Bifurcation Behaviour and Control on Chaotic Convection of...

Bifurcation Behaviour and Control on Chaotic Convection ofNanofluids with Fractional-Orders

KHALED MOADDY1, AHMAD RADWAN2,3, JADALLAH JAWDAT1 and ISHAK HASHIM1

1University Kebengsaan MalaysiaSchool of Mathematical Sciences

43600 UKM Bangi Selangor, [email protected]

2 Engineering Mathematics Dept., Cairo University, Egypt3 NISC research center, Nile University, Egypt

[email protected]

Abstract: In this paper, we study the effect of the fractional-order chaotic behaviour of nanofluids on in a fluid layerheated from below. Adams-Bashforth-Moulton predictor-corrector method was adopted to solve the fractionalnonlinear system. The synchronization based on the active control theory and Lyapunov stability theory and theeffective chaotic range of the fractional-order chaotic system for variation of the single control parameter have beendetermined. The transition to chaos occurs by a subcritical Hopf bifurcation in this fractional-order system. Theresults show that inhibition of chaotic convection with fractional-order can be observed when using nanofluids.Numerical simulations are provided to illustrate the effectiveness of the synchronization results derived in thispaper.

Key–Words: Fractional differential equations, Chaos, Bifurcation, Synchronization

1 Introduction

During the last few decades, there has been growinginterest in investigating the chaotic behavior and dy-namics of fractional order dynamic systems. Frac-tional order calculus has become a powerful tool todescribe the dynamics of complex systems which ap-pear frequently in several branches of science, engi-neering, viscoelasticity, robotics, feedback amplifiers,electrical circuits, control theory, electro analyticalchemistry, fractional multi-poles, chemistry and bio-logical sciences [1, 2, 3, 4, 5, 6, 7]. The scientificcommunity has found a large variety of applicationswhich can be modeled and more clearly understoodby using this branch of mathematics.

There is a great need to control or obtain accu-rate numerical results as chaos theory plays an im-portant role in industrial applications particularly inchemical reactions, biological systems [8], informa-tion processing, secure communications [9], electron-ics [10, 11], and with memristors [12]. Much attentionhas been devoted to the search for better and more effi-cient methods for the control or determination of a so-

lution, approximate or analytical, of chaotic systems.Recently, Chaotic convection behaviour has at-

tracted interest due to its wide application [13, 14, 15,16]. It has been observed in many natural systems,such as the time evolution of the magnetic field of ce-lestial bodies, molecular vibrations, the dynamics ofsatellite in the solar system, the weather, in ecologyand in neurons.

Very recently, Jawdat and Hashim [17] investi-gated the influence of nanofluids on chaotic convec-tion in a fluid layer heated from below. The truncatedGalerkin approximation was applied to the governingequations to deduce an autonomous system with threeordinary differential equations. Nanofluids, term pro-posed by Choi [18], are mixtures of base fluid suchas water or ethylene-glycol with a very small amountof nanoparticles, having dimensions from 1 to 100nm[19].

The synchronization of fractional chaotic sys-tems has began to attract much attention and hasraised up some problems [20, 21]. The consistencyof the improvement of models based on fractional-order differential structure have had increased repu-

Recent Advances in Mathematical Methods and Computational Techniques in Modern Science

ISBN: 978-1-61804-178-4 63

tation in the research of dynamical systems [22, 23].Yu et al. studied the synchronization of three chaoticfractional-order Lorenz systems with bidirectionalcoupling [24], Odibat et al. [25] investigated the chaossynchronization of two identical systems via linearcontrol, Bhalekar and Daftardar-Gejji [26] demon-strated that two different fractional order chaotic sys-tems can be synchronized using active control.

There are two aims for this paper, where thefirst aim is to study the proper fractional-orders rangewhich exhibit chaotic behavior for the chaotic systemof fractional orders. Several cases are investigatedfor different orders and changing only a single sys-tem parameter. Stable, periodic and chaotic responsesare shown for each system parameter but with dif-ferent fractional order ranges. The second aim is toinvestigate the synchronization of different variableand fractional-order chaotic systems while the effectof the variable and fractional orders has been studied,the numerical solutions of the master, slave and errorsystems using Adams-Bashforth-Moulton predictor-corrector algorithm are proposed.

