11421


FACTA UNIVERSITATIS 

Series: Electronics and Energetics Vol. 36, No 3, September 2023, pp. 365-378 

https://doi.org/10.2298/FUEE2303365C 

 

© 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper  

CHAOS SYNCHRONIZATION USING SUPER-TWISTING 

SLIDING MODE CONTROL APPLIED ON CHUA’S CIRCUIT  

Abdelilah Chibani1, Bachir Daaou2, Abdelmadjid Gouichiche1, 

Ahmed Safa1, Yacine Badaoui1, Zakaria Chedjara1  

1 Electrical Engineering and Plasma Laboratory LGEP, University of Tiaret, Algeria 
2Laboratory AVCIS, University of Mohamed Boudiaf, Oran, Algeria 

Abstract. Chua’s circuit is the classic chaotic system and the most widely used in 

serval areas due to its potential for secure communication. However, developing an 

accurate chaos control strategy is one of the most challenging works for Chua’s circuit. 

This study proposes a new application of super twisting algorithm (STC) based on 

sliding mode control (SMC) to eliminate or synchronize the chaos behavior in the 

circuit. Therefore, the proposed control strategy is robust against uncertainty and 

effectively regulates the system with a good regulation tracking task. Using the 

Lyapunov stability, the property of asymptotical stability is verified. The whole of the 

system including the (control strategy, and Chua’s circuit) is implemented under a 

suitable test setup based on dSpace1104 to validate the effectiveness of our proposed 

control scheme. The experimental results show that the proposed control method can 

effectively eliminate or synchronize the chaos in the Chua's circuit. 

Key words: Chaos control, Chua’s circuit, Control design Super-twisting 

1. INTRODUCTION 

In a natural phenomenon, such as varied weather, chaos is an evidently, stochastic 

motion in deterministic nonlinear systems that present a bounded unstable dynamic 

behavior. Chaos has been extensively investigated since Lorenz reported a sensitive 

dependence on initial conditions in an atmosphere prediction model and includes infinite 

unstable periodic motion. Therefore, attract the attention of research community in all the 

branches dealing with evolutionary processes: chemistry, biology, medicine, genetics, 

economics, sociology and electronic systems [1–7]. One of the most popular nonlinear 

electronic systems is Chua’s circuit which is a classical example of bifurcation and chaos 

in nonlinear circuits in many studies. Invented in 1983, Chua’s circuit has been extensively 

considered research subject to analyze chaotic phenomena. On the other side, random 

 
Received December 17, 2022; revised February 19, 2023; accepted February 23, 2023 

Corresponding author: Abdelilah Chibani 
Electrical Engineering and Plasma Laboratory LGEP, University of Tiaret, Algeria 

E-mail: abdelilah.chibani@gmail.com 



366 A. CHIBANI, B. DAAOU, A. GOUICHICHE, A. SAFA, Y. BADAOUI, Z. CHEDJARA 

 

motion of a system in a chaotic situation was considered to be unfavorable in engineering 

fields. Therefore, many efforts were given to eliminate or control chaos in the systems, 

which led to launching the work on chaos control in 1990 such as the OGY (Ott, Gerbogi 

and York) control method [8], linear feedback control [9], [10], time delay feedback 

control [4], [11], [12] and some others were reported: the control of chaos such as sliding 

mode control [13–17], fuzzy control[18–22], polynomial approach [23] high order sliding 

mode control [24] and harmonic approach [25], [26], etc. Many researchers have been 

addressed the control of Chua’s circuit, in several control methodologies. In [21] the 

authors present an H∞ tracking performance design scheme via fuzzy adaptive observer-

based control for chaotic Chua’s systems with output time delay. In [27], [28], the 

adaptive technique and high-gain methods were proposed respectively to achieve 

bounded synchronization in the presence of a norm-bounded perturbations. In [29], j. Yan 

et al. proposed an adaptive synchronization of modified Chua’s circuit, this technique 

based on adaptive switching surface in order to guarantee the occurrence sliding motion. 

In [30], the authors present a new technique for chaos synchronization based on quasi-

sliding mode control for Rikitake chaotic system. Global anti-synchronization of chaotic 

modified Chua’s circuits via linear feedback control is investigated in [31]. The 

investigations in [32] unveiled a novel robust chaotic controller for stabilizing uncertain 

time delay chaotic systems with input non-linearity. In [33], Dadras et al. propose a sliding 

mode controller for new chaotic dynamical system. In [34], In this study, authors focused 

more on the fractional order and adaptive finite-time sliding mode control in the financial 

risk chaotic system. However, despite the advantage of the controller, the chattering 

phenomenon associated with the classical sliding mode controller occurs and clear. 

