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Applied chaos theory

I was browsing around some cool circuits recently and I discovered Chua's circuit. How come I haven't come across chaotic circuits before?

I want to do a quick analysis on this non-linear circuit, and this post will ironically follow a chaotic fashion and begin upside down. Let's start with Chua's circuit non-linear behaviour. This is a non-linear circuit having a current-voltage relation resembling something like this:

Chua's classic circuit transfer function

So that you don't consider that I am describing a black hole, here is the classic chaotic Chua circuit diagram:

Chua's non-linear diode circuit diagram

Taking an initial look at Chua's circuit we can identify an LC resonance tank and an active non-linear element, which adds-up energy in the resonance loop and forming a special "energy injector" which injects current in the system depeding on the voltage's magnitude following a non-linear function as show on the first figure. The active non-linear element is effectively a negative impedance converter with a twist. The opamps in the Chua configuration act as adding or subtracting a varying voltage in series to the voltage drop across an equivalent positive impedance. The voltage polarity of the current direction of the element is reversed and thus a phase shift of 180 degrees between the voltage and the current is introduced.

Having a single negative impedance converter would simply keep the LC tank in resonance and would not introduce any form of chaos in the system. For this reason, Chua's circuit in the from as shown in the figure would normally contain two negative impedance converters in parallel with a slight slope offset.

I will now leave you with some pictures of the circuit in operation. The circuit was not very easy to tune and get into chaos with my setup but the wobbling sinewaves were quite fun to play with.

The spider's web
Plot X vs Y, circuit is out of chaos here
A semi-chaotic behavior starts to appear.
We're in chaos now
We're in chaos now
We're in chaos now

I started to wonder how would that sound as the frequencies generated are in the audible spectrum. It isn't hard to attach a speaker to it and listen to some chaos. Not very surprising or exciting sound, but still, here is how I heard it:

Too many pictures, too little explanations? Have a look at www.chuacircuits.com.

Date:Tue Jul 04 18:00:23 BST 2015