To paraphrase Supreme Court Justice Potter Stewart, while we might not be able to precisely define chaos, we know it when we see it! A good number of researchers (chaologists), however, have come up with some pretty concise definitions of chaos, several of which I happily present here:

"...largely unpredictable long-term evolution occurring in a deterministic, nonlinear dynamical system because of sensitivity to initial conditions."

Garnett P. Williams in Chaos Theory Tamed

" ...lawless behaviour governed entirely by law."

Ian Stewart in Does God Play Dice?

"Dynamics freed at last from the shackles of order and predictability....Systems liberated to randomly explore their every dynamical possibility.... Exciting variety, richness of choice, a cornucopia of opportunity."

Joseph Ford (quoted by James Gleick in Chaos)

Chaos presents itself in an incredibly diverse range of areas of science, including irregular heart rhythms studied by cardiologists (medicine); the erratic motion of some planetary satellites, such as Saturn's moon Hyperion (astronomy); the turbulent flow of fluids (fluid dynamics); the observed quirky behavior in electronic circuits (electronic engineering); and, of course, those eye-catching perpetual motion toys sold at airport gift counters (crass commercialism). Biology, mathematics, chemistry, economics, ecology, the stock market and, perhaps, even the quantum world all cross paths with chaos. Though we will try to avoid cardiac arrhythmias, there are many other areas that can be readily explored if approached with the right tools. First, we will have a quick look at some of those tools.

Mathematics

With little more than a few select equations and a computer, chaos can be explored in the comfort of your home. The results should be quite surprising. At right are the plots for three equations: a simple sine function; a more complicated hodgepodge of sines, cosines, tangents, and logs; and the deceptively simple-looking logistic map equation. (Note how complex the plot is for so simple a formula.) The logistic map equation was created in the mid-nineteenth century to deal with ecological population studies. By varying one of the parameters (often depicted as a Greek lambda or the English letter k), the one set to "4.0005," and by iterating the equation (that is, assigning the result of each calculation to x(t) and feeding it back for the next calculation) behavior ranging from periodic to chaotic can be observed.

In the plot shown, the value of 4.0005 forces the equation into the chaotic, infinity region. With a few more iterations, where each positive or negative peak is an iteration, the graph would fly off into infinity and off the chart! So, equations being evaluated for chaotic behavior need to be studied for a considerable number of iterations until they settle down and exhibit their characteristic behavior. Incidentally, the lambda (or k) parameter is usually varied from 1 to 4 with the initial value of x(t) set between 0 and 1.

Mathematics packages, such as Mathematica, Matlab, and Mathcad, make evaluating equations straightforward. There are also some great freeware packages currently available on the Internet. An example of that is Scilab (which comes bundled with a simulation package called Scicos). Scilab/Scicos is very similar to Matlab/Simulink and is very good for creating simulations. I used the Scilab/Scicos package to compute the logistic map equation and read the created file into DPlot. In fact, DPlot (not freeware but quite inexpensive) was used to create all of the above plots.

This is an example of Scicos being used to run a simulation of the logistic map equation. The sync clock acts to synchronize computational behavior in the simulation. The Mem (memory) block acts to prevent an algebraic loop, which would cause the simulation to immediately fail due to a fatal recursion during feedback. This occurs because the output of the simulation is being fed back as input, and the software has no way of knowing what the input is before calculating the output. Mem stores an initial x(t) value that will be modified after an initial computation.

Here the value of Lambda is 3.79. It is being multiplied by the previously calculated result for x(t). At the same time, the 1 is being added to the negative of the previously calculated result for x(t), and that sum is multiplied by the product of Lambda and the previous x(t). And so it goes.

Scicos produces a very nice oscilloscope plot, not shown, thanks to that green square (scope) function that performs the output plot. Optionally, Scicos can create a file that can be read in by another program, as was done for the above plot. Scicos has numerous palettes loaded with all sorts of nifty functions, including x-y plots, which are needed for other chaos studies.

Mathematica has available notebooks written by researchers to allow studying chaos. They simplify the process for creating bifurcation diagrams, phase space plots, statistical time-series calculations, etc. Any of these things can be programmed with other mathematics software, of course. It's just a matter of how much time you're willing to wait before getting started doing actual science.