Accordingly, this paper proposes a new and simple chaos control strategy designed to 

achieve the chaos synchronization or chaos suppression for Chua’s circuit. Therefore, we 

shall debate how to design the super twisting-based on the sliding mode technique for Chua’s 

circuit under parameter uncertainties. The super twisting sliding mode control stability of the 

closed loop system is proved by using the Lyapunov theory. Moreover, to validate our 

proposal we develop a test bench based on the dSpace 1104, for more detail see Fig.5. 

The rest of the paper is arranged as follows: the Chua’s circuit and its characteristics 

are given in section 2, section 3 presents the mathematical development of the proposed 

controller, experimental results are reported in section 4. Finally, some concluding 

remarks are discussed in section 5. 

2. PROBLEM STATEMENT AND PRELIMINARIES 

2.1. Chua’s circuit 

The chaotic Chua’s circuit, as shown in Fig.1, is a simple electronic circuit 

that consists of one inductor IL, two capacitors C1; C2, one linear resistor R, and a 

nonlinear resistor NLR [35]. The Chua’s circuit can be described by a third-order 

nonlinear differential equation, according to the electronics theory, the mathematical 

model of Chua’s circuit is given by: 



 Chaos Synchronization Using Super-Twisting Sliding Mode Control Applied on Chua’s Circuit 367 

 

 

( ) ( )

( )

1

1 2 1 1

2

2 1 2

2

( )
( ) ( ) ( )

( )
( ) ( ) ( )

( )
( )

1

1
L

L

d t
C t t g t

dt R

d t
C t t i t

dt R

di t
L t

dt


  


 




= − −




= − +



= −


 (1) 

 

Fig. 1 Chua’s circuit 

Where V1, V2 respectively are the voltages across capacitors C1 and C2; iL is the 

current through inductor L ; stands for the current through the nonlinear resistor NLR 

and can be expressed by a piecewise-linear function.  

 
1 1 1 1( ) 0.5( ) 1 1NR b a bi g G G G   = = + − − + − −‖ ‖ ‖ ‖  (2) 

For simplicity, a sole circuit system is described as, ( ( ), )
dx

f x t t
dt

= ,  

Where,  1 1( ) ( ), ( ), ( ) ,
T

Lx t t t i t =  1( ( ), ) ( ) ( ( )),0,0
T

f x t t Ax t g t = −      

1 1

1
,

C R
 =

1

1
,

C
 =

2 1 2

1 1 1
, , ,

C R C L
  = = =

0

.

0 0

A

 

  



− 
 

= −
 
 − 

 

Table.1 represent the parameters adopted for Chua’s circuit. However, our objective 

after this simulation is to validate the proposed parameters and to build a prototype 

(PCB), for more detail see Fig. 2. 

Table 1 Parameters adopted for Chua’s circuit 

Parameters  Values Unite 

R1 R2 220 Ω 

R3, R4, R6 2200 Ω 

R5 3300 Ω 

Amplificator TL084 / 

Variable resistor 2200 Ω 

C1 10 nF 

C2 10 nF 

Indictor 9 mH 

 



368 A. CHIBANI, B. DAAOU, A. GOUICHICHE, A. SAFA, Y. BADAOUI, Z. CHEDJARA 

 

 

Fig. 2 Chua’s circuit prototype: (a) schematic under Proteus, (b) PCB design 

Fig. 2a presents the electronic circuit under the proteus platform. however,  this phase 

we need to validate the functioning of Chua's circuit. Fig. 2b presents the PCB prototype 

of  the circuit. 

 

Fig. 3 Simulation results in open loop using proteus platform: (a) voltage capacitor Vc1, 

(b) voltage capacitor 
2Vc , (c) current Inductor 

LI  

Fig. 3 focuses on three components: Fig. 3a the voltage across a capacitor Vc1, Fig. 3b the 

voltage across another capacitor Vc2, and Fig. 3c the current through an inductor IL. The 

simulation results show that the behavior of these components presents a chaotic 

phenomenon, meaning they exhibit unpredictable and seemingly random behavior over time. 

2.2. Objectives 

The main objectives of this study are as follows: 

▪ Construct chaos control strategy u(t) based on the sliding mode technique 

moreover, guarantee the synchronization stability under the healthy condition or in 

the case of perturbations. 

▪ Performing the proposed control scheme physically, so that the closed-loop 

Chua’s circuit can be implemented by circuits (TL084) and the dSpace 1104 

card. 



