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Kyungpook Mathematical Journal 2021; 61(2): 353-369

Published online June 30, 2021

Copyright © Kyungpook Mathematical Journal.

Synchronization of Non-integer Chaotic Systems with Uncertainties, Disturbances and Input Non-linearities

Ayub Khan and Nasreen*

Department of Mathematics, Jamia Millia Islamia, New Delhi, India
e-mail : akhan12@jmi.ac.in and nasreen899@gmail.com

Received: March 14, 2019; Revised: November 22, 2020; Accepted: November 25, 2020

In this paper, we examine and analyze the concept of different non-integer chaotic systems with external disturbances, uncertainties, and input non-linearities. We consider both drive and response systems with external bounded disturbances and uncertainties. We also consider non-linear control inputs. For synchronization, we introduce the adaptive sliding mode technique, in which we establish the stability of the controlled system by a control which estimates uncertainties and disturbances, and then applies a suitable sliding surface to control them. We use computer simulations to established the efficacy and adeptness of the prospective scheme.

Keywords: chaos synchronization, adaptive sliding mode technique, unknown disturbances, model uncertainties and input non-linearity

In recent times, non-linear dynamical systems have become a hot topic among researchers. Discovered by Henri Poincaré[26], chaos is a complex phenomenon found in most non-linear dynamical systems, describing the sensitive dependence of the evolution of the system on the initial conditions. Poincaré observed that two neighboring points in state space can very quickly become isolated. The phenomenon of chaos has deep applications in viscoelasticity [14], dielectric polarization, electromagnetic waves[9], diffusion, signal processing, mathematical biology and, of course, chaotic systems. Different procedures have been used to investigate the behavior of the chaotic non-linear systems that surround us. Among these procedures are plotting phase portraits, poincaré sections, or bifurcation diagrams, or finding Lyapunov exponents.

To understand the behaviour of non-linear systems and to stabilize their control, Pecora and Carroll established the idea of synchronization. Under synchronization, trajectories of coupled systems evolve together in a usual pattern. Based on different control techniques such as adaptive backstepping [27], linear and nonlinear feedback synchronization [3], active control [25], sliding mode control [6], adaptive sliding mode technique[12], and time delay feedback[5], researchers have developed many synchronization schemes- schemes such as complete and anti synchronization [10, 21], phase and anti-phase synchronization [20], projective and hybrid function projective synchronization[13, 22], generalised synchronization[28].

To combine the concept of integer order differentiation and n-fold integration, Leibnitz and L'Hospital in 1675 gave the theory of integrals and derivatives of arbitrary order. Systems represented by non-integer differential equations[23] have been studied extensively in recent years. These studies focus on real-life systems and have many multidisciplinary applications. Specifically, it has been seen that non-integer systems, which generalize many well-known integer order systems, have chaotic and hyper-chaotic behavior. Some such systems are Lorenz systems[8], Chen systems, Rössler systems[15], Liu-systems[6], Genesio-Tesi systems[7], Chua systems[29], complex t-system [19] and complex Lu-systems[24].

In this manuscript, we synchronize two different non-integer chaotic systems. We treat a non-integer chaotic Liu-system[6] as drive a system, and a chaotic Genesio-Tesi system[7] as response a system. We do so with model uncertainties, and external bounded disturbances, but also with non-linear input. The considertion of these elements together seems to be novel in the literature. As uncertainties and disturbances introduce a dreadful change in chaotic systems, dynamics, and synchronization reduce this. Researchers have introduced various schemes [4, 12] to examine the synchronization of chaotic systems with various disturbances and uncertainties. Generally the sliding mode control technique is an efficient approach for dealing with uncertainties and disturbances.

In practice when we encounter a controller in a real-life systems, physical limitations cause some non-linearity in the control inputs. It has been shown that non-linear input can cause a severe decay in the system performance. If the controller is poorly designed, then system failure becomes worse. Therefore, non-linear input effects must be taken into consideration when evaluating and implementing a control scheme for chaotic systems. Researchers have designed various techniques to synchronize integer-order chaotic systems with non-linear inputs [2, 16]. It seems, however, synchronization among non-integer chaotic systems in the presence of external disturbances, model uncertainties and non-linear input has not been discovered. In our paper, we investigate the synchronization under these perturbations. We introduce an adaptive sliding mode control scheme to synchronize the considered systems. We estimated the disturbances and uncertainties through a adaptive control rule and we chose a suitable sliding surface to counter their effect. We design the appropriate controllers using known control techniques and Lyapunov stability theory.

