y x + c − az x2 − u1,Dαy = y x − 1 − u2,Dαz = −b + ax2
z − u3
25
Thus, system (25) is rewritten as
DαX t =CX t + h X t −U , 26
where X t = x y z T ,
C =1 0 00 −1 00 0 −b
27
is a constant parameter matrix, and the linear controller U
iswritten as
U =u1
u2
u3
=MKX, 28
1.41.2
10.8 0.7 0.8
0.9y x11.1 1.2
1.3 1.4
0
0.005
0.01
0.015
0.02
z
(a)
10080 9070603010 40 50200t
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x, y, z
x
y
z
(b)
Figure 2: The trajectory of system (8) with the parameter set a,
b, c = 0 8,3,0 07 and fractional parameter α = 0 98 is attracted by
a limitcycle: (a) view in xyz-space; (b) the solution versus
time.
5Complexity
-
where U is to be designed later, and h X t is also defined
as
h X t =−xy + cx2 − ax2z
xy
ax2z
29
Obviously,
limX t →0
h X tX t
= limX t →0
−xy + cx2 − ax2z 2 + x2y2 + a2x4z2
x2 + y2 + z2
≤ limX t →0
x2 −y + cx − axz 2 + y2 + a2x2z2
x2
= limX t →0
−y + cx − axz 2 + y2 + a2x2z2 = 0
30
So, according to Lemma 3, the zero steady state x − x∗,y − y∗, z
− z∗ of the controlled system (26) is LAS if the
linear control matrix is chosen such that arg λi C −U> απ/2,
i = 1,2,3.
The controlled Samardzija-Greller model (25) is numer-ically
integrated as a = 3, b = 4,c = 10, and α = 1 1 and linearcontrol
functions
u1 = k1 x − x∗ ,u2 = k2 y − y∗ ,u3 = k3 z − z∗ + ε x − x∗ ,
31
where k1, k2, k3 ≥ 0 and ε is a real constant. Thus,
theselection k1 > 1, k2 > 0, k3 > 0 ensures that the
conditionsarg λi C −U > απ/2, i = 1,2,3, hold. Figures
6(a)–6(c)show that system (26) is controlled to the steady states
E0,E1, E2 as using (k1 = 100, k2 = 150, k3 = 100, and ε = 100),(k1
= 100, k2 = 150, k3 = 150, and ε = 0 3), (k1 = 100, k2 =100, k3 =
150, and ε = 0 3), respectively.
7.2. Chaos Control of Samardzija-Greller System (8) via theFRH
Criterion. The FRH criterion is employed to controlchaos in system
(8) using linear feedback control techniquewhich is more easy of
implementation, cheap in cost, and
25
20
15
10
5
01.5
1
0.50 −2 0
24
x6 8
10 12
y
z
(a)
6
45
21
3
0y
z
x0
24
68
02468101214
(b)
05
43
21y
z
x0 0 510
15 2025
30
510152025303540
(c)
Figure 3: Phase diagrams of Samardzija-Greller model (8) with a
= 8/10, b = c = 3, and α equals: (a) 1.1, (b) 1.00, and (c)
0.99.