2 Problem FormulationConsider an infinite horizontal fluid layer subjectto gravity and heated from below with influence ofnanofluids. A Cartesian co-ordinate system is usedsuch that the vertical axis z is collinear with gravity,i.e. eg = −ez . The thermophysical properties of thenanofluids, considered in this study, given in Table (1),are assumed constant except for the density variation,which is determined based on the Boussinesq approx-imation and effected only by the gravity term in themomentum equation. Also, it is assumed that the basefluid and the nanoparticles are in thermal equilibriumand no slip occurs between them. Subject to theseconditions, the governing equations can be written as

∇ · V∗ = 0, (1)[∂V∗∂t∗

+ V∗ · ∇V∗]

=−1

ρnf∇p∗ + νnf∇2V∗

−(ρβ)nfρnf

−→g (T∗ − Tc) , (2)

∂T

∂t∗+ V∗ · ∇T = αnf∇2T, (3)

where V∗, T , and p∗ are the velocity, temperature,and pressure respectively. The effective density of the

Table 1: Thermophysical properties of water andnanoparticles [27]

H2O Ag Al2O3

ρ(kgm−3) 997.1 10500 3970Cp(Jkg

−1K−1) 4179 235 765k(Wm−1K−1) 0.613 429 40β × 105(K−1) 21 1.89 0.85

Table 2: Values of v, β and α for nanoparticlesv β α

Ag 0.770 0.675 1.181Al2O3 0.989 0.834 1.116

nanofluid, ρnf , is given as:

ρnf = (1− φ)ρf + φρnp, (4)

and φ is the solid volume fraction of nanoparticles.The thermal diffusivity of the nanofluid is:

αnf = knf/(ρCp)nf , (5)

where, the heat capacitance of nanofluid is given by:

(ρCp)nf = (1− φ)(ρCp)f + φ(ρCp)np, (6)

The thermal expansion coefficient of nanofluid can bedetermined by:

(ρβ)nf = (1− φ)(ρβ)f + φ(ρβ)np, (7)

The effective dynamic viscosity of the nanofluid isgiven by:

µnf = µf/(1− φ)2.5, (8)

The thermal conductivity of the nanofluid can be de-termined by:

knfkf

=knp + (n− 1)kf − (n− 1)φ(kf − knp)

knp + (n− 1)kf + φ(kf − knp), (9)

where n is an empirical shape factor for the nanopar-ticle. In particular n = 3/2 for cylindrical particlesand n = 3 for spherical ones (see [27]). In the present


ISBN: 978-1-61804-178-4 64

work n is set equal to 3 such that the results are re-stricted to spherical nanoparticles.

The following transformations will non-dimensionalize Eqs. (1)–(3):

V =H∗αf

V∗, p =H2∗

α2f

p∗, t =αfH2∗t∗, x =

x∗H∗

, (10)

y =y∗H∗

, z =z∗H∗

, T∆Tc = T∗ − Tc (11)

where t is the time, (T∗ − Tc) is the temperature vari-ations and ∆Tc = (TH − Tc) is the characteristictemperature difference. The fluid layer with stress-free horizontal boundaries is considered. Hence, thesolution must follow the impermeability conditionsV · en = 0 and the stress free condition ∂u/∂z =∂v/∂z = ∂2w/∂z2 = 0 on these boundaries, whereen is a unit vector normal to the boundary. The tem-perature boundary conditions are: T = 1 at z = 0,T = 0 at z = 1. The governing equations can berepresented in terms of a stream function defined byu = −∂ψ/∂z andw = ∂ψ/∂x, as for convective rollshaving axes parallel to the shorter dimension (i.e. y)when v = 0. Applying the curl (∇×) operator on Eq.(2) yields the following system of partial differentialequations from Eqs. (1)–(3):[

1

Pr

(∂

∂t− ∂ψ

∂z

∂

∂x+∂ψ

∂x

∂

∂z

)− ν∇2

](∇2ψ)

= βRa

(∂T

∂x

)(12)

∂T

∂t− ∂ψ

∂z

∂T

∂x+∂ψ

∂x

∂T

∂z= α

(∂2T

∂x2+∂2T

∂z2

)(13)

where Pr = νf/αf , Ra = βf∆Tcg∗H3∗/αfνf ,

are the Prandtl number, the Rayleigh number , respec-tively, and

ν =νnfνf

, β =(ρβ)nfρnfβf

, α =αnfαf

, (14)

