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PHYSC 3622
Experiment 2.7
17 June, 2017
Chaos in a Driven Resonant Circuit
Purpose
In this experiment, you will learn chaotic behaviors of a non-linear circuit, including
period-doubling route to chaos, state-space trajectory, and intermittency.
Equipment
Nonlinear circuit (SK3327 or 1N5450 varactor diode, 0.1 mH coil inductor), op-amp
differentiator, oscilloscope with a FFT module, two function generators (one with AM
and FM).
Background
This experiment involves measurements on a simple driven, damped oscillator circuit,
which shows many of the same features as the logistic map, which is used to study
chaos. The circuit is an RLC series circuit consisting of an inductor (with dc resistance)
and a varactor diode. Under reverse bias, the diode acts as a nonlinear capacitor, as
described in the attached references by Linsay (PSL) and Testa, Perez, and Jeffries
(TPJ). Some of the interesting effects to be studied in this experiment are: a voltagedependent resonance frequency, amplitude jumps when varying the frequency through
resonance, and a period-doubling sequence leading to chaos. This behavior is a
continuous-time approximate analog of the logistic map:
xn1  xn (1  xn ) ,
in which the diode voltage corresponds to Xn and the driving voltage corresponds to .
One cycle of the driving voltage corresponds to one iteration of the map. In the
experiment, you need to use the same circuit components throughout all
measurements. At each stage of the experiment, be sure you have a diagram showing
all connections to the circuit and among the instruments.
References:
P. S. Linsay, "Period Doubling and Chaotic Behavior in a Driven Anaharmonic
Oscillator," Phys. Rev. Lett. 47, 1349 (1981).
J. Testa, J. Perez, and C. Jeffries, "Evidence for Universal Chaotic Behavior of a Driven
Nonlinear Oscillator," Phys. Rev. Lett. 48, 714 (1982).
R. Van Buskirk & C. Jeffries, "Observations of Chaotic Dynamics of Coupled Nonlinear
Oscillators," Phys. Rev. A 31, 3332 (1985).
Procedure
1. Bifurcation diagram . Assemble the circuit as shown in the next page. The signal,
which will show the period-doubling route to chaos, is the diode voltage; this
arrangement will make it predominantly positive. Begin with a low amplitude
sinusoidal driving voltage (just large enough that the scope triggers on the diode
voltage) and find the resonant frequency by varying the drive frequency to maximize
the signal amplitude; make a quick measurement of this frequency using the cursors on
the scope. Increasing the driving voltage and notice how the resonant frequency
changes; do not go into the period-doubling regime, but note that beyond a certain
driving voltage the diode voltage amplitude jumps as you change the driving
frequency.
There is hysteresis in these jumps; for a range of drive amplitudes, the diode amplitude
can be large or small, depending on the history of the driving frequency. Verify the
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Experiment 2.7
17 June, 2017
hysteresis by measuring the jump-up and jump-down frequencies for some drive
amplitude below the period-two threshold. This will be studied further in Section 2.
Now set the drive frequency a bit (say 10%) above
the low-amplitude resonance (if you found the
resonance frequency to decrease with amplitude),
and observe the period-doubling transition to chaos
as you increase the driving amplitude. Note the
difficulty in triggering the scope to show highperiod behavior and, in the chaotic region, the
sensitive dependence on initial conditions evident
in the behavior after triggering at the highest level
possible (Question 1). You should be able to find a
P3 window, easily, and perhaps also a P5 window.
To see all these behaviors at once, use a triangle
wave (about 100 Hz) from another function
generator to amplitude-modulate the driving
generator and to trigger the scope; then you can
display a sort of bifurcation diagram for the circuit.
With this display, notice how the bifurcation structure, chaotic behavior, and
appearance of windows depend on the driving frequency (Question 2). Choose a
driving frequency that gives you a rich structure and use the B time sweep, triggering
after delay, to view the temporal behavior throughout the bifurcation diagram. Print
out the bifurcation diagrams, to be labeled with the different types of behavior
represented, for your report.
Adjust the bifurcation diagram display to show the main period-doubling region,
and change the driving frequency to optimize the main doubling sequence. You will
want to have as many doublings as possible, and to have them as clear as possible.
Get hard copies for your report.
2. Time series and state space. On the bifurcation display of the main period-doubling
region, use the method of TPJ (see their Fig. 4) to find a value for Feigenbaum's
constant . You may use the cursors to get a measurement of sufficient precision; do
this for as many bifurcations as you can measure. Do the same for the P3 doubling
sequence, resetting the frequency if necessary.
