1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l...
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CHAOTIC CIRCUITS The way the electron bounces
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Transcript of 1 CHAOTIC CIRCUITS The way the electron bounces. 2 TOPICS l Linear Circuits –Inductor –Diode l...
1
CHAOTIC CIRCUITS
The way the electron bounces
2
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
3
MAY POPULATION MODEL
4
Basic Population Model
POPULATION SEX
FOODSUPPLY
5
May’s Population Model
POPULATION SEX
FOODSUPPLY
FEEDBACK
6
Attractor for May model
7
8
LINEAR CIRCUITS
9
INDUCTOR
10
Input Voltage: 5 V
11
Input Voltage: 10 V
12
Input Voltage: 15 V
13
Input Voltage: 20 V
14
Inductor (Output vs. Input)
Input Voltage (peak to Peak)
Out
put
Vol
tage
(pe
ak t
o pe
ak)
15
DIODE
16
Input Voltage: 5 V
17
Input Voltage: 10 V
18
Input Voltage: 15 V
19
Input Voltage: 20 V
20
Diode (Output vs. Input Voltage)
Input Voltage (peak to peak)
Out
put
Vol
tage
(p
eak
to p
eak)
y = 0.4949 + 0.7762
21
NON-LINEAR CIRCUITS
22
How can we tell if a circuit is behaving non-linearly?
23
Period 1
PM3394A
ch1
ch2
CH1 1 V~ STOPCH2 1 V~ MTB5.00us ch1+
1
2
24
Period 2
PM3394A
ch1
ch2
CH1 2 V~ STOPCH2 2 V~ MTB5.00us ch1+
1
2
25
Period 4
PM3394A
ch1
ch2
CH1 5 V~ STOPCH2 2 V~ MTB5.00us ch1+
1
2
26
Diode-Inductor Circuit
27
Circuit Schematic
Inductor5.89 mH
Diode
VariableVoltageSource AC
28
Period 1
29
Period 2
30
Period 4
31
Period 8
32
Period 16
33
Chaos
34
Transistor Circuit
35
Circuit SchematicInductor
A.C.FunctionGenerator
Transistornpn type
Variable Resistor
D.C. Voltage Source
Load inductor
LoadVariableresistor
GroundGround
Feedback
36
FEEDBACK
SIGNAL AMPLIFIER
POWERSUPPLY
OUTPUT
Simplified Schematic
37
Period 1
38
Period 2
39
Period 4
40
Period 8
41
Period 16
42
Chaos
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BIFURCATION DIAGRAMS
44
Bifurcation Diagram (Inductor)O
utpu
t V
olta
ge (
peak
to
peak
)
Input Voltage (peak to Peak)
y = 1.3717x - 0.0724
45
46
Out
put V
olta
ge (
peak
to p
eak)
Input Voltage (peak to peak)
Bifurcation Diagram (Diode-inductor)
47
48
Bifurcation Diagram (Transistor)
Input Voltage (peak to peak)
Out
put V
olta
ge (
peak
to p
eak)
49
50
Mathematical model (May model)
51
52
Input Voltage (peak to peak)
Out
put V
olta
ge (
peak
to p
eak)
Chaotic Region (diode-inductor)
53
54
Out
put V
olta
ge (
peak
to p
eak)
Input Voltage (peak to peak)
Periodic Region Amplified (diode)
55
56
57
Feigenbaum Number
58
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
59
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)
60
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
61
Why are these systems chaotic ?
62
Water gate analogy of a diode
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Water rate corresponding to time constant
64
Simple Linear Amplifier
SIGNAL AMPLIFIER
POWERSUPPLY
65
Chaotic Amplifier Circuit
SIGNAL AMPLIFIER
POWERSUPPLY
FEEDBACK
66
ATTRACTORS
67
68
69
Inductor
70
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
71
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
72
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
73
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
79
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
80
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
84
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
86
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
88
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
90
Period 2 (anomaly) Input Voltage: 3.34 V
PM3394A
ch2: pkpk= 3.34 V
2
PM3394A
ch2: pkpk= 3.34 V
2
91
Period 4 Input Voltage: 4.19 V
PM3394A
ch2: pkpk= 4.19 V
2
PM3394A
ch2: pkpk= 4.19 V
2
92
Period 8Input Voltage: 4.38 V
PM3394A
ch2: pkpk= 4.38 V
2
PM3394A
ch2: pkpk= 4.38 V
2
93
Period 16 ?Input Voltage: 4.54 V
PM3394A
ch2: pkpk= 4.54 V
2
PM3394A
ch2: pkpk= 4.54 V
2
94
Approaching ChaosInput Voltage: 4.96 V
PM3394A
ch2: pkpk= 4.96 V
2
PM3394A
ch2: pkpk= 4.96 V
2
95
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