, see Table (2), and the boundary conditions for thestream function are ψ = ∂ψ

∂z = 0 on the horizontalboundaries. The set of partial differential equations,(12) and (13), form a nonlinear coupled system andtogether with the corresponding boundary conditionswill accept a basic motionless conduction solution. Inorder to obtain the solution to the nonlinear coupledsystem of partial differential equations in (12) and(13), we represent the stream function and tempera-ture in the form

ψ = A11 sin(κx) sin(πz), (15)

T = 1− z +B11 cos(κx) sin(πz) +B02 sin(2πz) (16)

This representation is equivalent to a Galerkin expan-sion of the solution in both the x- and z-directions.Unlike in the works of Vadasz [28], we rescale thetime and amplitudes with respect to their convectivefixed points of the form

X =A11√

βαλν

(R− αν

β

) , (17)

Y =B11(

νβ

)√βαλν

(R− αν

β

) , (18)

Z =−B02

(R− ανβ

), (19)

we have the following system of ordinary differentialequations

X = Prν(Y −X), (20)

Y =

(Rβ

ν

)X − αY −

(β

ν

)(R− αν

β

)XZ, (21)

Z = αλ(XY − Z), (22)

where R = RaRac

, Rac = (κ2+π2)3

κ2, λ =

8[(κ/κcr)2+2]

and the primes () denote time deriva-tives d()/dτ . System (20)–(22) are equivalent to theLorenz equations [29, 30], although with different co-efficients. By using the wavenumber corresponding tothe convection threshold, i.e. κcr = π√

2, in the defini-

tions of λ and Rac yields λ = 8/3.Fig.1 shows how the value of R affect the sys-

tem behaviour as shown from the X − Y projectionwhen the time step, and final time equal to 0.005Sec.,and 50Sec. respectively and for the integer order case.It is clear that when R is small the system exhibitssteady state response, and as R increase, the chaoticbehaviour appears for wide range as shown from Fig.1 then periodic responses can be obtained.

3 Numerical Method for SolvingFractional-Order System

Adams-Bashforth-Moulton predictor-corrector algo-rithm is proposed by Diethelm et al [31, 32] to solvefractional-order differential equations as follows


ISBN: 978-1-61804-178-4 65

-1.2 -1 -0.8-1.5

-1

-0.5

Y

-2 0 2-5

0

5

Y

-2 0 2-5

0

5

Y

X

-2 0 2-5

0

5

10

Y

-2 0 2-5

0

5

Y

-2 0 2-5

0

5

Y

X

R=100

R=150

R=30 R=200

R=225

R=250

Figure 1: The X − Y projection for different val-ues of R = 30, 100, 150, 200, 225, 250 of the nu-merical solutions Ag case for the integer order caseα1 = α2 = α3 = 1.

Dα(t) = f(t, y(t)), 0 ≤ t ≤ T (23)

yk(0) = y0(k), k = 0, 1, 2, . . . dαe − 1, (24)

where dαe denotes the integer part of α. Consider auniform grid {tj = jh : j = 0, 1, 2, . . . N} whereh = T/N , Eqs.(23) and (24) are equivalent to theVolterra integral equation [33]

y(t) =n−1∑k=0

y0(k) t

k

k!

+1

Γ(α)

t∫0

(t− τ)α−1f(τ, y(t))dτ (25)

the predictor-corrector formula for(3) is given as fol-lows

y(t) =

dαe−1∑k=0

y(k)0

tkn+1

k!+

hα

Γ(α+ 2)f (tn+1, y

ph(tn+1))

+hα

Γ(α+ 2)

n∑j=0

aj,n+1f(tj , yh(tj)) (26)

where nα+1 − (n− α)(n+ 1)α, j = 0 and (n− j +2)α+1 + (n − j)α+1 − 2(n − j + 1)α+1 when 1 ≤

j ≤ n. Using the one-step Adams-Bashforth-Moultonmethod, the predictor is defined as follows

yph(tn+1) =n−1∑k=0

y(k)0

tkn+1

k!+

1

Γ(α)

n∑j=0

bj,n+1f(tj , yh(tj)) (27)

where bj,n+1 = hα

α ((n− j + 1)α − (n− j)α) when0 ≤ j ≤ n. Now, system (20)–(22) can be written asa system of fractional order as follows