As motivated above, we summarize here the main aspects of the this paper.

  • 1. We propose a novel synchronization scheme for non-integer chaotic systems.

  • 2. We design non-linear control inputs for non-integer chaotic systems.

  • 3. We compare our proposed methodology with the previously published literature. We consider disturbances, uncertainties, and non-linear input. Even with this, our methodology yields better results than the previously publishedliterature.

  • 4. We ilustrate an application in secure communication in Section 5.

  • 5. We use numerical simulations to validate and visualize our results.

The fractional order derivative can be defined in various forms[23] such as Riemann-Lioville's derivative, Grünwald Letnikov's derivative, Caputo's derivative. Here we have taken Caputo's derivative defined as

t0Dtαf(t)=1Γ(nα)t0 t f (n)(τ) (tτ) αn+1 dτ,t>t0

where α+ and Γ(.) is the Gamma function.

Consider the drive system of n dimensions with model uncertainties and external disturbances

Dαui=fi(u1,u2....un)+Δfi(u1,u2....un,t)+di(t)

for i=1,,n, where u(t)=[u1,u2,....,un]Tn are state variables of the system (2.1), fi(u):n×1, are continuous functions, Δfi(u) are model uncertainties and di(t) are external disturbances for i=1,2,3,...,n.

Consider the response system of n dimensions with model uncertainties, external disturbances, and non-linear control inputs as

Dαvi=gi(v1,v2....vn)+Δgi(v1,v2....vn,t)+d i (t)+Φi(Ui)

for i=1,,n, where v(t)=[v1,v2,....,vn]Tn are state variables of the system (2.2), gi(v):n×1 are continuous functions, Δgi(v) are model uncertainties, di(t) are external disturbances, and Φi(Ui) are the non-linear control inputs for controller Ui i=1,2,3,...,n.

Assumption 1. The trajectories of non-integer chaotic systems are bounded so here we have assumed that the model uncertainties Δfi(u) and Δgi(v) are bounded. This implies that there exist constants ϑim>0 and ϑis>0 such that

|Δfi(u)|<ϑim   and   |Δgi(v)|<ϑis.

Consequently we have

|Δfi(u)Δgi(v)|<ϑi

for i=1,2,...,n.

Assumption 2. It is assumed that if external disturbances di(t) and di(t) are norm bounded then there exist constants νim>0 and νis>0 such that

|di(t)|<νim    and    |gi(v)|<νis.

Thus for i=1,2,...,n, we get

|Δfi(u)Δgi(v)|<νi.

Assumption 3. The control inputs Φi(Ui) are continuous non-linear functions and satisfy

ωiUi2UiΦi(Ui)ηiUi2

where i=1,2,...,n and ωi,η>0 are constant parameters.

In order to achieve synchronization, here we define synchronization error as ei=uivi, for i=1,2,....,n. The error dynamics is attained as

Dαei=gi(v1,v2....vn)+Δgi(v1,v2....vn,t)+d i (t)fi(u1,u2....un)Δfi(u1,u2....un,t)di(t)+Φi(Ui)

for i=1,,n.

To achieve synchronization, we have to establish that the error system (2.3) is stable. For that our aim is to design control laws for any two non-integer chaotic systems with model uncertainties, external disturbances, and non-linear input to established that it is stable asymptotically: limtei(t)=0,i=1,2,....,n.

To minimize the error, we choose the suitable sliding surface which is as follows:

si(t)=μiDα1ei(t)+0t e i(ξ)dξ

where s(t),s(t)=[s1,s2,...,sn]T and the sliding surface parameters μi,i=1,2,..,n are chosen in such a manner that they are positive.

To discuss the error system (2.3) at the chosen sliding surface (2.4), it is necessary that it should satisfy the following condition for i=1,2,..,n

si(t)=0,si.(t)=0

The derivative of (2.4) yields the following equation

si.(t)=μiDαei(t)+ei(t)

Then, by considering the necessary condition si.(t)=0, we obtain

Dαei(t)=1μiei(t)

Hence, the system (2.3) is asymptotically stable which shows that the slave system (2.2) can be tackled by the master system (2.1) by constructing the appropriate control inputs.

Our next step is to design the appropriate control inputs in order to stabilize the error system and attain synchronization on the chosen sliding surface s(t)=0 .