6 Complexity
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more appropriate of being designed in real-world
applica-tions.Moreover, wewill illustrate the role of fractional
param-eter α on controlling chaos in system (8). Consider
thefollowing system:
DαX = G X , 32
where α ∈ 0, 2 and X = x, y, z . System (32) has the con-trolled
form
DαX = G X − K X − X∗ , 33
where X∗ = x∗, y∗, z∗ is a steady state of (32) and K = diagk1,
k2, k3 , k1, k2, k3 ≥ 0 are the feedback control gains
10
8
6
4
y
0 0.5 1 x1.5 2.52
3 3.54.54
0
2
4
6
8
10
z
(a)
0.3
0.25
0.2
0.15
0.1
y
−1 0 12 3 4
5 6 78 9
x
0
2
4
6
8
10
12
14
16
z
(b)
02
5
10
15
20
z
1.5
1
0.5
y
00 2 4 x
6 8 1012 14
(c)
25
20
15
z
10
5
06
4
2
0
y
0 2 46
x8 10 12
14 16 18
(d)
15
10
5
z
06
5
4
3
2
1
y
0 1 23
x4 5 6 7
8 9 10
(e)
8
7
6
5
4
3
2
1
0
z
5
4.5
4
3.5
3
y
0 0.5 11.5 2 2.5
3 3.5 44.5
x
(f)
4.5
4
3.5
3
2.5
z
2
1.5
1
0.57
6
5
4
y
0.5 11.5
x
2 2.53 3.5
(g)
z
6
5
4
3
2
1
010
8
6
4
y
0 12
x3 4
5 6
(h)
6
5
4
3
2
1
0
z
10
8
6
4
y
0 12
x3 4
5 6
(i)
3
2.5
2
1.5z
1
0.510
8
6y
40.7 0.8 0.9
1 1.1 1.21.3 1.4 1.5
1.6
x
(j)
Figure 4: Phase diagrams of Samardzija-Greller model (8) with α
= 3, b = 4, c = 10, and α equals: (a) 1.1, (b) 1.05, (c) 1.00, (d)
0.98, (e) 0.95, (f)0.90, (g) 0.80, (h) 0.70, (i) 0.60, and (j)
0.45.
7Complexity
-
(FCGs) that are selected according to the FRH criterion insuch a
way that
limt→∞
X − X∗ = 0 34
So, controlled Samardzija-Greller system (8) is given as
Dαx = 1 − y x + c − az x2 − k1 x − x∗ ,Dαy = −1 + x y − k2 y −
y∗ ,Dαz = −b + ax2 − k3 z + k3z∗,
35
where 0 < α < 2 and X∗ = x∗, y∗, z∗ is a steady state of
sys-tem (8). Now, we are going to control chaotic system (8) tothe
steady state E1. Hence, system (35) becomes
Dαx = 1 − y x + c − az x2 − k1 x − 1 ,Dαy = −1 + x y − k2 y − 1
− c ,Dαz = −b + ax2 − k3 z
36
Then, system (36) has the following characteristic equa-tion
that is evaluated at the steady state E1:
P λ = λ3 + r1λ2 + r2λ + r3 = 0, 37
where
r1 = k1 + b − a + c ,r2 = 1 + c + ac − ak1 + bk1 − bc,r3 = 1 + c
b − a
38
System (36) is numerically integrated with a = 3, b = 4,c = 10,
α = 0 95, and k1 = 8 5, k2 = 1 5, k3 = 50. Consequently,the
fractional Routh-Hurwitz condition (iii) holds, that is,
thediscriminant of (37) is negative, r1, r2 are positive and r3
=r1r2. So according to Lemma 2, system (36) is stabilized to
the steady state E1 = 1,11,0 with the fractional parameterα only
in the interval 0, 1 . Figure 7(a) shows that the statesof
Samardzija-Greller model (36) approach the steady stateE1 = 1,11,0
. However, Figures 7(b) and 7(c) show that thestates of system (36)
are not stabilized to the steady stateE1 = 1,11,0 when α = 1 and α
= 1 1, respectively. Theseresults illustrate the role of the
parameter α on suppressingchaos in Samardzija-Greller model
(8).
Obviously, E0 satisfies the fractional Routh-Hurwitz con-dition
(i) as using the above selection of parameters andFCGs k1 > 1,
k2 = 1 5, k3 = 50 and k1 = 8 5, k2 > 0, k3 = 50.So, system (35)
is controlled to E0 for α = 0 95, k1 = 8 5,k2 = 1 5, k3 = 50 and
for α = 1 1, k1 = 8 5, k2 = 250, k3 = 50.The results are depicted
in Figure 8.
Also, E2 = 1 154700539,0,3 622008471 satisfies thefractional
Routh-Hurwitz condition (i) as using the aboveselection of
parameters and FCGs k1 > 0, k2 = 1 5, k3 = 50and k1 = 8 5, k2
> 0 1547005382, k3 = 50. Hence, system (35)is controlled to E2 =
1 154700539,0,3 622008471 for α =0 95, k1 = 8 5, k2 = 1 5, k3 = 50
and for α = 1 1, k1 = 8 5, k2 =250, k3 = 50. The results are
illustrated in Figure 9.
