““Multiscroll Chaotic Oscillators Design in Multiscroll Chaotic Oscillators Design in 1D, 2D 3D: Analysis, Theory and 1D, 2D 3D: Analysis, Theory and

ApplicationsApplications””

By:

Dr. Carlos Sánchez-López

Autonomous University of Tlaxcala, Mexico

Department of Electronics

1

Outline

• What is Chaos?What is Chaos?

• Previous WorkPrevious Work

• Design of Chaotic Oscillators in 1D Design of Chaotic Oscillators in 1D

• Analysis, Simulation and Experimental results Analysis, Simulation and Experimental results

• Design of Chaotic Oscillators in 2DDesign of Chaotic Oscillators in 2D

• Analysis, Simulation and Experimental results Analysis, Simulation and Experimental results

• Design of Chaotic Oscillators in 3DDesign of Chaotic Oscillators in 3D

• Analysis and Numeric Simulations Analysis and Numeric Simulations

• ApplicationsApplications

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What is Chaos?

Chaotic oscillators can be used in various applications such as medicine, biology, communications system, etc.

Traditionally, the nonlinear resistor is made with OPAMP, CFOA, CCII, OTA as the principally block.

Block Transmitter M-DCSKBlock Transmitter M-DCSK3

Previous Work

Chua´s circuit descomposition: a systematic design approach for chaotic oscillators. A.S. Elwakil, M.P. Kennedy, Journal of the Franklin Institute, pp. 251-265, February 2000.

On Current Feedback Operational Amplifier-Based Realizations of Chua´s Circuit. Recai Kilic, Circuits Systems Signal Processing,Vol. 22 No. 5 2003, pag. 475-491.

An Improved Chua’s Circuit and Its Use in Hyperchaotic CircuitGaurav Gandhi, Analog Integrated Circuits and Signal Processing, Vol. 46, pag. 173-178, 2006.

Improved implementation of Chua´s Chaotic Oscillator using Current Feedback Opamp. A.S. Elwakil and M.P. Kennedy, IEEE Transactions on Circuits and Systems I, Vol. 47 No 1 January 2000.

Inductorless Realization of Chua OscillatorO. Morgul, ELECTRONICS LETTERS, 17th August 1995, Vol 31, No 17.

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Chua’s Circuit

Cruz, J.M. Chua, L.O. [1993] “A CMOS IC nonlinear resistor for Chua's circuit,” “A CMOS IC nonlinear resistor for Chua's circuit,” IEEE Trans. Circuits Syst.-I 39, 985-995.

α=10, β=14.87, m0=-0.68, m1=-1.27, the system produces a double-scroll chaotic attractors

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Kennedy M.P. [1992] “Robust Opamp Realization of Chus’a Circuit,” “Robust Opamp Realization of Chus’a Circuit,” Frequenz, vol. 46, no.3-4, pp. 66-80

Chua’s Circuit: Nonlinear resistor with Opamp

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Nonlinear resistor with CFOA

Elwakil [2000] “Chua’s circuit descomposition: a systematic design approach for chaotic oscillators,” “Chua’s circuit descomposition: a systematic design approach for chaotic oscillators,” Elsevier, pp. 251-265

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Nonlinear resistor with CCII

Gandhi G. [2006] “An improved Chua’s Circuit and Its use in Hyperchaotic Circuit” “An improved Chua’s Circuit and Its use in Hyperchaotic Circuit” Analog Integrated Circuits and Analog Integrated Circuits and Signal Processing, 46, 173-178Signal Processing, 46, 173-178

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Nonlinear Resistor with UGCs

Sanchez, et.al. [2007] “Low-Voltage Chaotic Oscillator Using Voltage and Current Followers,” “Low-Voltage Chaotic Oscillator Using Voltage and Current Followers,” IEEE CERMA pp. 159-163. 9

Grounded Inductor with UGCs

Sanchez, et.al. [2008] “Simulation of Chua’s Chaotic Oscillator Using Unity-Gain Cells” “Simulation of Chua’s Chaotic Oscillator Using Unity-Gain Cells” IEEE ICCDCS, Cancun Mexico..

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CFOA Connected as UGCs

Zin=∞ Zout=0

Zout=∞ Zin=0

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Nonlinear Resistor with UGC

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Nonlinear Resistor with UGCs

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Experimental and Simulation Results

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Experimental and Simulation Results

15Sanchez, et.al. [2008] “Experimental Verification of the Chua’s Circuit Designed with UGCs” “Experimental Verification of the Chua’s Circuit Designed with UGCs” IEICE Electron. Express, vol. 5, no. 17, pp. 657-661

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1D-Multi-scroll Chaotic Oscillator: Analysis, Simulation and Experimental

Results

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1D Multi-scroll Chaotic Oscillator

• Attractors via PWL Functions.Attractors via PWL Functions.

• Generalized Chua’s circuit.Generalized Chua’s circuit.

• Sine Function ApproachSine Function Approach

• Nonlinear Transconductor Nonlinear Transconductor MethodMethod

• Cubic FunctionCubic Function

• Piecewise-quadratic Function.Piecewise-quadratic Function.

• Saturated Functions.Saturated Functions.

• Hysteresis series method.Hysteresis series method.

• Sawtooth Function.Sawtooth Function.

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f(vc1)=h1vc1+h2v3c1

f(vc1)=pvc1+qvc1|vc1|

Transconductor no-lineal

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Saturated Functions

xdfczbyaxz

zy

yx

22

+

_

Rz

Ry

Rx

RRi

Rf

+

_

+

_

RRi

Rf

+

_

+

_

NLF

C

Cz-y

y-x

x

C

ix

yd

dx

z

d

dy

))(( Raxi

Ra

Rz

Rz

Ry

Ry

Rx

Rx

R

d

dz

+

_

Rf

Rin

U=Ei

Rc u

+

_S(u-Ei)

+

_is(u-Ei)

Saturated Functions

BC

c1

c2

c3

c4

BC

BC

BC

inputi(u)

N - Basic Cellcn

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Saturated Functions

-0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.1

-0.05

0

0.05

0.1

X

Y

-0.1 -0.05 0 0.05 0.1 0.15-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

X

Z

-0.2 -0.1 0 0.1 0.2 0.3-0.1

-0.05

0

0.05

0.1

X

Y

-0.2 -0.1 0 0.1 0.2 0.3-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

X

Z

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Ri

Rf

+

_ Rcxi(x)

Ri

Rf

+

_ Rc

Ri

Rf

+

_ Rcxi(x)

Ri

Rf

+

_ Rc

+E 1

-E 1

Saturated Functions

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Ri

Rf

+

_ Rc

Ri

Rf

+

_ Rcxi(x)

Ri

Rf

+

_ Rc

+E 1

-E 1

Ri

Rf

+

_ Rc

+E 2

Ri

Rf

+

_ Rc

-E 2

Saturated Functions

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2D-Multi-scroll Chaotic Oscillator: Analysis, Simulation and Experimental

Results

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Saturated Functions

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Saturated Functions with CCII+

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Saturated Functions

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Generalized Chua’s Circuit

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Generalized Chua’s Circuit

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Generalized Chua’s Circuit

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3D-Multi-scroll Chaotic Oscillator: Analysis and Numeric Simulation

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Saturated Functions

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Saturated Functions

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Saturated Functions

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Applications to Secure Communications

38

Applications

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Applications: Duffing and Chua’s Systems

TransmitterReceiver

Synchronization Circuit

Applications

Applications

42

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Thanks you for

your attention