1 chaotic circuits the way the electron bounces. 2 topics l linear circuits –inductor –diode l...
TRANSCRIPT
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CHAOTIC CIRCUITS
The way the electron bounces
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TOPICS Linear Circuits
– Inductor– Diode
Non-Linear Circuits– How can we tell if a circuit is behaving non-linearly ? – Diode and inductor– Transistor
Bifurcation Diagrams The Feigenbaum number Why are these systems chaotic ? Attractors
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MAY POPULATION MODEL
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Basic Population Model
POPULATION SEX
FOODSUPPLY
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May’s Population Model
POPULATION SEX
FOODSUPPLY
FEEDBACK
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Attractor for May model
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LINEAR CIRCUITS
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INDUCTOR
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Input Voltage: 5 V
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Input Voltage: 10 V
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Input Voltage: 15 V
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Input Voltage: 20 V
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Inductor (Output vs. Input)
Input Voltage (peak to Peak)
Out
put
Vol
tage
(pe
ak t
o pe
ak)
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DIODE
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Input Voltage: 5 V
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Input Voltage: 10 V
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Input Voltage: 15 V
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Input Voltage: 20 V
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Diode (Output vs. Input Voltage)
Input Voltage (peak to peak)
Out
put
Vol
tage
(p
eak
to p
eak)
y = 0.4949 + 0.7762
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NON-LINEAR CIRCUITS
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How can we tell if a circuit is behaving non-linearly?
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Period 1
PM3394A
ch1
ch2
CH1 1 V~ STOPCH2 1 V~ MTB5.00us ch1+
1
2
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Period 2
PM3394A
ch1
ch2
CH1 2 V~ STOPCH2 2 V~ MTB5.00us ch1+
1
2
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Period 4
PM3394A
ch1
ch2
CH1 5 V~ STOPCH2 2 V~ MTB5.00us ch1+
1
2
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Diode-Inductor Circuit
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Circuit Schematic
Inductor5.89 mH
Diode
VariableVoltageSource AC
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Period 1
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Period 2
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Period 4
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Period 8
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Period 16
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Chaos
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Transistor Circuit
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Circuit SchematicInductor
A.C.FunctionGenerator
Transistornpn type
Variable Resistor
D.C. Voltage Source
Load inductor
LoadVariableresistor
GroundGround
Feedback
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FEEDBACK
SIGNAL AMPLIFIER
POWERSUPPLY
OUTPUT
Simplified Schematic
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Period 1
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Period 2
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Period 4
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Period 8
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Period 16
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Chaos
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BIFURCATION DIAGRAMS
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Bifurcation Diagram (Inductor)O
utpu
t V
olta
ge (
peak
to
peak
)
Input Voltage (peak to Peak)
y = 1.3717x - 0.0724
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Out
put V
olta
ge (
peak
to p
eak)
Input Voltage (peak to peak)
Bifurcation Diagram (Diode-inductor)
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Bifurcation Diagram (Transistor)
Input Voltage (peak to peak)
Out
put V
olta
ge (
peak
to p
eak)
![Page 49: 1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l Non-Linear Circuits –How can we tell if a circuit is behaving](https://reader030.vdocuments.us/reader030/viewer/2022032710/56649f155503460f94c2ada7/html5/thumbnails/49.jpg)
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Mathematical model (May model)
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Input Voltage (peak to peak)
Out
put V
olta
ge (
peak
to p
eak)
Chaotic Region (diode-inductor)
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Out
put V
olta
ge (
peak
to p
eak)
Input Voltage (peak to peak)
Periodic Region Amplified (diode)
![Page 55: 1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l Non-Linear Circuits –How can we tell if a circuit is behaving](https://reader030.vdocuments.us/reader030/viewer/2022032710/56649f155503460f94c2ada7/html5/thumbnails/55.jpg)
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Feigenbaum Number
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Out
put V
olta
ge (
peak
to p
eak)
Input Voltage (peak to peak)
4.7 V
1.1 V 1 = 4.7 / 1.1 = 4.27
Diode-Inductor Circuit
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Transistor Circuit1.2 V
0.28 V
1 = 1.2 / 0.28 = 4.285
Transistor Circuit
Input Voltage (peak to peak)
Out
put V
olta
ge (
peak
to p
eak)
![Page 60: 1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l Non-Linear Circuits –How can we tell if a circuit is behaving](https://reader030.vdocuments.us/reader030/viewer/2022032710/56649f155503460f94c2ada7/html5/thumbnails/60.jpg)
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Feigenbaum Number
Theoretical value: ∞= 4.669201
Experimental value: = 4.27 (diode-inductor)
Experimental value: = 4.285 (transistor)
Other experimental values recorded:
Electrical circuit (varactor) = 4.257
Fluid Mechanics (Convection) = 4.4
![Page 61: 1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l Non-Linear Circuits –How can we tell if a circuit is behaving](https://reader030.vdocuments.us/reader030/viewer/2022032710/56649f155503460f94c2ada7/html5/thumbnails/61.jpg)
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Why are these systems chaotic ?