8. Chaos Synchronization of Samardzija-GrellerSystem (8) via
Nonlinear Feedback Control
The drive system is introduced as
Dαx1 = 1 − y1 x1 + c − az1 x21,Dαy1 = −1 + x1 y1,Dαz1 = −b +
ax21 z1,
39
and the response system is presented as
Dαx2 = 1 − y2 x2 + c − az2 x22 + v1,Dαy2 = −1 + x2 y2 + v2,Dαz2
= −b + ax22 z2 + v3,
40
where α ∈ 0, 2 and v1, v2, v3 are the nonlinear feedback
con-trollers. Then, we suppose that
e1 = x2 − x1,e2 = y2 − y1,e3 = z2 − z1
41
By subtracting (39) from (40) and using (41), we get
Dαe1 = e1 − x2e2 + y1e1 + c x2 + x1 e1− a x22e3 + z1 x2 + x1 e1
+ v1 e1, e2, e3 ,
Dαe2 = −e2 + x2e2 + y1e1 + v2 e1, e2, e3 ,
Dαe3 = −be3 + a x22e3 + z1 x2 + x1 e1 + v3 e1, e2, e342
Theorem 2. The drive chaotic system (39) and the response
1.10
0.05
0.1
0.15
0.2
0.25
Max
imal
lyap
unov
expo
nent
s
0.55 0.65 0.75 0.85 0.95 1.050.45Alpha
Figure 5: The MLEs of system (8) for different values of
theparameter α and a, b, c = 3, 4, 10 .
8 Complexity
-
chaotic system (40) are synchronized for 0 < α < 2, if the
con-trol functions are chosen as
v1 e1, e2, e3 = u − cv + aw − k1e1,v2 e1, e2, e3 = −u − k2e2,v3
e1, e2, e3 = −aw − k3e3,
43
where
u = x2e2 + y1e1,v = x2e1 + x1e1,w = x22e3 + x2z1e1 + x1z1e1,
44
and the FCGs must satisfy
k1 > 1,k2 > 0,k3 > 0
45
Proof 2. The Jacobian matrix of system (42) with the
control-lers (43) evaluated at the origin steady state is described
by
A 0,0,0 =1 − k1 0 00 −1 − k2 00 0 −b − k3
46
So, if we select k1 > 1, k2 > 0, k3 > 0, then all the
eigenvaluesof the Jacobian matrix (46) have negative signs. Thus,
accord-ing to condition (6), the origin steady state of system (42)
isLAS for 0 < α < 2. Consequently, the proof is
completed.
The fractional-order systems (39) and (40) are numer-ically
integrated using the system’s parameters a = 8/10,b = 3, and c = 3,
the controllers (43) with k1 = 20, k2 = 20,k3 = 20, and the
fractional parameters α = 0 99 and α = 1 1,respectively. The
results are depicted in Figure 10.
8.1. The Role of Fractional Parameter on Synchronizing Chaosin
System (8) via Nonlinear Feedback Control Method. To
−0.1
0
0.1
0.2
0.3
0.4
0.5x, y, z
10 20 30 40 50 60 70 80 90 1000t
(a)
−2
0
2
4
6
8
10
12
x, y, z
0 20 30 40 50 60 70 80 90 10010t
(b)
0
0.5
1
1.5
2
2.5
3
3.5
4
x, y, z
10 20 30 40 50 60 70 80 90 1000t
(c)
Figure 6: The states of the controlled Samardzija-Greller model
(25) converge to the steady state: (a) E0 = 0, 0, 0 , (b) E1 = 1,
11, 0 , and (c)E2 = 1 154700539, 0, 3 622008471 , when a = 3, b =
4, c = 10, and α = 1 1 and using the linear controllers (31).
