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\title{Nonlinear Dynamics Lab}
\author{March 18--19, 2013}
\date{Report due April 2, 2013}
\maketitle

This weeks lab sessions study \emph{Chua's Circuit}, a nonlinear
electric oscillator which can be assembled with standard electronics
parts.  For background and circuit diagrams, consult the attached
article by Hobson and Lansbury.

In practice, inductors have a non-negligible resistance which we denote
$R_L$ which has significant ramifications for the functioning of the
circuit.  It can be analyzed as a resistor in series with an ideal
inductor. 

Consequently, the differential equations describing Chua's Circuit
read
\begin{align}
  C_1 \, \frac{\d V_1}{\d t}
  & = \frac{V_2 - V_1}{R_\c} - I_\nl \,, \\
  C_2 \, \frac{\d V_2}{\d t}
  & = \frac{V_1 - V_2}{R_\c} + I_L \,, \\
  L \, \frac{\d I_L}{\d t}
  & = - V_2 - R_L \, I_L \,.
\end{align}
The resistance $R_\c$ is the coupling resistor, labeled $1/G$ in
Hobson and Lansbury's article.  A short derivation of these equations
should be contained in your lab report.

\paragraph*{Lab tasks}

\begin{enumerate}
\item The inductor we have readily available is one with $10 \,
\mathrm{mH}$. Take a multimeter and measure its resistance $R_L$.

\item Assemble Chua's circuit.  Since the inductor is different
from the one used in \emph{Hobson and Lansbury}, some resistances
should be chosen differently from the circuit diagram: take $R_1 = 2.2
\, \mathrm{k\Omega}$ and $R_4 = 1 \,\mathrm{k\Omega}$, the other values
as in the diagram.  For the variable resistance $R_\c$ we use a
multiturn potentiometer which allows fine control of its resistance.

\item Determine experimentally the value of $R_\c$ which corresponds
to the onset of chaos.  Do you see a sharp transition, or a period
doubling cascade as for the logistic map?

\item Measure the response curve of each of the nonlinear inverse
resistors.  To do so, use a third operational amplifier as a voltage
follower attached to the noninverting inputs of the first two
operational amplifiers.  To the output of this voltage follower, you
can attach the oscilloscope ground.  The X- and Y-channels of the
oscilloscope can then be attached to ground to measure $-V_1$ and to
the far end of $R_3$ (or $R_6$).  The voltage across $R_3$ resp.\
$R_6$ can be converted into the current response of the corresponding
nonlinear resistor via Ohm's law.

\item If you have time: Use an inductor with $100 \, \mathrm{mH}$,
also available in the lab, and try if you can---potentially modifying
the values of the capacitors or resistors as well---find a regime as
well.  If this is successful, you will likely obtain less hysteresis
in the response curve of the nonlinear resistors, and consequently
better agreement of theory and experiment.  (Note: the $100 \,
\mathrm{mH}$ inductor might have too big a resistance to excite
nonlinear oscillations, so this is not guaranteed to work!)

\end{enumerate}

\paragraph*{Report items}

\begin{enumerate}

\item Analyze the ``inverse resistor'' consisting of $R_1$, $R_2=R_3$,
and the operational amplifier.  Write out an expression for the
current response as a function of the input voltage, assuming that the
operational amplifier saturates at output voltage $\pm V_{\max}$.
Write out and plot an expression for the overall response curve of the
two parallel inverse resistors used in Chua's circuit.

\item Use the two experimentally response curves to obtain an
approximate measured response curve.  Plot the theoretical curve and
the measured curve in one coordinate system.

\item Write a program which simulates Chua's Circuit over a time
interval $T=0.02$.  Plot $V_1$ vs.\ $V_2$ for times $t=[T/2,T]$,
thereby discarding transients.  Find the value for $R_\c$ at the onset
of chaos.  (Both for the theoretical response curve and for the
reconstruction of the measured curve where you may ignore hysteresis
effects.) 

\item Compare the experimental with the numerical results and discuss
possible differences.


\end{enumerate}

The experiment and the lab report may be done in groups of two.


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