Xn+1 = X0 +hα1

Γ(α1 + 2) n∑j=0

a1,j,n+1 [Prv(Yj −Xj)]

+[Prv(Y p

n+1 −Xpn+1)

])(28)

Yn+1 = Y0 +hα2

Γ(α2 + 2) n∑j=0

a2,j,n+1

[R

(β

v

)Xj − αYj

−(β

v

)(R− αv

β

)XjZ1

]

+

[R

(β

v

)Xpn+1 − αY

pn+1

−(β

v

)(R− αv

β

)Xpn+1Z

pn+1

])

Zn+1 = Z0 +hα3

Γ(α3 + 2) n∑j=0

a3,j,n+1 [αλ(YjXj − Zj)]

+[αλ(Y p

n+1Xpn+1 − Z

pn+1)

])(29)

where

Xpn+1 = X0 +

1

Γ(α1)

n∑j=0

b1,j,n+1 [Prv(Yj −Xj)]

(30)

Y pn+1 = Y0 +1

Γ(α2)

n∑j=0

b2,j,n+1

[R

(β

v

)Xj − αYj

−(β

v

)(R− αv

β

)XjZ1

])(31)


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Figure 2: Block diagram of the proposed system

Zpn+1 = Z0 +1

Γ(α3) n∑j=0

b3,j,n+1 [αλ(YjXj − Zj)]

(32)

where the estimated error of this method is e =max |X(tj) − Xh(tj)| = Ø(hp) where p =min(2, 1 + α), j = 1, 2, · · · , N.

4 Synchronization NumericalMethod for Solving Fractional-Order System

In this section we provide a general technique for anychaotic system to adapt its response and follow an-other chaotic response. Fig. 2 shows the general blockdiagram which describes the proposed technique. As-sume we have two different chaotic systems, one ofthem is the master system, and the other is the slave.We need to change the response of the slave systemto synchronize with the master chaotic system via ac-tive control functions. These functions affect only theslave system without making any loading on the mas-ter chaotic response. When ν = β = α = 1, (i.e.φ = 0), system (20)–(22) reduces to the Vadasz sys-tem [28]. We define the drive (master) and response(slave) systems as follows

Dα1X1 = Pr(Y1 −X1) + S1ux(t), (33)Dα2Y1 = R1X1 − Y1 − (R1 − 1)X1Z1 + S1uy(t), (34)

Dα3Z1 = αλ(Y1X1 − Z1) + S1uz(t). (35)

Dα1X2 = Prv (Y2 −X2)− S2ux(t), (36)

Dα2Y2 = R2

(β

v

)X2 − αY2 − S2uy(t)

−(β

v

)(R2 −

αv

β

)X2Z2 (37)

Dα3Z2 = αλ (Y2X2 − Z2)− S2uz(t). (38)

where S1 and S2 are on-off parameters (digital bit)which either are ”1” or ”0” according to the requireddependence between both systems as shown in Fig.2.The unknown terms (ux, uy, uz) are active controlfunctions to be determined, and define the error func-tions as

ex = X2 −X1, ey = Y2 − Y1, ez = Z2 − Z1, (39)

Eq. (39) together with (33) and (36) yields the errorsystem

Dα1ex = Prv(ey − ex) + (Prv − Pr)(Y1 −X1)

+(S2 + S1)ux(t), (40)

Dα2ey = R2

(β

v

)ex +

(R2

(β

v

)−R1

)X1 − αey

+(1− α)Y1 −(β

v

)(R2 −

αv

β

)X2Z2

+(R1 − 1)X1Z1 + (S2 + S1)uy(t) (41)Dα3ez = −αλe3 + (λ− αλ)Z1 + αλX2Y2

−λX1Y1 + (S2 + S1)uz(t). (42)

We define active control functions ux(t), uy(t) anduz(t) as

(S2 + S1)ux(t) = Vx − (Prv − Pr)(Y1 −X1), (43)

(S2 + S1)uy(t) = Vy −R2

(β

v

)ex −(

R2

(β

v

)−R1

)X1 − (1− α)Y1

+

(β

v

)(R2 −

αv

β

)X2Z2

−(R1 − 1)X1Z1, (44)(S2 + S1)uz(t) = Vz − (λ− αλ)Z1 −

αλX2Y2 + λX1Y1. (45)