 Chaos Synchronization Using Super-Twisting Sliding Mode Control Applied on Chua’s Circuit 369 

 

3. SUPER TWISTING ALGORITHM 

The sliding mode control is one of the most accurate and robust control techniques. 

The typical first-order sliding mode control causes undesirable high-frequency chattering 

problem. The chattering phenomena can be eliminated by using the second order sliding 

mode control. However, Different second order sliding mode topology such as: the super 

twisting algorithm, drift algorithm, sub-optimal algorithm, and prescribed convergence 

algorithm are discussed in literature [16], [36]–[38]. Therefore, The Super-twisting 

algorithm is a second order sliding mode controller is more suitable for the system with 

relative degree one. the main advantage of this algorithm: first, it does not require the 

time derivative information of the sliding variables. While, measurement of derivative of 

sliding surface is required which may increase noise in the system. Second, reduce the 

charting phenomena. 

The super twisting strategy present the control law as a combination between two 

functions: the continuous sliding variable function and the integral of a discontinuous 

sliding variable function. Furthermore, we have chosen the super-twisting algorithm since 

the relative degree of our system and we seek for the finite time convergence which is 

one of its features. In this section, we present the synthesis of super-twisting approach in 

order to control the Chua’s circuit. 

We consider the nonlinear system with the following dynamics: 

 ( ) ( ) ( )
dx

f x b x u t
dt

= +  (3) 

Where 
1, ( ) , 0x R f x R b R and b R−      

The main objective is to lead the state vectors to track their references vector. 

Therefore, the error vector ei (t) = 0 tends to zero vector. The block in diagram Fig. 4 

represents the general methodology for our application. 

 

 

Fig. 4 Block diagram of the controller scheme 

The control law 1 2 3[ ]Tu u u u= is designed to drive the system Eq. (1) to the desired 

surface 1 2 3[ ]Ts s s s=   



370 A. CHIBANI, B. DAAOU, A. GOUICHICHE, A. SAFA, Y. BADAOUI, Z. CHEDJARA 

 

The sliding surface is selected as 

 

1 1 1 1

2 2 2 2

3 3

ref

ref

ref

L L

s e

s s e

s e i i

 

 

 −   
    

= = = −    
     −     

 (4) 

Where S is the synchronization error and the sliding surface respectively. 
 

Proposition 1  
 

Consider system Eq. (1) and the sliding surface S  defined in Eq. (4). Let introduce 

the super twisting algorithm. 

 

1

s ( )

s ( )

p
i n n

i n

u s ign s

ign s dt

 

 


= +


= 

 (5) 

Where 
i and 

i  are a positive constant number. 

Proposition 2  
 

Consider the system Eq. (1), the sliding surface present in Eq. (4) and the super-

twisting proposition 1 in Eq. (5), let’s the control input in closed-loop system describe by 

 eq swu u u= +  (6) 

With, 

 

1 1 1

2 2 2

3 3 3

eq sw

eq sw

eq sw

u u u

u u u u

u u u

 + 
  

= = +  
   +   

 (7) 

Where, 
sw

iu : is the switching control Function. The switching control function drives the 

system in any initial state to reach the sliding manifold in finite time, which are 

calculated through the application of the super-twisting algorithm, see system Eq. (5). 
q

i

eu : is the equivalent control function. The equivalent control function drive the 

system to move over the sliding manifold under ideal conditions. One of its features, can 

speed-up the response of the system and reduce the steady state errors [39]. The 

equivalent control function is calculated by setting the derivative of the sliding surface 

tend to 0, 0ids

dt
= . 

 

1 1 1 1

2 2 2 2

3 3

0

0

0

ref

ref

ref

L L

s e
ds d d d

s e
dt dt dt dt

s e i i

 

 

 −     
      

= = = − =      
      −      

 (8) 



 Chaos Synchronization Using Super-Twisting Sliding Mode Control Applied on Chua’s Circuit 371 

 

 

( )

( )

( )

1

2 1 1

1

2

2 1 2

3

2

( ) ( ) ( )

( ) (

( ) ( )

)( ) (

( )

)

ref

eq

ref
eq eq

i L

eq

ref

L

d
t t g t

dt
u

d
u u t t i t

dt
u

di
t

dt


    


   



 
− − − 

  
  

= = − − +  
  

   
− − 

 

 (9) 

Based on the equations Eq. (1), (5), (6), (7) and Eq. (8) the super-twisting controller is 

given by 

 

( )

( )

1

2 1 1 1

1

2

2 1 2 2

3

3

2 3(

( ) ( )

(

( ) ( ) ( )

)( ) ( ) (

(

)

))

ref

ref

i L

ref

dx
g

dt
u

dx
u u i

d

t

t
u

dx

dt

t t

t t t

t

    

   



 
− − − + 

  
  

= = − − + +  
  

   
− − + 

 

 (10) 

Where, 

1 11 1 1 12 1( ) ( ) ,
p

k s sign s k sign s dt = +  2 21 2 2 22 2( ) ( )
p

k s sign s k sign s dt = +   

33 32 3( )
p

k s sign s dt =   

3.1. Stability proof 

Consider the super-twisting controller given by Eq. (6), the tracking errors [ ]i ie e= =  

[ ]i i

refx x= −  are globally asymptotically stable. 

