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. MEASURABLE QUANTATIES 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’ 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. Figure 1. Example Logistic Map* APPARATUS & SET-UP Hewlett Packard Function Generator Variable Inductor Diode Oscilloscope Figure 2. Circuit Diagram** METHOD 1. Checked circuit set up, made diagram 2. Found the resonant frequency (f, Hertz) of the circuit for different initial voltages (V 0, Volts) 3. Estimated the capacitance (C, Farads) of the diode 4. Took data by adiabatically changing the value of the initial voltage (V 0, Volts) 5. Observed bifurcations and chaos on oscilloscope 6. Analyzed data Graphically Estimated Feigenbaum’s constants NUMERICAL RESULTS L = 25 mH V 0 = 20mV f = 45.75 kHz C = 475 pF Q = 40.3 δ n=3 = 1.4 α n=2 = 4.5 ; α n=3 = 2.14 Initial Settings Measured/Calculated Properties Feigenbaum’s Constants, GRAPHICAL RESULTS 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. 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. 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). 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. Melissinos, A. C. 3.7 Chaos, Experiments in Modern Physics, 2nd Edition. Academic Press, CA: 2003. (pp. 133-144). “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/7- Diode/Lab07%20Diode.pdf