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

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

3. 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-DCSK 3

4. 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 Circuit Gaurav 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 Oscillator O. Morgul, ELECTRONICS LETTERS, 17th August 1995, Vol 31, No 17. 4

5. Chua’s Circuit “A CMOS IC nonlinear resistor for Chua's circuit,” Cruz, J.M. Chua, L.O. [1993] “A CMOS IC nonlinear resistor for Chua's circuit,” IEEE Trans. Circuits Syst.-I 39, 985-995. α=10, β=14.87, m 0 =-0.68, m 1 =-1.27, the system produces a double-scroll chaotic attractors 5

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

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

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

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

10. Grounded Inductor with UGCs “Simulation of Chua’s Chaotic Oscillator Using Unity-Gain Cells” Sanchez, et.al. [2008] “Simulation of Chua’s Chaotic Oscillator Using Unity-Gain Cells” IEEE ICCDCS, Cancun Mexico.. 10

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

12. Nonlinear Resistor with UGC 12

13. Nonlinear Resistor with UGCs 13

14. Experimental and Simulation Results 14

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

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

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

18. 18

19. f(v c1 )=h 1 v c1 +h 2 v 3 c1 f(v c1 )=pv c1 +qv c1 |v c1 |

20. Transconductor no-lineal

21. 21 Saturated Functions

22. 22 Saturated Functions

23. 23 Saturated Functions

24. 24 Saturated Functions

25. 25 Saturated Functions

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

27. 27 Saturated Functions

28. 28 Saturated Functions with CCII+

29. 29 Saturated Functions

30. 30 Generalized Chua’s Circuit

31. 31 Generalized Chua’s Circuit

32. 32 Generalized Chua’s Circuit

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

34. 34 Saturated Functions

35. 35 Saturated Functions

36. 36 Saturated Functions

37. 37 Applications to Secure Communications

38. 38 Applications

39. 39 Applications: Duffing and Chua’s Systems

40. Transmitter Receiver Synchronization Circuit Applications

42. 42

43. 43 Thanks you for your attention