““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
2
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.
4
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
5
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
6
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
7
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
8
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..
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
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
16
1D-Multi-scroll Chaotic Oscillator: Analysis, Simulation and Experimental
Results
17
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.
18
f(vc1)=h1vc1+h2v3c1
f(vc1)=pvc1+qvc1|vc1|
Transconductor no-lineal
21
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
23
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
24
Ri
Rf
+
_ Rcxi(x)
Ri
Rf
+
_ Rc
Ri
Rf
+
_ Rcxi(x)
Ri
Rf
+
_ Rc
+E 1
-E 1
Saturated Functions
25
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
26
2D-Multi-scroll Chaotic Oscillator: Analysis, Simulation and Experimental
Results
27
Saturated Functions
28
Saturated Functions with CCII+
29
Saturated Functions
30
Generalized Chua’s Circuit
31
Generalized Chua’s Circuit
32
Generalized Chua’s Circuit
33
3D-Multi-scroll Chaotic Oscillator: Analysis and Numeric Simulation
34
Saturated Functions
35
Saturated Functions
36
Saturated Functions
37
Applications to Secure Communications
38
Applications
39
Applications: Duffing and Chua’s Systems
TransmitterReceiver
Synchronization Circuit
Applications
Applications
42
43
Thanks you for
your attention