The Analog Attractor

The Analog Attractor

 C.P. Ravikumar, Texas Instruments

Pankaj Bongale and Abhimanyu Srivastava, interns at Texas Instruments India, were attracted to do a project on analog system design.  To understand this attraction, we must look at their definition of an attractor. 

“An attractor is a set of states -- points in the phase space -- invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution,” explained Pankaj.

“Aha! That makes perfect sense!” I said, without batting an eyelid.

Not convinced that I was speaking the truth, he explained further. “Suppose we place a ball bearing at the edge of a bowl. The ball bearing will move around the bowl and eventually come to rest at the lowest point. We can say that the ball bearing is attracted to that point. Each part of the bowl can be regarded as leading to that specific point, and the whole bowl is what we call the basin of attraction.”

“Indeed,” I said and this time I really meant it.

“There are what are called strange attractors. The dynamics of a strange attractor is chaotic, and the system never returns to the same place. ”

If there is one thing that I fully understand, it is chaos, and anyone who has seen my desk will immediately understand this.  I can privately share with you that my e-mail inbox is yet another example of a chaotic system. The new wisdom that Abhimanyu had imparted to me clearly explained so many things about my dynamic behavior.  At least a ball bearing will gravitate towards some point at the bottom of the bowl, but look at me. I started on this blog a month ago, then moved in some other diametrically opposite directions, and I have now returned to the blog again, and if the magnetic attraction of my inbox does not overpower me or a phone call does not take me away from it, I have some hope of getting to the bottom.

The Lorenz Attractor (see the Wikipedia entry ‘Lorenz System’) is based on three differential equations, three constants, and three initial conditions.

dx/dt = a(y - x)  .......................................... (1)

dy/dt = x(b - z) – y ..................,,,,,,,,,,,,,,,,,.. (2)

dz/dt = xy – cz ………………………….…(3)

This chaotic system is sensitive to the initial conditions. No matter how close two sets of initial conditions are, the time-domain output signals for the two cases will look very different. Pankaj and Abhimanyu first solved these equations in Matlab using a = 10, b = 28, c = 8 / 3. They also tried another set of constants, a = 28, b = 46.92, c = 4. When plotted in the three dimensions and then projected onto the XZ plane, the plot resembles a butterfly (Figure 1 from [1]).

Figure 1 : Three-D plot of the solution of the system of differential equations (1), (2), (3) for a = 10, b = 28, c = 8 / 3 (borrowed from [1])

 Once their Matlab simulation was in place, Pankaj and Abhimanyu proceeded to implement the Lorenz attractor on the Analog System Lab Starter Kit (see Reference [3] for a repository of information on the “ASLK”).   See Figure 2, where I have reproduced the implementation reported in [1].

 In Equation (2), the RHS involves the multiplication of signals x and (b-z). Similarly, in Equation (3), the RHS involves the multiplication of signals x and y.  The operational amplifiers and analog multipliers available on the Analog System Lab Kit come in handy to implement the Lorenz attractor. The capacitors and resistors on the ASLK can be used in the implementation, but a few extra resistors become necessary; the “breadboard” area of the ASLK can be used to insert them. 

I leave it as an exercise (how convenient) for the reader to verify that the values of passive components selected in Figure 2 will implement a = 10, b = 28, and c = 8 / 3 in the set of differential equations. Remember that the multiplier has a scaling factor of 1/10.

Figure 2: Differential Equation Solver built using ASLK Starter Kit (borrowed from [1])

After showing me the plots of x(t), y(t) and z(t) separately on the oscilloscope, Pankaj declared - “Now we will plot x(t) as a function of y(t) on the oscilloscope.” I felt butterflies in my stomach as the dot on the oscilloscope began to move to the left and to the right in a rhythmic pattern. Figure 3 shows what I saw on the screen of the oscilloscope. No novel that I have read had such an intricate plot.

Figure 3 – The plot

References

 [1] Pankaj Bongale and Abhimanyu Srivastava. Analog Implementation of Lorenz Attractor.  Report submitted as part of design contest held for interns. Available upon request.

 [2] Lorenz System – Wikipedia Entry. http://en.wikipedia.org/wiki/Lorenz_system

 [3] Texas Instruments. Analog System Lab Starter Kit. http://www.uniti.in/teaching-material/100