What about those garlic cloves?

Davide Liessi

You may have noticed that the CDLab logo includes two shapes that many people liken to garlic gloves. Where do they come from? Is garlic somehow related to the CDLab?

Those shapes actually come from an ordinary differential equation, and no, garlic has nothing to do with it!

While surfing the web for ideas on a CDLab logo, I came across a picture of some curves in the plane which resembled capital letters «C» and «D». The picture was taken from [1], where a variant of Chua’s circuit (the simplest electronic circuit exhibiting chaos) is presented and analyzed both numerically and experimentally. It showed the phase portrait in the \(x\)\(y\) plane of a solution of the equation

\[ \left\{ \begin{aligned} \dot{x} &= \alpha (y - x + (a - b u^2) x), \\ \dot{y} &= x - y - z, \\ \dot{z} &= \beta y, \\ \dot{u} &= - \gamma x - \epsilon u. \end{aligned} \right.\]

More specifically, it was Figure 13.a, depicting two coexisting limit cycles of period \(2\) for parameters \(\alpha = 8.73\), \(\beta = 28\), \(\gamma = 37\), \(\epsilon = 12\), \(a = 1.6\), \(b = 0.16\).

Phase portrait in the \(x\)–\(y\) plane of the two coexisting limit cycles.

I thought something on that line would fit perfectly with what we had in mind for the logo, so I experimented with the parameters to produce some pictures close enough to the limit cycles to suggest the shape of «C» and «D» but with some added spiral-like curves, typical of solutions approaching stable cycles. Then with some TikZ incantations I obtained the current logo.

  • [1] Q. Xu, Y. Lin, B. Bao and M. Chen, Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit, Chaos Solitons Fractals, 83 (2016), pp. 186–200, DOI: 10.1016/j.chaos.2015.12.007.