Now measure the L and R values of the inductor. Both PSL and TPJ give a description
of the supposed dependence of diode capacitance on voltage. From the value of L and
the voltage-dependent resonance frequency, find C and how it varies with the
amplitude of the diode voltage. The resonance frequency as a function of the diode
voltage amplitude is given approximately by the frequency and amplitude of the signal
just before the amplitude jumps down (Questions 3 and 4). You can display the
nonlinear resonance curve, complete with amplitude jumps and hysteresis, by using a
triangle wave (about 100 Hz) from the other function generator to frequency-modulate
your driving generator and to drive the x-axis of the scope display (in x-y mode). Do
this and print out your results. Observe the behavior at other frequencies, especially at
half the resonance frequency. Note what happens when the driving amplitude is large
enough to cause period-doubling (Question 5).
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Experiment 2.7
17 June, 2017
The driven nonlinear resonant circuit is described by a second-order non-autonomous
differential equation,
L 2 Q / t 2  RQ / t  Q / C (V )  Vd cos t , or
 2V / t 2  [( 2c'Vc' ' ) /( c  Vc' )]( V / t ) 2  ( R / L)V / t  V /[ L(c  Vc' )]
 Vd cos t /[( L(c  Vc' )],
which may be written as three first-order equations:
D / t  [( 2c'Vc' ' ) /( c  Vc' )]D 2  ( R / L) D  V /[ L(c  Vc' )]
 Vd cos /[( L(c  Vc' )],
V / t  D,
 / t   ,
where c'  c / V . The state of the system is now completely specified (for all time)
by giving the values of all three variables at any one time. The evolution of the system
is represented as a trajectory or orbit in a three-dimensional state space, one dimension
corresponding to each variable. You can display a two-dimensional projection of the
system's trajectory on the scope in x-y mode with the derivative of the diode voltage
signal going into the x-input (channel 1) and the diode voltage into y (channel 2). Do
this and observe what happens during the period-doubling sequence. Keep in mind
that the trajectory cannot cross itself, because any point uniquely determines the entire
orbit. This will show you why the state space has to be three-dimensional in order to
have chaos. You may want to print out your results here also.
The time behavior and the state-space trajectory can be strobed once every driving
cycle to reproduce a sequence of points whose values are like those of successive
iterations of a map. To do this set up one function generator as a pulse generator,
synchronized with the driving generator, and driving the z-axis (trace intensity) of the
scope. Notice how, as the system goes through its period-doubling sequence, each
strobed orbit point bifurcates; this is true for every point in the orbit.
3. Power spectrum and intermittency. Use a single un-modulated drive generator
again, and notice how the power spectrum of the diode voltage signal relates to its
temporal behavior. Note specifically the sinusoidal behavior at very low amplitude
(single frequency), the distorted waveform at higher amplitudes (harmonics), and the
period-doubled waveform (sub-harmonics). Be sure that the vertical sensitivity of the
scope is such that the diode voltage signal never goes off scale. This will keep the
spectrum analyzer input at a safe level. The power spectrum makes it much easier to
determine the drive amplitude needed for bifurcation. By monitoring the drive
amplitude (as long as the drive frequency is not too close to resonance, the generator
readout may be used; observe the drive output on the scope to verify this), you can find
the values of Vn (drive voltage for the onset of P2n behavior). Repeat this measurement
for as many bifurcations as you can. From these data, find a value for Feigenbaum's
as done by PSL and TPJ.
You can also determine Feigenbaum's  from the power spectrum, as indicated by PSL.
Record the heights of the various spectral peaks and find a few values for . It may be
helpful here to display the spectrum out to the second harmonic in order to have two
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period-two peaks. Compare this value with the one determined in Section 2. The
power spectrum is useful in finding other behavior as well. Notice the P3 window and
the period-doubling sequence observable there. If you can find P12 you can make
another measurement of  and also of . Also notice the inverse period-doubling
sequence beyond the accumulation point; note how the P2n spectral peaks are washed
out in inverse order by the broad chaotic spectrum. Find the P5 spectrum, and note
that it seems to be a noisy period five because of the spectral background. This is a
result of intermittency, as you can easily observe. With the P5 spectrum as quiet as
possible, set the scope's time base to display many cycles of oscillation. Now trigger the
scope manually in the single-shot mode, and note how the regular periodic behavior is
intermittently punctuated by bursts of chaos. Print out your results. This is also
supposed to occur near the onset of period-three behavior; see if you can find it there.
Questions
1. Why does the scope trigger allow you to observe the sensitive dependence on initial
conditions?
2 How would you explain this frequency dependence?
3 Why does this method allow you to find the approximate resonance frequency?
4.What problems do you see in this method for finding the voltage dependence of C?
5. How do you explain the behavior observed at half the resonance frequency?
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