The control inputs are designed as follows

Ui=1ωi 1μi|ei |+|gi fi |+ϑ^i +ν^i +λi sign(si)=ζisign(si)

where ϑi^ and νi^ are estimates of ϑi and νi respectively and λi are switching gain.

The adaptive laws are chosen as

ϑ^˙i= ν^˙i=μi|si|,i=1,2,..,n.

Theorem 2.1. For the error system (2.3) with control laws (2.8) and adaptive laws (2.9), if the following condition is fulfilled:

(μiηiλi)<0,

then the synchronization error converges to si=0. Thereby the synchronization between (2.1) and (2.2) can be achieved.

Proof. Consider the Lyapunov function given as

Vi=12 i=1N[si2+(ϑ^iϑi)2+(ν^iνi)2]

The derivative of Vi is

V˙i= i=1N[si s ˙ i+( ϑ^ iϑi) ϑ ^ ˙ i+( ν^ iνi) ν ^ ˙ i]

By substituting the value of s˙i,

V˙i= i=1N[si(μiDαei(t)+ei(t))+( ϑ^ iϑi) ϑ ^ ˙ i+( ν^ iνi) ν ^ ˙ i]

Using adaptive laws (2.9) and substituting the values of Dαei(t), we obtain

V˙i=i=1N[si(μi(gi+Δgi+d i(t)fiΔfidi(t)+Φi(Ui))+ei(t))+(ϑ^iϑi)(μi|si|)+(ν^iνi)(μi|si|)]

Using Assumption (2) Φi(Ui)ηiUi and siφi(Ui)ωζi|si|, where Ui=ζisign(si) and (2.3), we have

V˙ii=1N[si(μi(gi+Δgi+d i(t)fiΔfidi(t)+Φi(Ui))+μiηi+ei(t))+(ϑ^iϑi)(μi|si|)+(ν^iνi)(μi|si|)]i=1N[|si||ei|+|si|μiηi+|si|μi|gifi|siμiΦi(Ui)+ϑ^iμi|si|+ν^iμi|si|]i=1N[|si||ei|+|si|μiηi+|si|μi|gifi|μiωζi|si|+ϑ^iμi|si|+ν^iμi|si|]

Substituting ζi=1ωi1μi|ei|+|gifi|+ϑ^i+ν^i+λi into the above inequality:

V˙ii=1N[|si||ei|+|si|μiηi+|si|μi|gifi|    μiω1ωi1 μi|ei|+|gifi|+ϑ^i+ν^i+λi|si|+ϑ^iμi|si|    +ν^iμi|si|]

Using equations (2.8), (2.9) and (2.10) and simplifying, we get the following inequality:

V˙ii=1N[(μiηiλi)|si|]=i=1N[(λiμiηi)|si|]=i=1N[Θi|si|]=Θi|si|=Ωi(ξ)0.

Integrating the above equation from 0 to t yields

Vi(0)Vi(t)+0t Ω i(ξ)dξ. 

Since Vi.(t)<0,Vi(0)Vi(t)0 is positive and finite, the limit limtΩi(ξ) exists and is finite (i.e. limtΩi(ξ)=Vi(0)Vi(t)0). Using the Barbalat Lemma ([11, lemma 8.2]), limt0t Ωi(ξ)dξ=0, which implies |si|=0. Thus the error dynamical system is asymptotically stable. Hence, synchronization is achieved between non-integer chaotic systems on the considered stationary surface. This completes the proof.

Remark 2.2.([1, 17]) The signum function behaves as a rigid switcher in the prospective control law and it can cause chattering. Therefore, we modify the controller to prevent chattering:

Ui=1 ωi 1 μi|ei |+|gi fi |+ϑ^i +ν^i +λi tanh(δisi)

where δi>0 is a constant.

Remark 2.3. After substituting the controller (2.11) into V˙i for error dynamical system (2.3), we have

V˙i i=1N[(μiηiλi)tanh(δisi)].

Using condition (2.10) and (μiηiλi)|tanh(δisi)|0, we have

V˙i i=1N[(μiηiλi)|tanh(δisi)||si|]0.

Using Theorem 2.1 together with the Barbalat Lemma, we obtain |si|=0 subsequently the error system is stable with controllers (2.11) and adaptive laws (2.9).

