1. 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 Signal Generator Diode Circuit To Oscilloscope Inductor From Signal Generator 1 2 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 indistinguish able 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 x n+1 = r x n (1 – x n ), using the initial values of x o = 0.7, r = {2.5 : 0.015 : 4.0} 2 4 3 Signal Generator 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. 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