INVESTIGATION INTO CHAOTIC OSCILLATOR_Public Copy.pdf

This project involved the study, analysis and simulation of the Chua's Circuit and different techniques of implementing Chua's Circuit. A comparative analysis of those existing circuits have been made. Their characteristics and outputs were simulated using Ltspice. After observing the behaviour of existing techniques , a new method was proposed in one of the circuits where we replaced the passive inductor with GIC and the circuit was analysed for all required parameters and then compared with the available circuit characteristics.

1. INVESTIGATION INTO CHAOTIC
OSCILLATOR


KISHAN GUPTA ISHU ARORA SANJANA YADAV BHOMIT

UNDER THE GUIDANCE OF
PROF. RAJ SENANI SIR
7/22/2019
1

2. RECAPPING WORK
During our coursework, we captured the following essence in our work to
investigate the chaotic oscillators:

1.Introduction to Chaotic Oscillator

2.Implementation of Chaotic Oscillator using Chua’s Circuit

3.Characteristics of our circuit.

4.Results and Inferences

3. CHAOS THEORY
Chaos theory is the branch that deals with the complex systems
whose behavior is highly sensitive to slight changes in conditions, so
that small alterations can give rise to unintended consequences.
Chaos theory is the science of surprises, and not always pleasant
surprises.
Edward Lorenz officially coined the term Chaos theory. Lorenz
studied chaos theory in the context of weather systems. His
observations on chaos theory in weather systems led to his famous
talk, which he entitled ,”Predictability : Does the flap of Butterfly’s
wings in Brazil set off a tornado in Texas?” In reference to this talk,
chaos theory has also been described as the “BUTTERFLY EFFECT”.
Butterfly effect is the idea that small things can have non-linear
impacts on a complex systems.

Picture reference here

4. INTRODUCTION TO CHAOTIC OSCILLATORS
An oscillator circuit that generates aperiodic signal that is the signal that never repeats itself, is termed as
chaotic or non-periodic or non-linear oscillator.
Chaotic Criteria :- An electronic circuit made from standard components that is resistors, capacitors and
inductors can show chaotic behaviour, if they satisfy the following three criteria :

Criteria for Chaotic Behaviour
At least one non- linear element One or more active resistor Three or more energy storage elements

5. IMPLEMENTATION OF CHAOTIC CIRCUIT USING CHUA'S CIRCUIT




What is Chua’s circuit?
Chua’s circuit is a simple electronic circuit that exhibits classical chaotic behavior. This means roughly that
it’s a “non-periodic oscillator”; it produces an oscillating waveform that unlike an ordinary electronic
oscillator, never repeats.
Why Chua’s circuit?
Chua's Oscillator also known as Chua's Circuit is the simplest electronic circuit that satisfies the minimum
required criteria for a circuit to show chaotic behaviour.


1983
RW Newcomb

proposed his method of generating chaos with the title “RC Active Chaos
Generator”
1985
Leon Chua

Proposed an electronic circuit for generating chaos, which latter was known as Chua’s
Oscillator

6. CHARACTERISTICS OF OUR CIRCUIT
MAIN CIRCUIT
NIC
Here, the first two criteria (i) non-linear
element (ii) active register , are
combined together and has been
implemented by parallel combination
of two negative impedance convertors.

The state equations are as follows:


INDUCTORLESS
The third criteria of three energy
storage elements is full filled by the
two capacitors C1 & C2 and the
inductor L.

Vc1Vc2
C1C2
Fig: Chua’s circuit

7. NIC- NEGATIVE IMPEDANCE CONVERTOR
The non-linear part of the Chua’s circuit, also known as Chua’s Diode is implemented by the parallel
combination of two NICs. Here the NIC is used as the active resistor for the circuit hence acting as
the power source.

8. OUR NIC SIMULATION RESULTS

9. GIC- GENERAL IMPEDANCE CONVERTOR
The inductor of our circuit is realised using GIC, hence making it an inductorless circuit.