To check the applicability and efficacy of the proposed control scheme, we consider the following two different non-integer chaotic systems:

Drive system[6]

Dαu1=au1eu22+0.1cosπu1+0.5sintDαu2=bu2ku1u3+0.1cos2πu2+0.5sintDαu3=cu3+mu1u2+0.1cos3πu3+0.5sint

where u1,u2,u3 are state variables. The parameters a,b,c,d, and e are non-negative constants. For the parameter values a=1,m=4,b=2.5,c=5,e=1,k=4,m=4, initial conditions (u1(0),u2(0),u3(0))=(0.2,0,0.5)., and fractional order α=0.95 the system (3.1) exhibits chaotic behaviour, as seen in Fig.1.

Figure 1. Phase Portraits of fractional order Liu chaotic system for α=0.95 (a)u2-u1 axis (b)u2-u3 axis (c)u3-u1 axis (d)u1-u2-u3 axis.

The uncertainties and disturbances for the drive system are taken as

Δfi=0.1cos(iπui) and di=0.5sint

for i = 1,2,3.

Response system[7]

Dαv1=v20.1cosπv10.5sint+Φ1(U1)Dαv2=v30.1cos2πv20.5sint+Φ2(U2)Dαv3=fv1gv2hv3+iv120.1cos3πv30.5sint+Φ3(U3)

where v1,v2,v3 are state variables. Parameters f, g, h, and i are non-negative constants. For the parameter values f=1,g=1.1,h=0.4,i=1 and initial conditions (v1(0),v2(0),v3(0))=(0.3,0.1,0.2). and for fractional order α=0.95 the system (3.2) shows the chaotic behaviour in Fig.2.

Figure 2. Phase Portraits of fractional order Genesio-Tesi chaotic system for α=0.95 (a) v1-v2 axis (b) v2-v3 axis (c)v1-v3 axis (d)v1-v2-v3 axis.

The uncertainties and disturbances for response system are taken as

Δfi=0.1cos(iπvi) and di=0.5sint

for i = 1,2,3.

The non-linear control inputs are taken as Φi(Ui)=[5+3sint]Ui. Also, it is assumed that ωi=1,ηi=4.

The error dynamical system can be written as

Dαe1(t)=au1eu22+0.1cosπu1+0.5sintv2+0.1cosπv1    +0.5sintΦ1(U1)Dαe2(t)= bu2ku1u3+0.1cos2πu2+0.5sintv3+0.1cos2πv2    +0.5sintΦ2(U2)Dαe3(t)=cu3+mu1u2+0.1cos3πu3+0.5sint+fv1+gv2    +hv3iv12+0.1cos3πv3+0.5sintΦ3(U3)

Choosing a suitable sliding surface (2.4) using control law (2.11) and adaptive law (2.9), we take μi=0.1, λi=0.5 and δi=200. This yields (μiηiλi)=0.1*40.5=0.1<0.

Using Theorem 2.1 and (2.11), we get the following control laws.

U1= [10|e1|+|au1eu22v2|+ϑ^1+ν^1+0.5]tanh(200s1)U2= [10|e2|+|bu2ku1u3v3|+ϑ^2+ν^2+0.5]tanh(200s2)U3= [10|e3|+|cu3+mu1u2+fv1+gv2+hv3iv12|  +ϑ^3+ν^3+0.5]tanh(200s3)

For the systems (3.1) and (3.2) take (u1(0),u2(0),u3(0))=(0.2,0,0.5) and (v1(0),v2(0),v3(0))=(0.3,0.1,0.2). Also take ϑ^1(0)=0.1,ϑ^2(0)=0.1,ϑ^3(0)=0.1 and ν^1(0)=0.1,ν^2(0)=0.1,ν^3(0)=0.1.

Figures 1 and 2 show the phase portraits of system (3.1) and (3.2). Figure 3 shows the synchronized state trajectories of system (3.1) and (3.2). Figure 4 shows the synchronized error and that the sliding surface converges to zero at approximately time t=2 seconds. Figure 5 shows the estimated values of uncertainties and disturbance bounds.

Figure 3. Synchronized state trajectories which are synchronized at time t=2 unit(approx.)(a)u1-v1 (b)u2-v2 (c)u3-v3
Figure 4. (a) Synchronization error (b) sliding surface converging to zero at time t=2 unit(approx.).
Figure 5. Estimated values of (a)uncertainties bounds (b)disturbances bounds.

1. First of all, we compared our synchronization result to the integer-order Liu system and Genesio-Tesi system for the same set of parameter values and initial conditions. For α=1, synchronization was achieved at approximately t=15 seconds. as seen in Fig.6. Therefore, from Fig.4(a) and Fig.6, we see that our scheme gives better results for non-integer chaotic systems.

