Read before class:
- Textbook section 3.7
- Taylor Classical Mechanics, chapter 12
We will attempt to observe the onset of chaos in a series diode-R-L circuit. The input to the circuit will be a sinusoidal driving voltage V_0 cos(omega t).
Find the resonant frequency of the circuit. This should be close to 1/[2 pi sqrt(LC)]. Look for resonance in the frequency range of 100-300 kHz.
When tuning the circuit to resonance, use a small value of V_0 (~20 mV) so that the circuit is in the linear regime. To observe resonance, examine the output amplitude V_b as the inductance of the variable inductor is varied.
Familiarize yourself with the picoscope 6 (USB based oscilloscope). Learn to change the vertical and horizontal scales for channels A and B. Learn to change the trigger source and other trigger features. Learn to turn on the pulse generator. Make sine wave and square pulses. Connect the function generator to either input A or B. Learn how to move the A or B traces on the screen. Learn to place markers and make measurements.
Now examine the frequency spectra of sine waves and square waves using the FFT (Fast Fourier Transform) feature. Learn to change the FFT vertical scale between the linear and logarithmic settings.
Now examine the waveforms of the chaotic circuit on the picoscope As V_0, the driving voltage, is varied, look for changes in the behaviour of the output of the circuit. When examining the picoscope FFT (or in the spectrum mode), look for new peaks BELOW the fundamental frequency.
Try to find the first period doubling (what is the expected frequency for this bifurcation ?) Look for the second and third period doubling. (what are the expected frequencies here ?) Look for the onset of chaos.
To observe period doubling, you can use the FFT (Fast Fourier Transform) feature or the spectrum mode of the picoscope to look at the output of the signal in frequency space. What signature are you looking for in the FFT ?
Now calculate
- delta^1 = (lambda_2 – lambda_1)/(lambda_3-lambda_2)
- delta^2 = (lambda_3 – lambda_2)/(lambda_4-lambda_3)
Are the results approaching the Feigenbaum number ? Save the waveforms of the V_b at period doublings. Also save the frequency spectrum at each period doubling. These should be included in your report.
Technical notes:
1) The USB connection to the picoscope should be connected to the USB slot on the PC (a Windows feature).
2) Need a high quality pulser (borrowed from the IDL) to generate the sinsuoidal driving voltage. 2018: Use the Rigel pulse generator.
Self-Study Questions for Chaos
1. What is meant by “Chaos” in Physics ? <br>
2. What are some examples of Chaos in physical systems ? <br>
3. What is period doubling (or a bifurcation) ? <br>
4. What is the Feigenbaum number ? <br>
5. For the logistic map, where does the chaotic regime begin ? <br>
6. For a demonstration of chaotic behaviour with an electric circuit, what kind of circuit element is required <br> (Hint: do linear systems exhibit chaos ?)
7. Examine the chaotic pendulum. How does it differ from the simple pendulum ? <br>
8. What is an example of period doubling in the onset of turbulence in fluids ? <br>
9. What is the characteristic of period doubling in an electric circuit ? How can this be observed using a Fourier transform ? <br>
10. After the onset of chaos, is there a return to orderly deterministic behavior ? Explain.
Last updated: Nov 2, 2018
Tom Browder