Let’s using eq swu u u= + , In this case, the matter of the stability condition is expressed as: 

 0 0
ds

s and
dt

= =  (11) 

The dynamics of the system Eq. (2) is subjected to the following: 

 
ds s dx

dt x dt


=


 (12) 

Since 

 ( ) ( ) ( )
dx

f x b x u t
dt

= +  (13) 

By using Eq. (6) and Eq. (13) we have 

  ( ) ( ) ( )
ds s

f x b x u t
dt x


= +


 (14) 



372 A. CHIBANI, B. DAAOU, A. GOUICHICHE, A. SAFA, Y. BADAOUI, Z. CHEDJARA 

 

 [ ( ) ( ) ] [ ( ) ]eq sw

ds s s
f x b x u b x u

dt x x

 
= + +
 

 (15) 

By setting 0
ds

dt
=  ds dt = 0 and 0swu = we get 

 

[ ( )]

[ ( )]
eq

s
f x

xu
s

b x
x



= −




 (16) 

Using Eq. (15) and Eq. (16) we get 

 ( ) sw

ds s
b x u

dt x


=


 (17) 

We consider the Lyapunov candidate function 

 21

2
V S=  (18) 

The derivative of along the trajectories of the system is given by: 

 31 2

1 2 3

dsds dsdV dS
S s s s

dt dt dt dt dt
= = + +  (19) 

To ensure the condition of global asymptotic stability, we have two issues: Firstly, we 

must verify the decrease of the Lyapunov function to zero. Secondly, it’s needful to 

ensure the derivative of Lyapunov function is negative. For our purpose its sufficient to 

verify that its derivative is negative. 

 0 0
dV dS

S
dt dt

    (20) 

 ( ) sw

dS s
S S b x u

dt x


=


 (21) 

 1 2 331 2

1 2 3( ) ( ) ( ) 0SW SW SW

ss sdS
S S b x u S b x u S b x u

dt x x x

 
= + + 

  
 (22) 

From Eq. (10) we have 

 

1 11 1 1 12 1

2

21 2 2 22 2

3

31 3 3 32 3

( ) ( )

( ) ( )

( ) ( )

p

sw
pi

sw sw

p
sw

k s sign s k sign s dt
u

u u k s sign s k sign s dt

u
k s sign s k sign s dt

 +
   
   = = +   
     +  







 (23) 

By replacing Eq. (23) in Eq. (22) we obtain 



 Chaos Synchronization Using Super-Twisting Sliding Mode Control Applied on Chua’s Circuit 373 

 

 

1

1 11 1 1 12 1

2

2 21 2 2 22 2

3

3 31 3 3 32 3

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) 0

p

p

p

sdS
S S b x k s sign s k sign s dt

dt x

s
S b x k s sign s k sign s dt

x

s
S b x k s sign s k sign s dt

x


 = +
 


 + +
 


 + + 
 







 (24) 

Let’s introduce the nonlinear term as 

 ( )i i is sign s s=  (25) 

 

1

11 1 1 12 1

2

21 2 2 22 2

3

31 3 3 32 3

( ) ( )

( ) ( )

( ) ( ) 0

p

p

p

sdS
S b x k s s k sign s dt

dt x

s
b x k s s k sign s dt

x

s
b x k s s k sign s dt

x


 = +
 


 + +
 


 + + 
 







 (26) 

 

1 1

1 11 1 1 12 1

2 2

2 21 2 2 22 2

3 3

3 31 3 3 32 3

( ) ( ) 0

( ) ( ) 0

( ) ( ) 0

p

p

p

ds s
s b x k s s k sign s dt

dt x

ds sdV
s b x k s s k sign s dt

dt dt x

ds s
s b x k s s k sign s dt

dt x

    +     
   

    = = + 
    
   

    + 
       







 (27) 

Where 
11 12 21 22 31 32, , , , ,k k k k k k are a constant number.  