9Complexity
-
explain the role of fractional parameter α on the
synchroniza-tion process of this model, we present the following
theorem:
Theorem 3. The trajectories of the drive chaotic
Samardzija-Greller model (39) asymptotically approach the
trajectoriesof the response chaotic Samardzija-Greller model (40)
justwhen the fractional parameter α is less than one, if the
controlfunctions are chosen as
v1 e1, e2, e3 = b − 1 − k1 e1 − e2 − ae3 + u − cv + aw,v2 e1,
e2, e3 = 1 + c e1 + 1 − k2 e2 − u,v3 e1, e2, e3 = −aw − k3e3,
47where a = 3, b = 4, c = 10, k1 = 3 9, k2 = 0 1, and k3 =
10.
Proof 3. The Jacobian matrix of system (42) with the
control-lers (47) evaluated at the origin steady state is described
by
A 0,0,0 =b − k1 −1 −a1 + c −k2 00 0 −b − k3
48
and has the following characteristic equation:
P λ = λ3 + a1λ2 + a2λ + a3 = 0, a1 = k1 + k2 + k3,
a2 = −b2 + b k1 − k3 + c + 1 + k1k2 + k2k3 + k1k3,a3 = b + k3 c
+ 1 − b2k2 + bk2 k1 − k3 + k1k2k3
49
0
5
10
15
20
25
30
35x, y, z
5 10 15 20 25 30 35 400t
xyz
(a)
−5
0
5
10
15
20
x, y, z
10 20 30 40 50 60 800 70t
xyz
(b)
−5
0
5
10
15
20
x, y, z
80100 20 30 60 705040t
xyz
(c)
Figure 7: The states of the controlled Samardzija-Greller model
(36) (a) converge to the steady state E1 = 1, 11, 0 when α = 0 95,
(b) are notstabilized to the steady state E1 = 1, 11, 0 when α = 1,
(c) are not stabilized to the steady state E1 = 1, 11, 0 when α = 1
1, using theparameter values a = 3, b = 4, and c = 10 and FCGs k1 =
8 5, k2 = 1 5, and k3 = 50.
10 Complexity
-
Using the parameters a = 3, b = 4, and c = 10 and the
con-trollers (47) along with the selection k1 = 3 9, k2 = 0 1, k3 =
10, the characteristic equation (49) satisfies the
fractionalRouth-Hurwitz condition (iii), since its discriminant is
nega-tive, a1 ∈ℜ+, a2 ∈ℜ+, and a3 = a1a2. So according to Lemma2,
the origin steady state of system (42) is LAS only forall α ∈ 0, 1
. Consequently, the states of Samardzija-Greller model (39)
asymptotically approach the states ofSamardzija-Greller model (40)
as the controllers (47) areemployed. Since the characteristic
equation (49) has a pair
of purely imaginary eigenvalues, the condition of
asymptoticstability of the origin steady state 0,0,0 of (42) is not
satis-fied as 1 ≤ α < 2. Thus, the theorem is now proved.
Therefore, as the parameter α is decreased, the
nonlinearfeedback control technique has a stabilizing effect on
thesynchronization process in the chaotic
Samardzija-Grellersystems. These results are illustrated in Figure
11.
On the other hand, the origin steady state 0,0,0 of
(42)satisfies the fractional Routh-Hurwitz condition (i) as usinga
= 3, b = 4, and c = 10 and FCGs k1 > 3 9, k2 = 0 1, k3 = 10.
xyz
0
2
4
6
8
10
12
14x, y, z
5 10 15 20 25 30 35 400t
(a)
xyz
−5
0
5
10
15
20
25
30
x, y, z
0 20 30 40 50 60 70 80 9010 100t
(b)
Figure 8: The trajectories of Samardzija-Greller model (35) with
a = 3, b = 4, and c = 10 are controlled to E0 = 0, 0, 0 as (a) α =
0 95, k1 = 8 5,k2 = 1 5, and k3 = 50; (b) α = 1 1, k1 = 8 5, k2 =
250, and k3 = 50.
