CHAOS IN A DIODE: PERIOD DOUBLING AND CHAOTIC BEHAVIOR IN A DIODE DRIVEN CIRCUIT

Maxwell Mikel-Stites University of Rochester, Rochester, NY, 14627

Chris Osborn, for the code involved in producing the chaos model and Dan Richman and Chris Osborn for their assistance in gathering data.

Overall, the data gathered accurately demonstrates only the progress of increasing bifurcations leading to chaos, and also showed that the inductance of the system also greatly affects the voltages at which the bifurcations are observed. In this way, by increasing the inductance value, one could cause the bifurcation pattern to emerge earlier and earlier. Similarly, one could decrease the inductance to cause bifurcations to appear at later intervals. In order to improve the lab, it would be beneficial to obtain better equipment, such as more reliable inductors and a digital oscilloscope.

The purpose of this experiment was to examine period doubling and chaotic behaviour in a diode-driven circuit. This allows us to learn more about the physics behind the diode and its interaction with the circuit, as well as the manifestation of chaos in a system as a result of increasing bifurcation in the diode voltages. It was possible to observe this type of behaviour clearly, even with varying data quality; even in the worst case, the bifurcations leading into chaos were clearly defined and were relatively easy to examine.

The Diode Itself•Constructed out of a combination of a p and n type semiconductors

•Diode not perfect; causes finite time for current reversal.•Causes forward and reverse bias to alternate•Forward; diode acts as a resistor•Reverse; causes diode to act as a capacitor

•The interactions between these parameters with increasing voltage causes the signal to bifurcate as it is read from the diode.

inductor

Oscilloscope

SignalGenerator

Diode Circuit

To Oscilloscope

Inductor

From Signal Generator

12 4

3

1) Diode

2)2400 ohms

3)185 ohms

4)590 ohms

To the left is the Lissajous graph of the third bifurcation, since the diode voltage graph is nearly indistinguishable from the second bifurcation in many cases.

Procedure•In order to find the resonant frequency of the circuit, fix the amplitude to a low value (~20mV) and fix a low frequency (~30khz) for some chosen inductance. Next, increase the frequency until the peak to peak voltage stops increasing and begins to decrease. This point is the resonant frequency for the circuit.

•In order to examine chaos, increase the amplitude in small increments (0.20V or less) at this resonant frequency and measure the peak to peak voltages displayed on the oscilloscope.

•Bifurcations should appear at successively closer voltages, at varying voltages depending on the inductance chosen.

•With the experimental setup provided, it was only possible to observer the first three bifurcations before the system developed into chaos in observed cases.

The above pictures, from left to right, show the oscilloscope graph of the peak to peak diode voltage v.s. time for the initial voltage, first voltage bifurcation and the Lissajous graph of driving voltage v.s. diode voltage. For the first bifurcation, this was found at approximately 3.20 volts for one run and approximately 3.6 volts for another.

Above are the graphs of the second bifurcation, found at approximately 8.20 volts and 10.6 volts for the two runs.

Shortly after the third bifurcation, the graph evolves into chaos, with the leftmost picture the peak to peak diode voltage, the center the Lissajous graph of chaos and the rightmost a closeup of the chaos visible in each peak of the graph, as the increasing number of bifurcations overlap each other.

This model demonstrates what would happen if we were able to observe bifurcations beyond third order with our diode/oscilloscope setup. As the bifurcations increase, in addition to the readily observable chaos, there are island of stability appearing at systematic intervals. In order to observe this behaviour experimentally, one would have to continue to increase the voltage extremely carefully in order to observe one, as the peaks for stability are quite narrow.

The equation used to produce the below graph was xn+1 = r xn (1 – xn), using the initial values of xo = 0.7, r = {2.5 : 0.015 : 4.0}

2

4

3Signal Generator

GroundGround Ground

Ground

Circuit Diagram

The below photos detail the setup for the entire lab and the provided equipment. For the experiment, the inductor was set to approximately 10mH, since it was the lowest inductance value possible with the given inductor.

Diode Voltage vs Driving Voltage

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25

Driving Voltage (V)

Dio

de

Vo

lta

ge

(V

)

Diode Voltage vs Driving Voltage

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12

Driving Voltage (V)

Dio

de v

olt

ag

e (

V)

•Resonant frequency= 73.7khz•Inductance=~10 ohms. There is some uncertainty due to problems with the inductor.•This caused earlier bifurcations.

•Change in bifurcations consistent with higher inductance; system likely not at resonance.•Feigenbaum’s constant evaluated to be approximately 4.167.•Probably due to error in the equipment, since system not at resonance for a higher inductance.

•Resonant frequency=73.7khz•Inductance=10 ohms•Bifurcations at 3.6, 10.6, 15.1V•Chaos at ~18V•Fiegenbaum’s constant=1.56; too few bifurcations to measure accurately.

Measurable Quantities•Feigenbaum’s Constant:The ratio of the difference between the bifurcations; as the number of bifurcations goes to infinity, it approaches 4.6669.•Driving voltage•Peak to peak diode voltage•Frequency •Inductance