Visualizing the logistic map with a microcontroller. utku writes – [via]

The logistic map is one of the simplest nonlinear dynamical systems that clearly exhibit the route to chaos. In this paper, we explored the evolution of the logistic map using an open-source microcontroller connected to an array of light emitting diodes (LEDs). We divided the one dimensional interval [0, 1] into ten equal parts, and associated and LED to each segment. Every time an iteration took place a corresponding LED turned on indicating the value returned by the logistic map. By changing some initial conditions of the system, we observed the transition from order to chaos exhibited by the map.

**Visualizing the logistic map with a microcontroller-** [Link]

juho-eric.blogspot.com writes:

I had been reading on chaos generators in the past, and this e-mail got me interested again. Their design is somewhat complicated, with a total of 13 stages of 7 different circuit blocks. You can read all about it in the PDF they published

A much simpler chaos generator that can be built with a lot less components is called Chua’s circuit. The downside with this simple circuit is that you need an inductor which you might have to wind up yourself. Also there should be much less variety than in the waveforms of the Elektor circuit. It has so many stages that can saturate to give different waveforms.

Still, Chua’s circuit is very interesting to fiddle with. I haven’t built the circuit yet, but I did some simulations with LTSpice IV and here’s what I found out.

**LTSpice simulation of Chua’s circuit –** [Link]

What is Chaos theory?** Chaos theory** is the theory that describes systems that appear disordered, but also this theory is about finding underlying *order* in apparently random data. According to Wikipedia:

Chaos theory describes the behavior of certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, the behavior of chaotic systems

appears to be random.

Surpisingly, chaotic effects appear on systems that are deterministic, that means it appears on systems without random elements involved. According to another definition:

A chaotic system is one where the system’s variable quantities satisfy deterministic mathematical equations (i. e., present values may be calculated from past ones), while at the same time being highly irregular, but still contained within a finite region (the “strange attractor”).

Did you know that chaos can be represented using simple electronic circuits and appear on your oscilloscope’ s screen? A circuit that can differentiate a voltage is able to represent the differential equations describing a chaotic system thus appear “attractors” on your oscilloscope. On the site bellow you will find some simple chaos-generating electronic circuits that will “blow your scope”!!

**Schematics representing Chaos theory –** [Link]