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Water gate analogy of a diode
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Water rate corresponding to time constant
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Simple Linear Amplifier
SIGNAL AMPLIFIER
POWERSUPPLY
![Page 65: 1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l Non-Linear Circuits –How can we tell if a circuit is behaving](https://reader030.vdocuments.us/reader030/viewer/2022032710/56649f155503460f94c2ada7/html5/thumbnails/65.jpg)
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Chaotic Amplifier Circuit
SIGNAL AMPLIFIER
POWERSUPPLY
FEEDBACK
![Page 66: 1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l Non-Linear Circuits –How can we tell if a circuit is behaving](https://reader030.vdocuments.us/reader030/viewer/2022032710/56649f155503460f94c2ada7/html5/thumbnails/66.jpg)
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ATTRACTORS
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Inductor
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Input Voltage: 3.23 V
PM3394A
ch1: pkpk= 3.23 V
CH1 2 V~ X= CH2 2 V~
1
PM3394A
ch1: pkpk= 3.23 V
CH1 2 V~ X= CH2 2 V~
1
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Input Voltage: 4.44 V
PM3394A
ch1: pkpk= 4.44 V
CH1 2 V~ X= CH2 2 V~
1
PM3394A
ch1: pkpk= 4.44 V
CH1 2 V~ X= CH2 2 V~
1
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Input Voltage: 5.74 V
PM3394A
ch1: pkpk= 5.74 V
CH1 2 V~ X= CH2 2 V~
1
PM3394A
ch1: pkpk= 5.74 V
CH1 2 V~ X= CH2 2 V~
1
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Input Voltage: 7.11 V
PM3394A
ch1: pkpk= 7.11 V
CH1 2 V~ X= CH2 2 V~
1
PM3394A
ch1: pkpk= 7.11 V
CH1 2 V~ X= CH2 2 V~
1
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Input Voltage: 9.18 V
PM3394A
ch1: pkpk= 9.18 V
CH1 2 V~ X= CH2 2 V~
1
PM3394A
ch1: pkpk= 9.18 V
CH1 2 V~ X= CH2 2 V~
1
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Input Voltage: 11.0 V
PM3394A
ch1: pkpk= 11.0 V
CH1 2 V~ X= CH2 2 V~
1
PM3394A
ch1: pkpk= 11.0 V
CH1 2 V~ X= CH2 2 V~
1
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Input Voltage: 13.0 V
PM3394A
ch1: pkpk= 13.0 V
CH1 2 V~ X= CH2 2 V~
1
PM3394A
ch1: pkpk= 13.0 V
CH1 2 V~ X= CH2 2 V~
1
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Diode-Inductor
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Period 1Input Voltage: 2.16 V
PM3394A
ch2: pkpk= 2.16 V
CH2 1 V~ X= CH1 1 V~
2
PM3394A
ch2: pkpk= 2.16 V
CH2 1 V~ X= CH1 1 V~
2
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Butterfly Diagram (Period 2)Input Voltage: 2.75 V
PM3394A
ch2: pkpk= 3.10 V
CH2 1 V~ X= CH1 1 V~
2
PM3394A
ch2: pkpk= 3.10 V
CH2 1 V~ X= CH1 1 V~
2
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Period 4Input Voltage: 7.45 V
PM3394A
ch2: pkpk= 6.82 V
CH2 1 V~ X= CH1 1 V~
2
PM3394A
ch2: pkpk= 6.82 V
CH2 1 V~ X= CH1 1 V~
2
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Period 8Input Voltage: 8.55 V
PM3394A
ch2: pkpk= 7.23 V
CH2 1 V~ X= CH1 1 V~
2
PM3394A
ch2: pkpk= 7.23 V
CH2 1 V~ X= CH1 1 V~
2
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CHAOSInput Voltage: 8.5 V
PM3394A
ch2: pkpk= 8.85 V
STOPCH2 2 V~ X= CH1 2 V~
2
PM3394A
ch2: pkpk= 8.85 V
STOPCH2 2 V~ X= CH1 2 V~
2
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Transistor
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Input Voltage: 0.259 V
PM3394A
ch2: pkpk= 259mV
2
PM3394A
ch2: pkpk= 259mV
2
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Input Voltage: 0.709 V
PM3394A
ch2: pkpk= 709mV
2
PM3394A
ch2: pkpk= 709mV
2
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Input Voltage: 1.63 V
PM3394A
ch2: pkpk= 1.63 V
2
PM3394A
ch2: pkpk= 1.63 V
2
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Begin shifting phaseInput Voltage: 1.86 V
PM3394A
ch2: pkpk= 1.86 V
2
PM3394A
ch2: pkpk= 1.86 V
2
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Phase shift completeInput Voltage: 1.98 V
PM3394A
ch2: pkpk= 1.98 V
2
PM3394A
ch2: pkpk= 1.98 V
2
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Butterfly Diagram (Period 2) Input Voltage: 3.06 V
PM3394A
ch2: pkpk= 3.06 V
2
PM3394A
ch2: pkpk= 3.06 V
2
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Period 2 (anomaly) Input Voltage: 3.34 V
PM3394A
ch2: pkpk= 3.34 V
2
PM3394A
ch2: pkpk= 3.34 V
2
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Period 4 Input Voltage: 4.19 V
PM3394A
ch2: pkpk= 4.19 V
2
PM3394A
ch2: pkpk= 4.19 V
2
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Period 8Input Voltage: 4.38 V
PM3394A
ch2: pkpk= 4.38 V
2
PM3394A
ch2: pkpk= 4.38 V
2
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Period 16 ?Input Voltage: 4.54 V
PM3394A
ch2: pkpk= 4.54 V
2
PM3394A
ch2: pkpk= 4.54 V
2
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Approaching ChaosInput Voltage: 4.96 V
PM3394A
ch2: pkpk= 4.96 V
2
PM3394A
ch2: pkpk= 4.96 V
2
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CHAOS Input Voltage: 7.34 V
PM3394A
ch2: pkpk= 7.34 V
2
PM3394A
ch2: pkpk= 7.34 V
2
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More Chaos
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Acknowledgments M.J. Murdock Charitable Trust Fluke Corporation Thomas J. Holthaus Pacific Lutheran University Dr. Keith Clay Lori Briggs Jana Steiner Christian Dilley