Transcript Slide 1

Emily Veit, Chris McFarland, Ryan Pettibone
The purpose of this experiment is to observe the
approach to chaos and chaotic behavior in a physical
system using a sinusoidally-driven RLC circuit.
Assessments will be made of several characteristics of
the circuit. Period doubling bifurcations will be
observed and the results will be used to develop a
logistic map showing the approach to chaos.
Approximations of Feigenbaum’s constants will be
used to measure error and validity.
•The data graphed in Fig. 3, ‘Driving Voltage vs
Diode Voltage’, is a nice example of a logistic plot
for the RLC circuit. The first three bifurcations
are easily seen at 3.1, 8, and 11 Volts.
APPARATUS & SET-UP
•Hewlett Packard Function Generator
•Variable Inductor
•Diode
•Oscilloscope
•On the oscilloscope, at points of chaotic activity
the signal appears either to jump between two
points or blurs into a wide line. This trend makes
it very easy to recognize chaos, but sometimes
very difficult to tell exactly where the bifurcations
occur.
The RLC circuit is composed of a function generator, a
variable inductor, a diode, and an oscilloscope.
When we put a diode into a simple circuit why does it
show bifurcations and the eventual approach to chaos?
Our diode is constructed of p- and n- type
superconductors. Like all diodes, depending on the
sign of the voltage across the diode, it acts as either
a capacitor or a conductor.
• Positive voltage across the diode = conduction phase
•It acts as a voltage drop
•The diode is ‘forward biased’.
• Negative voltage across the diode = acts as a
capacitor
•No current flow
•Draws a charging current
•Said to be ‘reversed biased’
METHOD
Figure 2. Circuit Diagram**
1. Checked circuit set up, made diagram
2. Found the resonant frequency (f, Hertz) of the circuit for
different initial voltages (V0, Volts)
3. Estimated the capacitance (C, Farads) of the diode
4. Took data by adiabatically changing the value of the initial
voltage (V0, Volts)
5. Observed bifurcations and chaos on oscilloscope
6. Analyzed data
• Graphically
• Estimated Feigenbaum’s constants
A real diode does not switch instantly from the
forward-biased condition to reversed bias.
After the current has gone to zero the diode continues
to conduct for a small time. Because of this the
maximum current in the following cycle depends on
the current value in the previous cycle.
This leads to period doubling which eventually leads
to chaos.
•Beginning at the third bifurcation (11 V) the
system begins to dissolve into chaos. Any of
points with driving voltage greater than 11.5 are a
best estimate of the stable portion of the range.
This uncertainty also occurs very close to each
of the earlier bifurcation points, but is much
smaller in scale. As the driving voltage increases
the system moves ever more quickly to chaos.
Past about 12.5 V it is very difficult to
differentiate exact points. After 16 V the system
dissolves totally into chaos.
As stated above, past 16 V it is impossible to
differentiate specific points and the system is
totally chaotic; this is because our scope
resolution is simply not high enough. However, if
we were able to more closely examine this area
we would see that it is made of increasingly
smaller bifurcations.
The calculated values of Feigenbaum’s constants
are very far off from the expected values. These
constants are evaluated at the limit as the
bifurcations go to infinity (n → ∞). Unfortunately
we were only able to reliably take data on three
bifurcations (n = 3).
NUMERICAL RESULTS
L = 25 mH
Initial Settings
V0 = 20mV
Measured/Calculated Properties
f = 45.75 kHz
C = 475 pF
Q = 40.3
δn=3 = 1.4
αn=2 = 4.5 ; αn=3 = 2.14
Feigenbaum’s Constants
Advice for future researchers:
Fully understanding what it is you are seeing and
measuring on the oscilloscope is one of the most
difficult tasks. The data taking process is lengthy
and tedious but it is possible to get very good
data. Try measuring the bifurcation voltages for
differing values of inductance.
Figure 1. Example Logistic Map*
GRAPHICAL RESULTS
MEASURABLE QUANTATIES
Driving Voltage vs. Diode Voltage
C = Capacitance of the diode (F, Farads)
Q = quality factor =
0L
R
energy stored
average power dissapated
1
L
R
C
4
Melissinos, A. C. 3.7 Chaos, Experiments in Modern Physics,
2nd Edition. Academic Press, CA: 2003. (pp. 133-144).
3.5
= 2 f , driving frequency (Hz, Hertz)
3
R = resistance ( , Ohms)
C = capacitance (F, Farads)
f = resonant frequency of the RLC circuit (H, Hertz)
V = Bifurcation Voltages (V, volts)
for both driving voltage (V0 ) and diode voltage (Vd )
,
= Feigenbaum's constants
= 4.6692
n
=2.5029
“Chaos in Physical Systems: Period Doubling and Chaotic
Behavior in a Diode”. University of Rochester Department of
Physics and Astronomy.
http://www.pas.rochester.edu/~advlab/7Diode/Lab07%20Diode.pdf
2.5
2
1.5
1
0.5
accepted values (where n = number of terms):
n
Diode Voltage (V)
L = inductance (H, Henrys)
,
n
= 2.5029...
0
0
2
4
6
8
10
Driving Voltage (V)
12
14
16
18