Analog Simulation

One of the first Heathkits I built, and the one that actually worked, was the model EC-1 analog computer. It was probably around 1961. I can remember carrying the kit back home from the store. Naturally, I couldn't wait to start building it. Months later, when I finally finished, there was a gorgeous instrument boasting nine operational amplifiers (op amps) with three initial condition supplies. After patching in the demo programs (patching is how you program most analog computers) to simulate a bouncing ball, a fired projectile, and such, the unit kind of just sat there. The EC-1 was really a very sophisticated instrument. This sophistication was not so much in the electronics end but, rather, in its problem-solving capabilities. Analog computers thrive on solving differential equations, which are the equations governing (or, rather, describing) natural phenomena, such as wave motion, atmospheric turbulence, electronic circuits, biological rhythms, chaotic phenomena, etc. If one is not familiar with differential equations, and I wasn't, analog computers are just nice things to look at. But once you study differential equations, you will find that worlds open, curtains part, salaries skyrocket, and people usually treat you better.

The EC-1 used vacuum tubes for the op amps. The tube amplifiers had gains of 1000 (a measure of accuracy—the higher the better) and were very prone to drift. Drifting meant frequent amplifier balancing. Conveniently, the amplifiers were capable of outputting +/- 60 volts. I no longer have the Heathkit computer as I gave it away years ago, before eBay was born. To this day, I still walk around with a "kick me" sign on my back!

Analog computers are practically synonymous with operational amplifiers. Today, op amps have little drift, very high open and closed loop gains, low power requirements, and they don't weigh a ton! However, analog computers seem to have taken a back seat to the ubiquitous desktop digital computer. Whereas digital computers can solve problems to any specified precision, they do so discretely in computed chunks. As fast as digital computers may be, it takes time to solve differential equations. With an analog computer, once a problem is "patched in" the solution is revealed. It's also great fun modifying system parameters in real time. While digital computers in theory can achieve precision to a specified number of decimal places, analog computers use voltages to achieve precision to within the level of system noise and beyond! Analog computers are fantastic for simulating natural phenomena if you know the differential equations governing the natural. There are now plans in the works that may allow a successful bridging of the two types of computers in ways previously undreamt of! Although hybrid computers (part analog and part digital) have been around for a long time, special chips are being developed that may allow digital computation to mesh with the virtually instantaneous computational speed of analog computers.

Incidentally, analog computers played a part in other instrumentation produced by Heath (in its Malmstadt-Enke line), including a dropping mercury polarograph. (I actually purchased the polarograph, but the dropping mercury electrode was so precariously constructed that I never had the nerve to fill it with mercury!) At least one other company, McKee-Pedersen, employed analog computer circuitry in its line of laboratory instrumentation for chemical analysis. Sadly, both Heath and McKee-Pedersen are now just footnotes to history.

Analog computers employ feedback for proper operation, and they can operate nonlinearly, meaning the output is not linearly related to the input. Feedback and nonlinearity are also characteristic features of chaotic phenomena. And so, analog computers and op amps are ideal simulators for chaos. Until recently, one company still making the intrepid analog computer was Comdyna. I acquired a used system on the Internet. At right is a picture taken of the unit in operation, along with an X-Y monitor overlay of the computer's output. The unit's eight amplifiers, two of which are just sign inverters, are easily patched and set up to perform simulations.

In order to work with breadboarded circuits where externally wired integrated circuits (i.e., multipliers, A/D converters, and sensors) can be readily incorporated into op amp circuits, I assembled a number of Tektronix modules. Basically, there are two TM 506 power modules sitting one on the other with a group of six AM 501 operational amplifiers in the lowest row. The AM501 incorporates dual 741 op amps. The second row up includes a DC 503A universal counter/timer, an FG 503 function generator, an MR 501 X-Y monitor (cute as a bug), a DM 502 digital multimeter, and two PS 503 dual-power supplies.

Finally, at the very top is a model 604 X-Y monitor. To its left is a homemade interface box that allows adjusting the x- and y- voltages being fed to the 604.

Chua's Circuit

Invented by Leon Chua back in 1983, one version of the basic circuit is shown below both as a schematic and as its breadboard realization. Over the years, there have been many modifications and application-specific variations made to the original circuit. The circuit was designed from first principles by Chua, and it reveals a remarkable broad spectrum of chaotic behavior. Later, when we get to actual experiments, we will look at some of the results in phase space output by the electronics. In the photos below, the first depicts a Chua circuit drawn and simulated by Electronics Workbench v5.12 software. Here, the overlaid simulated scope output shows the chaotic signal generated with the specified component values. The two pictures that follow show an honest-to-goodness, real-life breadboarded circuit along with an equally real oscilloscope output. Both the simulated and breadboarded circuits use components of nearly identical values. I was pleasantly surprised (actually, shocked) that electronic design software could simulate the chaotic state produced in Chua's circuit. It is probably unrealistic to expect this kind of compliancy across-the-board in chaos simulations.