0
1
2
3
4
5
6
7
x, y, z
5 10 15 20 25 30 35 400t
xyz
(a)
0
1
2
3
4
5
6
7
8
x, y, z
5 10 15 20 25 30 35 400t
xyz
(b)
Figure 9: The trajectories of Samardzija-Greller model (35) with
a = 3, b = 4, and c = 10 are controlled to E2 = 1 154700539, 0, 3
622008471as (a) α = 0 95, k1 = 8 5, k2 = 1 5, and k3 = 50; (b) α =
1 1, k1 = 8 5, k2 = 250, and k3 = 50.
11Complexity
-
So, systems (39) and (40) are synchronized for α = 1 1, k1 =15,
k2 = 0 1, k3 = 10 as shown in Figure 12.
9. Concluding Remarks
In this paper, nonlinear dynamics, conditions of chaos con-trol,
and synchronization are studied in the fractionalSamardzija-Greller
population model with fractional orderbetween zero and two. To the
best of the authors’ knowl-edge, dynamical behaviors and chaos
applications in thismodel have not been investigated elsewhere when
the
fractional order lies between zero and two. This kind ofstudy
has great importance to predator-prey ecosystemssince it provides
greater degrees of freedom in modelingsuch systems. In addition,
increasing the interval of frac-tional parameter helps to raise the
complexity in the systemsince the fractional parameter is in a
close relationship withfractals which are abundant in ecosystems.
Furthermore, asthe fractional parameter α is increased, more
accuratedescription of the whole time domain of the process
isachieved and the system will show rich dynamics such asthe
existence of chaos.
1
e1e2e3
2 3 4 5 6 7 8 9 100t
−0.1
−0.05
0
0.05
0.1
0.15
0.2Sy
nchr
oniz
atio
n er
rors
(a)
e1e2e3
−0.05
0
0.05
0.1
0.15
0.2
Sync
hron
izat
ion
erro
rs
1 2 3 4 5 6 7 8 9 100t
(b)
Figure 10: The synchronization errors (41) converge to zero as
using the parameter values a = 0 8 and b = c = 3, the controllers
(43), the FCGsk1 = k2 = k3 = 20, and fractional order (a) α = 0 99
and (b) α = 1 1.
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Sync
hron
izat
ion
erro
rs
5 10 15 20 25 35 35 400t
e1e2e3
(a)
10 20 30 40 50 60 70 80 90 1000t
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Sync
hron
izat
ion
erro
rs
e1e2e3
(b)
Figure 11: The synchronization errors (41) with a = 3, b = 4,
and c = 10 and the controllers (47) with FCGs k1 = 3 9, k2 = 0 1,
and k3 = 10 (a)converge to zero when α = 0 95 and (b) oscillate
about zero when α = 1 0.
12 Complexity
-
By using the fractional Routh-Hurwitz criterion, localstability
in this model has been demonstrated as α ∈ 0, 2 .Moreover, the
chaoticity of this model has been numericallyexamined by
calculating the maximal Lyapunov exponentsusing the Wolf’s
algorithm. The calculations show that chaosexists in this system
when the fractional parameter lies in theintervals (0, 1) and (1,
2).
Chaos control in this system has been achieved via anovel linear
control technique as α ∈ 1, 2 . Furthermore,the role of fractional
parameter α on controlling chaos in thismodel has also been
illustrated using a linear feedback con-trol technique. It has been
proved that using the appropriateFCGs, the trajectory of
Samardzija-Greller population modelis stabilized to the steady
state E1 = 1, 1 + c, 0 as the frac-tional parameter α lies between
zero and one.
Chaos synchronization has been achieved in this systemvia
nonlinear feedback control method. It has also beenshown that,
under certain choice of nonlinear controllers,the fractional-order
Samardzija-Greller model is synchro-nized only when the fractional
parameter α is less than one.Thus, the obtained results illustrate
that the parameter αhas also a stabilizing role on the
synchronization process inthis system.
All the analytical results have been verified using numer-ical
simulations.
Data Availability
The data used to support the findings of this study are
avail-able from the corresponding author upon request.
Conflicts of Interest
The authors declare that this article content has no conflictof
interest.
Acknowledgments
This work is supported by the Deanship of Scientific Researchat
King Saud University, Research group no. RG-1438-046.
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