Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 7, No. 1, 2023 

 

149 

Research	on	the	Characteristics	of	Typical	Chaotic	
Circuits	 	
Yong Deng, Bowen Li 

Southwest Petroleum University, Chengdu 610065, China 

 

Abstract: Chaotic circuit, also called nonlinear circuit. Up to now, there is no unified definition in academic circles. The most 
typical chaotic circuit is the Chua's circuit invented by the Chinese scientist Cai Shaotang. In order to study the chaos 
characteristics of Chua's circuit, this paper studies the hardware design and software simulation of Chua's circuit. In terms of 
hardware, in order to verify the feasibility of the components selected in this paper, the simulation is carried out on the basis of 
Multisim software. In the aspect of software, based on the best display effect of the result graph, this paper chooses Simulink in 
MATLAB software to simulate the model. Under the software simulation, the chaos phenomenon is observed. Through the 
modification of various parameters, the chaos imagination corresponding to the theory is obtained. Finally, the research results 
are analyzed and the conclusion is obtained. 

Keywords: Chaos, Chua's circuit, Simulation, Matlab, Characteristics. 
 

1. Introduction 
There is still no clear definition of chaos, which means that 

there is still a huge unknown waiting to be explored. As for 
the chaotic circuits, many improved chaotic circuits have 
appeared, but if we want to get more results on the chaotic 
circuits, we still need to keep on researching on them. The 
purpose of the research is to have a more complete and 
systematic results and conclusions for this area and to use 
them in practice. Chaotic circuits, which have a huge role in 
many applications, especially in the field of communication 
security, are particularly promising. In the very important area 
of communication security, the signals emitted by chaotic 
circuits have a very important characteristic, which is that 
they are extremely sensitive to certain conditions. Signals 
modulated by chaotic circuits as carriers have similar 
sensitivity. Therefore, even if the modulated chaotic signal is 
received by a non-receiver, it is very difficult to receive the 
correct signal, because the chaotic signal is very difficult to 
be cracked, which provides an important basic condition for 
the application of chaotic circuits to secure communications. 

1. theoretical foundation 
In 1983, Prof. Siu-Tong Tsai, a Chinese American scientist, 

invented Tsai's circuit. This is a kind of nonlinear circuit, and 
this circuit can exhibit the standard theoretical chaos 
phenomenon. The biggest advantage of this circuit is that it is 
simple in structure and easy to realize. It can be called the 
most typical chaotic circuit. The diagram of Tsai's circuit is 
shown below: 

 

 
Figure 1. Circuit diagram of Tsai 

The component Rn on the far right of the diagram is called 
a Zeiss diode and is not currently commercially available. 
Instead, we generally use other components. A combination 
of an operational amplifier and a resistor is usually used 
instead of a Chua's diode. The operational amplifier is not 
unique and is usually the TL082, but other operational 
amplifiers can be used. As shown in the figure below: 

 

 
Figure 2. Chua's Diode Substitution Diagram 

 
The equation of state for the Chua's circuit can be derived 

from the circuit diagram as follows: 
 

𝐶1
𝑑𝑣𝑐1

𝑑𝑡
𝐺 𝑣𝑐2 𝑣𝑐1 𝑔 𝑣𝑐1                  (1) 

 

𝐶2
𝑑𝑣𝑐2

𝑑𝑡
𝐺 𝑣𝑐1 𝑣𝑐2 𝑖𝐿                      (2) 

 

𝐿 𝑉                               (3) 

 
 
Vc1 represents the current flowing across C1, Vc2 

represents the current flowing across C2, Uc1 is the voltage 
across C1 and Uc2 represents the voltage across C2. In order 
to simplify the equations of the circuit and reduce the 
parameters, the following dimensionless equation is obtained 
as shown in the figure: 

 
𝑥 𝑎1 𝑦 𝑘 𝑥                                   (4) 

 



 