10. OUR GIC SIMULATION RESULTS
Transient Response of GIC circuit
V-I characteristics of GIC

11. IMPLEMENTATION OF CHUA’S CIRCUIT
Complete Chua’s Circuit

12. RESULTS
1.APERIODIC SIGNAL
Vc1 vs Time for L=18mH , C1= 10nF, C2=100nF,
R=1.73Kohm
2. DOUBLE SCROLL ATTRACTOR
Double scroll attractor Vc2 vs Ix(x1:N1) for R=1.73k;
C1=10nF; C2= 100nF and L=18.8mH

14. MULTIPLIER CIRCUIT
● AD633J is one of the most common
analog multiplier IC
● A dual op amp IC, which takes four
inputs , two to each opamp and the
result is the product of the difference
of set of inputs given to each opamp.
●The output thus obtained is scaled by a
factor of 10 internally to minimise the
effect of high gain of the opamp.
●The result from here is then passed
through another op amp connected in
negative feedback topology and a
reference voltage is added to this result
and the output of this opamp is the
final output of our multiplier circuits.

Internal architecture of AD633J

15. OUTPUT OF MULTIPLIER CIRCUIT
Basic multiplier circuit Response of Multiplier circuit

16. REQUIREMENT OF SMOOTH NON-LINEARITY
All properties and characteristics of a real circuit can not be
obtained or simulated using piecewise linearity.
So, we have now implemented the non linearity using
smooth curves. Smooth non linear curve basically means
that a curve with continuous function that is their plot does
not have any discontinuity or break points.

One of the major criteria to be fulfilled by the non linear
element is being odd symmetric about origin that is having
characteristics which lies in opposite quadrants about
origin.

The function which can be used reliably is the cubic
polynomial.

17. REALIZATION OF CUBIC POLYNOMIAL
1.Cubic Polynomial Function
can be obtained through
simple analog multiplier
circuits.
2. To obtain the degree three
polynomial function we have
to cascade two such analog
multiplier circuits.
3.The circuits are connected in
feedback topology.
Cubic polynomial as active resistor

18. GRAPH AND EQUATIONS
Response of Cubic polynomial as active resistor

19. ACTIVE RESPONSE OF CUBIC POLYNOMIAL
It is necessary for the cubic polynomial
circuit to be active as well. This
characteristic can be achieved by
connecting a negative impedance
converter (NIC) circuit with the cubic
polynomial circuit. The NIC is
connected in parallel with the second
multiplier circuit. So, now it has
become an active element.
Active response of Cubic polynomial

20. CHUA’S CIRCUIT USING CUBIC POLYNOMIAL

21. 1
2
3
1.Double scroll
attractor Vn001 vs
Vn002*
2.Double scroll
attractor Vn002 vs
Ix*
3.Aperiodic signal
generated by
Chua’s circuit.*

*(For R1=1.9k, C1=7nF,
C2=78nF, L=18.8mH)

22. WHAT IS MEMRISTOR?
THE FOURTH FUNDAMENTAL ELEMENT
A memristor is an electrical component that
limits or regulates the flow of electrical current in
a circuit and remembers the amount of charge
that has previously flown through it.
Memristors are important because they are
non-volatile, meaning that they retain memory
without power.
Conceptual symmetries of resistor, capacitor, inductor, and memristor.

Picture reference here

23. MEMRISTOR PROPERTY AND CHARACTERISTICS
Its special property is that its
resistance is programmable (resistor
function) and can be stored
subsequently (memory function).
Unlike others that exist today in
modern electronics, memristors are
the stable memories and keeps
remembering their state even if the
electronic circuit has lost their power.

24. IMPLEMENTATION OF MEMRISTOR WITH CUBIC POLYNOMIAL
V-I characteristic of memristor
shows the property of hysteresis
and lies in the first and the third
quadrant. Now we have integrated
the cubic polynomial with the
memristor. Since we know that the
cubic polynomial is an active
circuit as it contains NIC so the
memristor hysteresis loop will
now lie in the second and fourth
quadrant.
There will be a cubic relation
between Q and Φ.

25. GRAPH AND EQUATIONS
V-I characteristic of memristor with cubic polynomial

26. CHUA’S CIRCUIT USING MEMRISTOR
Conventional Chua's Circuit contains one non-linear element. We will use memristor as that non-linear
element.One of the major characteristic of this circuit is that it is a physical chaotic device that employs
the four fundamental circuit elements - capacitor, resistor,inductor and memristor.