The terms Vx, Vy, and Vz are linear functions of theerror terms ex, ey,and ez . With the choice of ux, uy,and uz . The error system between the two chaoticsystems (39) becomes

Dα1ex = Prv(ey − ex) + Vx, (46)

Dα1ey = −αey + Vy, (47)

Dα1ez = −αλez + Vz. (48)

In fact we don’t need to solve (46)–(48) if the so-lution converges to zero. Therefore, the control termsVx(ex), Vy(ey), and Vz(ez) can be chosen such that


ISBN: 978-1-61804-178-4 67

Figure 3: The synchronization betweenX1 andX2 forAg case with different fractional orders α1 = α2 =α3 equal to 0.8, 0.9, 0.96, 0.965, 0.97, 1.0

the system (46)–(48) becomes stable with zero steadystate. Vx

VyVz

= A

exeyez

, (49)

whereA is a 3×3 real matrix, chosen so that all eigen-values λi of the system (49) satisfy the following con-dition

|arg(λi)| >απ

2. (50)

We choose

A =

Prv − k −Prv 00 α− k 00 0 αλ− k

. (51)

Then the eigenvalues of the linear system (46)–(48)are equal (−k,−k,−k), which is enough to satisfythe necessary and sufficient condition (50) for all frac-tional orders α < 2 [34]. In the following cases, wetake k = 1 for simplicity.

We investigate two different cases as follows:

1. (S1, S2) = (0, 0), In this case, the two sys-tems are working independently (no synchro-nization). We validate the Adams-Bashforth-Moulton predictor-corrector method for the so-lution of both systems at α1 = 0.99, α2 = 0.98

10 20 30 40 50 60 70 80 90 100-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time

X 1

10 20 30 40 50 60 70 80 90 100-4

-3

-2

-1

0

1

2

3

4

Time

Y 1

10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

Z 1

Figure 4: Time waveforms of X1, Y1, Z1 for Al2O3

α1 = 0.99, α2 = 0.98, α3 = 0.97

and α3 = 0.97, with same initial conditionsX1(0) = X2(0) = 0.8, Y1(0) = Y2(0) = 0.8and Z1(0) = Z2(0) = 0.92195.

2. (S1, S2) = (0, 1), In this case system1 worksnormally and system2 adapts its response to fol-low system1, therefore the first system worksnormally without any loading effect, and thesecond system adapts its response to synchro-nize with the first system with initial conditionsX2 = 0.2, Y2 = 0.4 and Z2 = 0.1. Fig.3 studys the synchronization between the twosystems when R1 = 75, R2 = 150 for Ag,time step=0.003, final time=10, and the initialvalues are (0.2, 0.4, 0.1)T and (0.8, 0.8, 0.922)T

and for equal fractional orders α1 = α2 = α3 =0.8, 0.9, 0.96, 0.965, 0.97, and 1.0. Therefore,the two systems will be identical whatever thesystem responses. It is clear that the under the


ISBN: 978-1-61804-178-4 68

-2 -1 0 1 2 3-4

-3

-2

-1

0

1

2

3

4

5

X1

Y 1

-2-1

01

2

-4-2

02

4

0.5

1

1.5

X1Y1

Z 1

Figure 5: The 2D and 3D strange attractors for Al2O3

α1 = 0.99, α2 = 0.98, α3 = 0.97

10 20 30 40 50 60 70 80 90 100-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

X2

Y 2

-2 -1 0 1 2 3

-4-2

02

4

0.5

1

1.5

2

2.5

X2Y2

Z 2

Figure 6: The numerical solutions of Y2, X1, X2 andthe 3D strange attractor of system 2 for Al2O3 α1 =0.99, α2 = 0.98, α3 = 0.97

0 20 40 60 80 10010-5

10-4

10-3

10-2

10-1

100

Time

exeyez

Figure 7: The numerical errors between the two sys-tems for Al2O3 α1 = 0.99, α2 = 0.98, α3 = 0.97

previous systems parameters, the system behaveschaotically for all orders greater than 0.96. Inaddition, the time needed for the synchroniza-tion is short although the initial are away. Fig.4 and 5 show the time responses of x, y, z and thestrange attractors in 2D and 3D for the same sys-tems parameters when R1 = 75 and R2 = 150,α1 = 0.99, α2 = 0.98, and α3 = 0.97. The sys-tem 2 responses are shown in Fig. 6 which illus-trate an identical response except in the transientresponse as clear from the 3D attractor as com-pared with the 3D attractor of system 1 shown inFig. 5. Moreover, the error response in this caseis presented in Fig. 7 where the differences aredecaying exponentially as shown from the loga-rithmic Y -scale.

