Post on 18-Apr-2022
Chua Circuit
Experimental
Realization
Dany
Masante
PHY256: NCASO β Final Project
UC Davis Physics
6 June 2017
Chuaβs
Circuit
1
The equations of motion
ππ1ππ‘
= 1
π πΆ1( π2 β π1 β π β π(ππ)
ππ2ππ‘
= 1
π πΆ2( π1 β π1 β π β ππΏ)
πππΏππ‘
= 1
πΏ(βπ£2)
Where π π₯ is Chuaβs Diode
(piecewise function)
" π " β
" π " β
" π " β
y x
z
+ +
β β
Three Fixed Points
π1,2 = Β±βπΈ(πΊπβπΊπ)(ππΏ + π )
1 + (π + ππΏ)πΊπ,
βπΈ(πΊπβπΊπ)(ππΏ)
1 + (π + ππΏ)πΊπ,
+πΈ(πΊπβπΊπ)(ππΏ)
1 + (π + ππΏ)πΊπ
π0 = 0 , 0, 0
EXPERIMENTAL
REALIZATION
2
Experimental Realization
Where π 1 = 220 [Ξ©] π 2 = 220 [Ξ©] π 3 = 2.2 [πΞ©] π 4 = 22 [πΞ©] π 5 = 22 [πΞ©] π 6 = 3.3 [πΞ©] πΆ1 = 10 [ππΉ] πΆ2 = 100 [ππΉ] πΏ = 18 [ππ»]
And π is a potentiometer
(variable resistor)
Chua Diode
π π£1 =
π0π£1 + π0 βπ1 πΈ1, π£1 β€ πΈ1π1π£1, βπΈ1 β€ π£1 β€ πΈ1π0π£1 + π1 βπ0 πΈ1, π£1 β₯ πΈ1
Where
π0 =1
π 4β
1
π 3 , π1 = β
1
π 3β
1
π 6 , πΈ1 =
π 3
(π 2+π 3)πΈπ ππ‘
+
β
-6 -4 -2 0 2 4 6-3
-2
-1
0
1
2
3x 10
-3
V1 [V]
I [A
]
Chua Diode
List of Materials Device Stock Code Quantity Description
Capacitor Mapln Sc Series 2 Metallized Polyester Film
(MKT) Capacitors
Inductor PCH45X186KLT 1 Axial Lead Power Choke
Resistor TOPCOFRLD008 6 Metal Film full range
resistor
Potentiometer B01LYHMKNN 1 Logarithmic Dual Rotary
Potentiometer
Operational
Amplifier
TI6PCSUA741CP 2 General Purpose
Operational Amplifier
All elements are Rohs complilant
Experimental Realization
NUMERICAL
3
Numerical Realization β’ Sagemath
β’ Dimensionless Chua Eqs.
β’ Finite difference method
β’ Timestep = 0.000001
β’ Iterations = 100000
β’ Discarded (transtients)= 10%
β’ Initial Conditions= Aleatory negative initial conditions ( -10 to 0 )
RESULTS
4
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
(π΅ππ πππ ππππ πππππ πππππβ¦ )
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ%
π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
β= ππ. ππ% π = ππππ [π] π = ππππ [π]
Numerical VS Experimental
The outmost limit cycle
Experimental: Dancing around
the outmost limit cycle
Synchronization
5
Bidirectional Chuaβs Circuit
Experimental Realization
Chua 1 Chua 2
Synchronization Resistor
Bidirectional Chuaβs Circuit
Bidirectional Chuaβs Circuit
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[3] Jakob LavrΓΆd, βThe Anatomy of a Chua Circuitβ, Bachelor Thesis Equivalent to 15 ECTS. Division of Mathematical Physics. Lunds Universitet.
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[10] Dedieu, H., M.P. Kennedy and M. Hasler, Chaos shift keying: Modulation and demodulation of a chaotic carrier using self-synchronizing Chuaβs circuits. IEEE Transactions on Circuits and Systems II 40(10), 634β642. 1993.
[11] Heagy, J.F., T.L. Carroll and L.M. Pecora. Synchronous chaos in coupled oscillator systems. Physical Review E 50(3), 1874β1886. 1994
[12] Christine A. Skarda & Walter J. Freeman. How brains make chaos in order to make sense of the world. Behavioral and Brain Sciences 10. 1987
[13] Garfinkel A. A mathematics for physiology. Am J Physiol. 1983