27. CHUA’S CIRCUIT WITH MEMRISTOR
Chua’s circuit with memristor as non-linear element

28. RESULTS
Limit cycle as observed from Chua’s circuit with memristor
(For R=110k; C1=6.8nF; C2=68nF; L=18.8mH)
Limit cycle as observed from Chua’s circuit with memristor
(For R=100k; C1=6.8nF; C2=68nF; L=18.8mH)

29. HARDWARE IMPLEMENTATION
Hardware implementation of Chua’s Circuit with
NIC as Chua’s Diode
Aperiodic signals for Vc2 vs time
for R=1.7k; C1=10nF; C2= 100nF;
L=18.8mH

30. Aperiodic signals for Vc1 vs time for
R=1.7k; C1=10nF; C2= 100nF; L=18.8mH
Aperiodic signals for R=1.7k; C1=10nF; C2= 100nF;
L=18.8mH

31. VIDEOS OF OUR SIMULATIONS
Simulation for aperiodic signal

32. Simulation for attractor

33. CONCLUSION

●Building blocks like operational amplifiers (AD712) and multipliers (AD633J) were
studied in detail. These circuits were analysed to study the theory of chaos and chaotic
circuits in detail.
●Firstly, Chua’s circuit was stimulated with Negative Impedance Converter as the non
linear element on LTSpice and the results were verified. Double scroll attractors were
obtained in the output.
●Another circuit (Chua’s circuit) with Memristor as the nonlinear element was analysed.
One of the major characteristics of this circuit is that it is a physical circuit that takes
into all the 4 fundamental circuit elements- resistor, capacitor, inductor and memristor.
●Memristor is implemented with the cubic polynomial. We started with the multiplier
circuit (AD633J) that was analysed first and with its help cubic polynomial is simulated.
And finally the memristor was simulated and the results were verified.
●All the existing methods have used the passive inductor as their third element, but in
our circuits we have used Generalized Impedance Converter to realise the inductor.
This project aimed at the study of chaotic circuits with chua’s circuit as the basic one
which further can be used in different fields like Communications , Detection of weak
signals etc.

34. APPENDIX
1.KENNEDY, M.P.: “Robust op-amp realization of Chua’s circuit”, Frequenz, 1992, 46, pp. 66-80
2.R. Senani and S. S. Gupta, “Implementation of Chua's chaotic circuit using current feedback op amps,” Electron. Lett., vol.
34, pp. 829–830, Apr. 1998.
3.R.W.Newcomb and S.Sathyan, “An RC-op-amp Chaos Generator”, IEEE Trans. Circuits and Systems, Vol.30, pp.54-56,1983.
4.ABUELMA ATTI, M.T.: “Chaos in an autonomous active-R circuit”, IEEE Trans. Circuits Syst. I: Fundam. theory appl., 1995,
42, (l), pp. 1-5
5.L. 0. Chua. “Global unfolding of Chua’s circuit”, DEICE Trans. Fund. Electr. Comm. Comput. Sci.. Vol. E76-A, pp. 704734,
1993.
6.Leon O. Chua , “The Genesis of Chua's Circuit”, AEO, Vol. 46 (1992), No.4
7.MATSUMOTO, T.: “A chaotic attractor from Chua’s circuit”, IEEE Trans. CAS, 1984, CAS-31, pp. 1055-1058
8.MORGUL, 0.: “Inductorless realisation of Chua’s oscillator”, Electron. Lett., 1995, 31, (17), pp. 1403-1404
9.Pao-Lung Chen , “An Inductorless Chua’s Circuit with Memristor”; 2nd International Conference on Information Science and
Control Engineering ; 2015 ;
10.BHARATHWAJ MUTHUSWAMY, “ IMPLEMENTING MEMRISTOR BASED CHAOTIC CIRCUITS”, International Journal of
Bifurcation and Chaos, Vol. 20, No. 5 (2010) 1335–1350 c World Scientific Publishing Company
DOI: 10.1142/S0218127410026514
11.S.Nakagawa and Toshimichi Saito , “An RC OTA Hysteresis Chaos Generator”; IEEE International Symposium on Circuits and
Systems. Circuits and Systems Connecting the world. ISCAS, 1996