Chaos is both a very interesting and important field of study as we observe that chaotic phenomena arises everywhere around us. Most people tend to shy away when they hear the word chaos, not realizing the beauty and potential that exists in it to enrich their lives. What we see here is the attractor for a chaotic circuit. While being of a unpredictable nature, there are symmetries which exists in the system (the axes of the graph are the voltages measured across two different places in the circuit).
This report describes setting up a pair of electronic circuits which model the behavior of a series of differential equations known as the Rossler equations, shown below. While voltage output of the circuits was chaotic, the two were coupled together in such a way that both acted in the same chaotic manner, which we call synchronization. That is, the difference between the two chaotic systems was zero (well close to it, taking noise into account)! This is a remarkable result, showing us that we do indeed have some control over chaos.
Below is the circuit schematic for the drive circuit. The circuit is fairly simple and consists of resistors, op-amps, capacitors, diodes, two switches, and a potentiometer (which serves as our bifurcation parameter as well as a means of varying one circuit with respect to the other).
A breadboard is used to put the circuits together and this is what the circuit actually looks like when all put together ...
Here is an overview of the setup.
There is a lot of potential for synchronizing chaotic systems, particuarly in the field of communications. The experiment described here in this paper is easy to setup and requires only having had some experience with crcuits, though it is easy enough to pick up along the way. Perhaps the hardest part is getting the power sources (two are needed) as the electrical components can be picked up at just about any electronic supply store and any old oscilliscope will work. While it is interesting to build a chaotic system and also show some means of control over it, the true value in building such circuits is the ability to illuminate previously unexplored paths and make exciting new discoveries. Click here for a printable version of this paper in pdf format.