Multiscroll Chaotic Oscillators Design in 1 D 2

“Multiscroll Chaotic Oscillators Design in 1 D, 2 D 3 D: Analysis, Theory and Applications” By: Dr. Carlos Sánchez-López Autonomous University of Tlaxcala, Mexico Department of Electronics 1

Outline • What is Chaos? • Previous Work • Design of Chaotic Oscillators in 1 D • Analysis, Simulation and Experimental results • Design of Chaotic Oscillators in 2 D • Analysis, Simulation and Experimental results • Design of Chaotic Oscillators in 3 D • Analysis and Numeric Simulations • Applications 2

What is Chaos? Ø Chaotic oscillators can be used in various applications such as medicine, biology, communications system, etc. Ø Traditionally, the nonlinear resistor is made with OPAMP, CFOA, CCII, OTA as the principally block. Block Transmitter M-DCSK 3

Previous Work q 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. q 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. q An Improved Chua’s Circuit and Its Use in Hyperchaotic Circuit Gaurav Gandhi, Analog Integrated Circuits and Signal Processing, Vol. 46, pag. 173 -178, 2006. q 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. q Inductorless Realization of Chua Oscillator O. Morgul, ELECTRONICS LETTERS, 17 th August 1995, Vol 31, No 17. 4

Chua’s Circuit α=10, β=14. 87, m 0=-0. 68, m 1=-1. 27, the system produces a double-scroll chaotic attractors 5 Cruz, J. M. Chua, L. O. [1993] “A CMOS IC nonlinear resistor for Chua's circuit, ” IEEE Trans. Circuits Syst. -I 39, 985 -995.

Chua’s Circuit: Nonlinear resistor with Opamp Kennedy M. P. [1992] “Robust Opamp Realization of Chus’a Circuit, ” Frequenz, vol. 46, no. 3 -4, pp. 66 -80 6

Nonlinear resistor with CFOA Elwakil [2000] “Chua’s circuit descomposition: a systematic design approach for chaotic oscillators, ” Elsevier, pp. 251 -265 7

Nonlinear resistor with CCII Gandhi G. [2006] “An improved Chua’s Circuit and Its use in Hyperchaotic Circuit” Analog Integrated Circuits and 8 Signal Processing, 46, 173 -178

Nonlinear Resistor with UGCs Sanchez, et. al. [2007] “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” IEEE ICCDCS, Cancun Mexico. . 10

CFOA Connected as UGCs Zin=∞ Zout=0 Zout=∞ Zin=0 11

Nonlinear Resistor with UGC 12

Nonlinear Resistor with UGCs 13

Experimental and Simulation Results 14

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

1 D-Multi-scroll Chaotic Oscillator: Analysis, Simulation and Experimental Results 16

1 D Multi-scroll Chaotic Oscillator • Attractors via PWL Functions. • Generalized Chua’s circuit. • Sine Function Approach • Nonlinear Transconductor Method • Cubic Function • Piecewise-quadratic Function. • Saturated Functions. • Hysteresis series method. • Sawtooth Function. 17

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f(vc 1)=h 1 vc 1+h 2 v 3 c 1 f(vc 1)=pvc 1+qvc 1|

Transconductor no-lineal

Saturated Functions 21

Saturated Functions 22

Saturated Functions 23

Saturated Functions 24

Saturated Functions 25

2 D-Multi-scroll Chaotic Oscillator: Analysis, Simulation and Experimental Results 26

Saturated Functions 27

Saturated Functions with CCII+ 28

Saturated Functions 29

Generalized Chua’s Circuit 30

Generalized Chua’s Circuit 31

Generalized Chua’s Circuit 32

3 D-Multi-scroll Chaotic Oscillator: Analysis and Numeric Simulation 33

Saturated Functions 34

Saturated Functions 35

Saturated Functions 36

Applications to Secure Communications 37

Applications 38

Applications: Duffing and Chua’s Systems 39

Applications Transmitter Receiver Synchronization Circuit

Applications

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r o f u o y s k n Tha n o i t n e t t a r u yo 43