Figure 6. Synchronization Error for integer order Liu system and Genesio-Tesi system.

2. In [6], the author adopted the active control and sliding mode control methods to analyse the complete synchronization between two identical non-integer Genesio Tesi systems with parameter values a=6,b=2.92,c=1.2 & d=1, initial conditions (x1(0),x2(0),x3(0))=(0.3,0.7,1.2), (y1(0),y2(0),y3(0))=(0.1,0.3,0.7) and α=0.97. The author achieved synchronization with the active control method at approximately t=20, and with the sliding mode method at time t=15. When we implemented our proposed methodology for the same systems with same set of parameter values and initial conditions in the presence of a set of disturbances, uncertainties, and non-linear input, we achieve synchronization at time t=4, as shown in Fig.7.

Figure 7. ynchronization Error for two identical non-integer Genesio-Tesi system.

3. In [18], the complete synchronization between two identical non-integer Genesio-Tesi systems with fifth order non-linearity based on the adaptive control method was studied with parameter values β1=2,β2=3.5,β3=0.3 & β4=-1, initial conditions (x1(0),x2(0),x3(0))=(-0.2,0.5,0.2), (y1(0),y2(0),y3(0))=(0.5,1,1) and α=0.95. They achieved synchronization at time t=75. Our scheme for the same systems with same parameter values and initial conditions in the presence of disturbances, uncertainties, and input non-linearities, achieves synchronization at time t=4, as shown in Fig.8.

Figure 8. Synchronization Error for two identical non-integer Genesio-Tesi system with fifth order non-linearity.

4. In Section 4 of [25], the author anti-synchronized two identical non-integer Genesio-Tesi systems using the active control method with parameter values a=6,b=2.92,c=1.2 & m=1, initial conditions (x1(0),x2(0),x3(0))=(2,3,4), (y1(0),y2(0),y3(0))=(1,6,6) and α=0.95. Anti-syncronization occured at time t = 6. When we applied our proposed scheme for the systems in [26] with same parameter values and initial conditions in the presence of disturbances, uncertainties and input non-linearities, we achieves synchronization at time t=3.5, as shown in Fig.9.

Figure 9. Synchronization Error for two identical non-integer Genesio-Tesi system.

5. In Section 5 of [25], the author anti-synchronized a non-integer Genesio-Tesi system and a Qi system using the active control method with parameter values a=6,b=2.92,c=1.2,m=1,p=35,q=8/3, & r=80 initial conditions (x1(0),x2(0),x3(0))=(2,3,5), (y1(0),y2(0),y3(0))=(1,1,2) and α=0.96. They acheived anti-synchronization at time t = 8. When we adopted our scheme for the same with same parameter values and initial conditions in the presence of disturbances, uncertainties, and input non-linearities, we achieved synchronization at time t=1, as shown in Fig.10.

Figure 10. synchronization Error for non-integer Genesio-Tesi system and Qi-system.

Our methodology significantly beats published liturature in these settings.

Non-integer, chaotic systems have applications in many fields, such as physics, chemical science, and secure communication.

In this manuscript, we show an application in secure communication.

Here we take a simple additive encryption masking scheme, to validate our proposed scheme.

The information input signal is selected as IS=sint+cost and u3 is a chaotic carrier. The chaotic encrypted signal CS=IS+u3. The original input information signal is regained by our proposed methodology- the decrypted signal is DS=CSv3. The results are shown in Fig. 11.

Figure 11. Additive encryption masking scheme (a) Information input signal(IS ) (b) Chaotic encrypted signal(CS ) (c) Decrypted signal(DS ) (d)Error between original signal and decrypted signal.

In this paper, a robust adaptive sliding mode technique has been used to achieve synchronization between two different fractional-order chaotic systems with model uncertainties, external disturbances, and non-linear inputs. Synchronization of non-integer chaotic systems in the presence of uncertainties, disturbances and non-linear control inputs has not been examined in the prior literature. We synchronized non-integer chaotic Liu-system and Genesio-Tesi systems. We chose a suitable sliding surface and estimated the bounded uncertainties and disturbances using update laws to achieve the desired synchronization and reduce the consequence of external uncertainties and disturbances and non-linear input. Then, using the considered control scheme and Lyapunov stability theory, we designed appropriate controllers. Although we have taken non-integer chaotic systems with uncertainties, disturbances, and non-linearities, we get better synchronization results. This scheme should perform a significant role to enhance security in communications. Computational methods were used to evaluate the efficiency of the considered scheme.

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