150 

𝑦 𝑥 𝑦 𝑧                                 (5) 

 

𝑧 𝑎2𝑦                                    (6) 

 

2. Hardware Circuit Simulation 
MULTISIM software has been chosen to simulate the 

analog circuit becausethescheme is clear and the circuit model 
is given directly below as follows: 

 
Figure 3. MULTISIM Chua's Circuit Modeling Diagram 

 
The model building is completed and the chaotic circuit 

phenomenon can be observed accurately and clearly, and the 
results are shown below: 

 

 
Figure 4. MULTISIM simulation result plot 

 

3. Software Modeling 
Simulink software is a powerful software from Mathworks, 

which is often used in combination with MATLAB software, 
and is usually embedded in MATLAB, equivalent to a 
functional module, which has powerful simulation and image 

display 
functions, so it is also one of the reasons for choosing it for 

this simulation. After the selection and connection of modules, 
the simulation model of the Chua's circuit was finally built. 
The built model is shown below: 

 



 

151 

 
Figure 5. Chua's circuit model diagram 

 

4. Test 

4.1. Changing the capacitance value 
In Chua's circuit there are two capacitors one is C1 and one 

is C2.C1=10nF, C2=100nF. This is fixed and generally people 
say that Chua's circuit is not changed by the parameters. And 
in this study, the fixed parameters in the Chua's circuit are 
only adjusted in order to observe different chaotic phenomena. 

For the Chua's circuit, changing the value of capacitance only 
requires changing the value of capacitance C1, and the change 
of C1 is related to the value of the gain module in the model. 
In the gain module, Gain2 represents the parameter a in the 
dimensionless equation of the Chua's circuit, i.e., if you 
change a, you change the capacitance. 

(1) When the capacitance value is 16.5nF, a=5.86. Run the 
simulation and the results are as follows: 

 

        
Figure 6. Stable state phase         Figure 7. Stable state timing 

 
As can be seen from the phase diagram, the chaotic 

phenomenon of the Chua's circuit is being stabilized at this 
point, as the trajectory is essentially smooth at this point, with 

little movement. 
(2) When C1=14.5nF, a=6.95. run the simulation with the 

following results: 
 

         
Figure 8. Single-cycle state phase          Figure 9. Single-cycle state Timing 



 

152 

 
The phase diagram at this point is a single cycle chaotic 

state. The phase diagramappears to be a ring, and the number 
of rings is not high enough to converge to one ring, so it can 

be determined that this is a single cycle chaotic state. 
(3) when C1=13.6, a=7.35, run the simulation and the 

results are as follows: 
 

            
Figure 10. Multiple Cycle state phase                 Figure 11. Multiple Cycle state Timing 

 
The phase diagram at this time is similar to the previous 

one when a=6.95, both of them are in the shape of ring, but 
the difference is that the number of rings in the phase diagram 
at this time is more, and there is obvious periodicity, so it can 

be judged that the chaotic state at this time is a multiplicative 
cycle chaotic state. 

(4)when C1=12.5nF, a=8, run the simulation and the results 
are as follows: 

 

           
Figure 12. Single vortex state phase            Figure 13. Single vortex state timing 

 
As you can tell from the phase diagram, this is a single 

vortex chaotic state, which can also be referred to 
academically as a Losle shape state. One of the single vortex 
states actually has a distinct periodicity, so sometimes the 

single vortex state is also categorized as the first cycle. 
(5)when C1=10.8nF, a=9.235, run the simulation and the 

results are as follows: 

 

           
Figure 14. Double vortex state phase   Figure 15. Double vortex state timing 

 
The phase diagram at this point was, quite obviously, 

another vortex added to the phase diagram of the single-
vortex state, which is academically called the double- vortex 
chaotic state. Although it looks chaotic, the periodicity is also 
more pronounced. 