Furthermore, another two cases can be discussedwhen (S1, S2) = (1, 0), Similarly the secondsystem works individually, and the first systemfollows exactly the second system and the mixedmode synchronization case, where the switchingparameters are a function of time.

5 Bifurcations with different controlparameter R

The fractional-order system was numerically inte-grated using the Adams-Bashforth-Moulton method.The scheme is coded in the computer algebra packageMATLAB. Fixing the values Pr = 10, λ = 8/3. Thestep size h is set to 0.001 in all the calculations donein this paper and we removed the first 500 points.

Now, let α1 = α2 = α3 = q and vary the value of


ISBN: 978-1-61804-178-4 69

(a)

(b)

Figure 8: Bifurcation diagrams of Z versus R repre-senting maxima and minima of the posttransient solu-tion of with Pr = 10, λ = 8/3 and q = 1 for Vadaszcase [28] and Ag case [17] with φ = 0.05.

R from 10 to 400. The initial states for the fractional-order system are taken as X(0) =, Y (0) = 0.8 andZ(0) = 0.92195 with step size of R is 0.5.

Fig. 8 presents the bifurcation diagrams, in termsof maxima and minima in the posttransient values ofZ versusR for Vadasz case [28] and Ag nanoparticlescase [17].

Comparing to Vadasz [28] and Jawdat [17] etal. cases, the critical value of R in each case isgreater than the critical value in Vadasz and Jawdat etal. cases. Thus the onset of chaotic convection usingthe fractional order is delayed.

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

R

Z

(a)

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

R

Z

(b)

Figure 9: Bifurcation diagram of the fractional-ordersystem with R when q = 0.99 for: (a) Ag ; (b) Al2O3

In the case of Ag, at the value ofR = 36.918 (thecritical value of R where the solution is limit cycle,see [17]), we obtain a solitary limit cycle when q = 1signifying the loss of stability of the steady convectionfixed points. When R = 150, the convection fixedpoints lose their stability and a chaotic solution takesover for integer and non-integer orders.

In the case of Al2O3, also the critical value ofR = 34.0316, we obtain a solitary limit cycle signify-ing the loss of stability of the steady convection fixedpoints for both integer and non-integer cases. WhenR = 75, the convection fixed points lose their stabil-ity and a chaotic solution takes over.

Comparing between Ag and Al2O3 cases as


ISBN: 978-1-61804-178-4 70

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

R

Z

(a)

0 50 100 150 200 250 300 350 4000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

R

Z

(b)

Figure 10: Bifurcation diagram of the fractional-ordersystem with R when q = 0.98 for: : (a) Ag ; (b)Al2O3

shown in the bifurcation diagrams in Fig. 9 and 10, thefractional-order system has the same tendency as itsfractional-order counterpart, but the attractor is differ-ent. It is chaotic over most of the range R ∈ [50, 150]for Al2O3 case. As R increase to 50, the systemabruptly becomes chaotic at about R ≈ 50, whereasa decrease in R from 400 causes the fractional-ordersystem to inter chaos by pichfork and period doublingbifurcations. While it is chaotic over most of the rangeR ∈ [60, 150] forAg case. In addition, the chaotic be-haviour is delayed from 50 to 60 for Al2O3 case andfrom 60 to 90 in Ag case when the fractional orderdecreases from 0.99 to 0.98.

6 ConclusionIn this paper, the effect of fractional-order in a fluidlayer subject to gravity and heated from below un-der the effect of nanofluids for low Prandtl numberhas been studied. Adams-Bashforth-Moulton methodhas been successfully applied to the purpose of findingthe numerical solutions of the fractional-order chaoticsystem. An active controller is designed to achieve thesynchronization between the fractional-order chaoticsystems. The bifurcations and dynamics of the frac-tional order system have been studied numerically byvarying the system parameter.

References:

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