Demonstrating Chaos: A Physics 407 Laboratory Experiment, Lab Reports of Physics

Download Demonstrating Chaos: A Physics 407 Laboratory Experiment and more Lab Reports Physics in PDF only on Docsity! revised 4/21/03 Demonstration of Chaos Advanced Laboratory, Physics 407 University of Wisconsin Madison, Wisconsin 53706 Abstract A simple resonant inductor-resistor-diode series circuit can be used to demon- strate universal properties of bifurcation and chaos. Quantitatively, the mea- surement of circuit voltages can be used to determine two universal chaos pa- rameters. 1. Introduction Consider the following recurrence relation: Vj+1 = Vj(1− Vj)Vin Suppose we chose some starting value for the variable Vj = V1, then use the formula above to generate V2, then V3, and so on. For some values of Vin, the value of Vj converges to some value V∞ when j continues toward infinity. And even more surprisingly, Vj does not converge to any value for other Vin. 1 In those nonconvergent cases, as j increases from one large integer to the next, Vj alternates from one apparent value of V∞ to another. It is as if Vj has two limits as j tends to infinity. For higher values of Vin, there may be four, eight, sixteen, or more apparent limits to Vj. These “limits” are called “attractors.” 1 Figure 1 shows a graph of the attractors versus Vin. 2 Notice that at certain values of Vin, lines break from one into two parts, each of which breaks into two more parts further to the right. Each break is called a “bifurcation.” 1 0 x 2 3 4 V∞ Vin 3. Equipment These are the specifics for the equipment used in the experiment: • Several banana and coaxial cables with appropriate connectors. Note the lengths of these cables, as the capacitance of the circuit depends on the amount of wire used (as well as the positions of these wires). • A Tektronix model 2465B (400 MHz) oscilloscope. Detailed documentation about this machine is provided in the packet on top of the oscilloscope. • A variable resistor. The HeathKit substitution box (model RS-1) will work well. • An inductor such as the General Radio Co. 10mH Standard Inductor (type no 1481-D). • A silicon signal diode. Almost any will do. • A Stanford Research Systems Synthesized Function Generator (model DS345). The appendix of this handout includes a short description of how to use this instrument. • A Wavetek frequency generator (model 184). Even if a Stanford frequency generator is available, a Wavetek frequency generator still will be useful while finding the resonant frequency of the circuit, since the Stan- ford frequency generator requires the user to push buttons to change voltage in steps, whereas the Wavetek frequency generator has a knob controlling the magnitude of the voltage. If the Wavetek frequency generator is used (as described below) for pro- ducing Vin, then a Keithley Autoranging Microvolt DMM should be attached across Vin, since the voltage scale on the Wavetek frequency generator isn’t very precise. In addition, a helipot may be attached to the Wavetek frequency generator to give finer control over Vin. The appendix of this handout contains a short description on how to use this setup. 4. Experimental Procedure 1. Before the chaos and bifurcations can be explored, f must be set to the resonant frequency. Initially, set R≈ 68Ω and L = 10 mH. The diode capacitance is C ≈ 100 pF. Thus, using the formula f =1/(2π √ LC), f ≈ 105 Hz. The circuit is shown in Fig. 3a. Vin L R Vscope Vin L R Scope Ch. 1 Trig. Scope Ch. 2 Fig. 3a and 3b Adjust f around the expected frequency until the voltage across the resistor is a maximum; this is resonance. 2. Switch to the circuit of Fig. 3b. Slowly increase the input voltage, recording qualitatively what the oscilloscope displays. You should see the bifurcations described in section 2 above. Make sure you have a good feel for what is happening at each input voltage. 3. Set the input voltage somewhat below the first bifurcation voltage x1, but high enough so that the baseline of the diode is evident. Where does this baseline come from? 4. Now, adjust the resistance R slightly and adjust f slightly to bring the circuit back to resonace; this may sharpen the oscilloscope trace, making it clearer, taller, or less wiggly. Continue adjusting R and f until you are satisfied that the oscilloscope trace is optimized. 5. You will now gather data so that you may make an approximation of δ and α. Record Vdiode vs. Vin in appropriate steps and slowly increase Vin until the first bifurcation is on the verge of occurring. This is x1. Photograph the oscilloscope trace. 6. Using the oscilloscope cursors, measure the vertical distance from the diode baseline to the highest voltage of the diode. This is y(1,1). 7. Estimate how accurately you measure x1 and y(1,1). Make certain that you can explain why these errors are appropriate. 8. Continue to record Vdiode vs. Vin until the second bifurcation is about to occur. Step sizes of ∆vin = 5 mV are probably sufficient although you should be able to step as small as ∆vin = 1 mV if necessary. 9. Repeat step 7 for higher bifurcations. You may get good data up to the fourth bifurcation. 10. You now have values for x and y for all observable bifurcations. These bifur- cations, of course, occur at j much lower than infinity; however, the limits for α and δ converge fast enough that you can use these low-order bifurcations to approximately determine the Feigenbaum parameters. Calculate Xj for each bi- furcation available (don’t forget to propogate the estimated errors from above). Calculate Yj using each reasonable combination of points, as described in the Special Note of section 1; that is, find each Yj multiple times, using the lowest two bifurcation points for each, then the third and fourth, then the fifth and sixth, etc. 11. Calculate a weighted mean of the Xj and Yj for α and δ (respectively); compare each to the theoretical values. 12. The first measurements of α and δ (X1 and Y2) will be probably be especially incorrect. Why? Calculate a mean for each Feigenbaum parameter, this time ignoring the first measurements. 13. Returning to the circuit, increase Vin until chaos is reached. Photograph this. 14. Further increase the input voltage until you find a node. Photograph it. How many peaks are there in the diode voltage (per cycle)? 15. Continue increasing the input voltage to search for other nodes. Describe the node(s) that you find. References [1] M. J. Feigenbaum, ”Quantitative Universality for a Class of Nonlinear Trans- formations,” J. Stat. Phys. 19 (1978) 25. [2] R.C. Hilborn, “Chaos and Nonlinear Dynamics”, (Oxford University Press, 1994), pp. 9–18, pp. 44–56.