4.2. Changing the resistance value 
For the Chua's circuit, a change in the value of the resistor 

R. A change in R is modeled as a change in the value of the 
gain module. In the gain module, Gain3 represents the 
parameter b in the dimensionless equation of the Chua's 



 

153 

circuit, i.e., if you change b, you change the resistance. In a 
Chua's circuit, the resistance of the non-Chua's diode part 
tends to be around 2KΩ. So adjusting the resistance would 

(1) When R=1.9KΩ, b=-20, run the simulation and the 
results are as follows: 

 

       
Figure 16. Stable state phase            Figure 17. Stable state timing 

 
 
As can be seen from the phase diagram, the chaotic state at 

this point is in a single vortex chaotic state, which is very 
similar to the phase diagram that appears when changing the 
value of the capacitance, although the difference is that their 

timing diagrams are a little different, the peaks of the timing 
diagrams that appear when changing the resistance are almost 
unchanged, while the other is periodically changing. 

(2) When R=1.6KΩ, b=-16. run the simulation and the 
results are as follows: 

 

     
Figure 18. Double vortex state phase          Figure 19. Double vortex state timing 

 
From the phase diagram at this point, it is known that the 

chaotic state is a double vortex state. The phase diagram is 
also almost the same as the phase diagram when changing 

capacitance. 
(3) When R=2.2KΩ, b=-24.2, run the simulation and the 

results are as follows: 

 

     
Figure 20. Multiple Cycle state phase      Figure 21. Multiple Cycle state Timing 

 
 
At this time, the phase diagram is ring-shaped, and the 

periodicity is obvious, the number of rings is more, it can be 

judged as a multiplicative cycle chaos state.  
(4) When R=2.45KΩ, b=-26.35, run the simulation and the 

results are as follows: 
 



 

154 

     
Figure 22. Single-cycle state phase           Figure 23. Single-cycle state Timing 

 
At this time, the phase diagram is ring-shaped, the number 

of rings is very small, infinitely converging to 1 ring, basically 
can be judged as a single cycle chaotic state. 

(5) When R=2.65KΩ, b=-35.5, run the simulation and the 
results are as follows:  

 

    
Figure 24. Stable state phase          Figure 25. Stable state timing 

 
As can be seen from the phase diagram at this point, the 

chaotic phenomenon of the Chua's circuit is stabilizing at this 
point, as the trajectory is essentially smooth at this point, with 
little movement. 

5. Summary 
In a Chua's circuit, there are two important parameters, the 

value of the resistanceoutside the non-Chua's diode and the 
value of the capacitor C1. Changes in both of these 
parameters affect the state of the Chua's circuit and the chaos 
phenomenon changes.As the value of capacitor C1 gradually 
varies from large to small, the value of gain a in the Chua's 
circuit model constructed in MATLAB also gradually varies 
from small to large, and the phase diagram of the chaotic 
phenomenon of the Chua's circuit becomes more and more 
divergent. 

The other parameter, the non-Cai diode resistor value, as 
this resistor value continues to get smaller, the absolute value 
of the gain b will also get smaller as the resistor value gets 
smaller. The operating state of the Chua's circuit will also 
slowly transition from an apparent steady state to a periodic 
vortex state. 

From the simulation results of this study, it seems that 
almost all chaotic states,except for the steady state, show 
significant 

The periodicity phenomenon of the circuit is also observed 
from the time domain waveform instead of the phase diagram. 
When a = 10 resistance value, this time from the time domain 

waveform, rather than the phase diagram, can also be 
observed from the obvious periodic phenomenon, indicating 
that, not only the phase diagram, the circuit's time domain 
waveform is also correspondingly periodic. However, only at 
this point in the operation of the circuit state of the most 
obvious periodicity, in other resistance values, the periodicity 
is less obvious. 

For the Chua's circuit, it is inherently very sensitive to 
changes in these two key parameters. So the chaotic state of 
the Chua's circuit is very unstable, but it exhibits a part of 
order and periodicity. 

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