Mitch Richling: Chua's Circuit

Chua's Circuit is probably the most well known chaotic, electronic circuit. The governing equations are:

\[\begin{align*} \frac{dx}{dt} & = \alpha\left[y-x-f(x)\right] \\ RC_2\frac{dy}{dt} & = x-y+Rz \\ \frac{dz}{dt} & = -\beta y \end{align*}\]

Some complexity is hidden in the function \(f(x)\). While many choices of \(f\) lead to qualitatively similar strange attractors, Chua used a peicewise linear function of the following form:

\[f(x)=m_1x+\frac{1}{2}(m_0-m_1)(|x+1|-|x-1|)\]

Where \(m_0\) & \(m_1\) are slopes.

In the schematics below, the voltages across the capacitors C1 and C2 are represented by the variables \(x\) and \(y\) above. The variable \(z\) is the current through the inductor L1.

Note this is the version using a gyrator as I had difficulty sourcing an appropriate inductor. This version of the circuit has the additional advantage that the voltage across R12 is proportional to the variable \(z\) above. This is doubly handy as R12 is grounded, and thus may be easily probed with an oscilloscope!

Also notice that I used potentiometers for R7 & R12 so I could play with the circuit a bit. As a secondary advantage, the potentiometers also allowed me to avoid expensive, high precision resistors.

• Sprott, Julien Clinton, and Wesley Joo-Chen Thio. "Elegant Circuits: Simple Chaotic Oscillators." New Jersey: World Scientific, 2022.
• Check out the chaos in electronics section of my reading list.
• For more about chaos in general, take a look at the fractals section of my reading list.