To sum up, the term ( )is
b x

x




is positive according to the system under consideration, 

where as the gains kij must be selected negative in order to satisfy the condition stability 

shown below. In the end, the requirement of global asymptotic stability is achieved. 

4. EXPERIMENTAL RESULTS 

A Chua’s circuit is developed to validate the analysis described above. Table.1 

contains the details of Chua’s circuit parameters. A suitable test setup built around the 

dSpace 1104 serves to implement the mathematical model which was established with 

closed-loop control. The experimental set-up for this work is depicted in Fig. 5. The 

following constitutes the test setup: 

▪ The Chua’s chaotic system. 

▪ A DC power supply is to deliver DC voltage (5V, +15V, -15v) from an AC 

network.  

▪ An interface Controller Board dSpace (model Ds 1104) with a power Pc 603e 

at 400 MHz and a fixed-point digital signal processor DSP TMS320F240. 

 



374 A. CHIBANI, B. DAAOU, A. GOUICHICHE, A. SAFA, Y. BADAOUI, Z. CHEDJARA 

 

 

Fig. 5 Experimental setup 

Fig. 5, presents a test setup depicted for Chua's chaotic system, The setup is designed to 
operate with a DC power supply that converts AC voltage from the network into a stable DC 
voltage of 5V, +15V, and -15V. The system also includes an interface controller board, 
referred to the dSpace model Ds1104. The dSpace board is equipped with a PowerPC 603e 
processor running at a clock frequency of 400 MHz. This is designed to handle real-time 
digital signal processing tasks and is an important component in the overall control and 
monitoring of Chua's chaotic system and validate our proposed control scheme. 

Fig. 6 presents the experimental results in open-loop of the Chua's circuit. As shown in 
Fig. 6a, the time response of voltage capacitor Vc1 is depicted, in Fig. 6b the variation of Vc2 is 
illustrated, and in Fig. 6c the time response of the Il is presented. From these figures, it can be 
observed that the Chua's circuit generates chaotic phenomena, which is a characteristic 
behavior of the Chua's circuit. The chaotic oscillations of Vc1 and Vc2 are clearly visible in 
Fig. 6a and Fig. 6b, respectively. Furthermore, the time response of Il in Fig. 6c. 

Fig. 7 demonstrates the experimental behavior of Chua's circuit under the application 
of the proposed super twisting control scheme. As shown in Fig. 7a and Fig. 7b, the 
evolution of capacitor voltage Vc1 and Vc2 are closely follows their references, which 
indicates that the synchronization design is guaranteed and effectively realized. Fig. 7c 
illustrates the experimental behavior of the static error, which confirms that the proposed 
scheme is both effective and convincing. To further investigate the performance of the 
super twisting sliding mode algorithm, we evaluated the proposed scheme under various 
conditions, including both fixed point and periodic orbit scenarios. As illustrated in 
Fig. 8.a and Fig. 8.b, the effectiveness of our proposed scheme is clearly demonstrated in 
closed-loop. Additionally, to evaluate the robustness of the proposed method, we also 



 Chaos Synchronization Using Super-Twisting Sliding Mode Control Applied on Chua’s Circuit 375 

 

tested its performance in the presence of disturbances and system parameter variations. 
The results show that our proposed method is effective in ensuring the control, 
synchronization and stability of the Chua's circuit system. 

 

Fig. 6 Experimental results in open loop: (a) current Inductor IL (b) voltage capacitor Vc1 

(c) voltage capacitor Vc2 

 

Fig. 7 Experimental time response of the state variables in closed loop with: (a) voltage 

capacitor 
1Vc , (b) voltage capacitor 

2Vc , (c)Time responses of the tracking errors 



376 A. CHIBANI, B. DAAOU, A. GOUICHICHE, A. SAFA, Y. BADAOUI, Z. CHEDJARA 

 

 

 

Fig. 8 Time response of the state’s variable with different reference (a) capacitor voltage 

1Vc , (b) capacitor voltage 
2Vc  

5. CONCLUSION 

In this paper, we presented a chaos controller algorithm for Chua’s chaotic system. A 

new application for super twisting sliding mode controller was proposed. The controller 

scheme aims to making the states of Chua’s circuit to track their desired states; therefore, 

we tested the ability of the proposed approach under different situations. However, 

synchronization results of the closed-loop Chua’s circuit have been achieved based on 

Lyapunov stability theory. Moreover, an experimental test setup is presented to 

demonstrate the effectiveness of the proposed method. 

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[2] Z. Wang and K. T. Chau, "Anti-control of chaos of a permanent magnet DC motor system for vibratory 

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