Note the nonlinear components highlighted in the schematic. As mentioned above, nonlinearity is a requirement for chaos. Though not sufficient in and of itself to guarantee chaos, nonlinearity must be present in the feedback loop. Note also the inductor at the lower left. While capacitors are available in a large range of values, inductors are not. Also, for many circuits the inductor must have a very low internal resistance. I specified a 6.5 ohm internal resistance, here shown as a resistor in series with the coil. Usually, inductors are fixed in value. However, by using a small but powerful magnet, it is possible to vary the inductance of a coil if it has been wound on a ferrite bobbin. (The inductance can be reduced and restored depending on the magnet's motion.) I use just such a magnet while exploring circuits that appear reluctant to enter a chaotic state. By approaching and withdrawing the magnet from the coil, I can usually find the sweet spot at which chaos appears. This assumes the design is fairly close to correct, of course. If you plan to do a lot of chaotic circuit experiments with real inductors, by all means try the magnet trick while tuning.

Next, a look at some laser-based tools for exploring chaos.



To paraphrase Supreme Court Justice Potter Stewart, while we might not be able to precisely define chaos, we know it when we see it! A good number of researchers (chaologists), however, have come up with some pretty concise definitions of chaos, several of which I happily present here: "...largely unpredictable long-term evolution occurring in a deterministic, nonlinear dynamical system because of sensitivity to initial conditions." Garnett P. Williams in Chaos Theory Tamed " ...lawless behaviour governed entirely by law." Ian Stewart in Does God Play Dice? "Dynamics freed at last from the shackles of order and predictability....Systems liberated to randomly explore their every dynamical possibility.... Exciting variety, richness of choice, a cornucopia of opportunity." Joseph Ford (quoted by James Gleick in Chaos) Chaos presents itself in an incredibly diverse range of areas of science, including irregular heart rhythms studied by cardiologists (medicine); the erratic motion of some planetary satellites, such as Saturn's moon Hyperion (astronomy); the turbulent flow of fluids (fluid dynamics); the observed quirky behavior in electronic circuits (electronic engineering); and, of course, those eye-catching perpetual motion toys sold at airport gift counters (crass commercialism). Biology, mathematics, chemistry, economics, ecology, the stock market and, perhaps, even the quantum world all cross paths with chaos. Though we will try to avoid cardiac arrhythmias, there are many other areas that can be readily explored if approached with the right tools. First, we will have a quick look at some of those tools.
Mathematics With little more than a few select equations and a computer, chaos can be explored in the comfort of your home. The results should be quite surprising. At right are the plots for three equations: a simple sine function; a more complicated hodgepodge of sines, cosines, tangents, and logs; and the deceptively simple-looking logistic map equation. (Note how complex the plot is for so simple a formula.) The logistic map equation was created in the mid-nineteenth century to deal with ecological population studies. By varying one of the parameters (often depicted as a Greek lambda or the English letter k), the one set to "4.0005," and by iterating the equation (that is, assigning the result of each calculation to x(t) and feeding it back for the next calculation) behavior ranging from periodic to chaotic can be observed. In the plot shown, the value of 4.0005 forces the equation into the chaotic, infinity region. With a few more iterations, where each positive or negative peak is an iteration, the graph would fly off into infinity and off the chart! So, equations being evaluated for chaotic behavior need to be studied for a considerable number of iterations until they settle down and exhibit their characteristic behavior. Incidentally, the lambda (or k) parameter is usually varied from 1 to 4 with the initial value of x(t) set between 0 and 1.
Mathematics packages, such as Mathematica, Matlab, and Mathcad, make evaluating equations straightforward. There are also some great freeware packages currently available on the Internet. An example of that is Scilab (which comes bundled with a simulation package called Scicos). Scilab/Scicos is very similar to Matlab/Simulink and is very good for creating simulations. I used the Scilab/Scicos package to compute the logistic map equation and read the created file into DPlot. In fact, DPlot (not freeware but quite inexpensive) was used to create all of the above plots.
	This is an example of Scicos being used to run a simulation of the logistic map equation. The sync clock acts to synchronize computational behavior in the simulation. The Mem (memory) block acts to prevent an algebraic loop, which would cause the simulation to immediately fail due to a fatal recursion during feedback. This occurs because the output of the simulation is being fed back as input, and the software has no way of knowing what the input is before calculating the output. Mem stores an initial x(t) value that will be modified after an initial computation. Here the value of Lambda is 3.79. It is being multiplied by the previously calculated result for x(t). At the same time, the 1 is being added to the negative of the previously calculated result for x(t), and that sum is multiplied by the product of Lambda and the previous x(t). And so it goes. Scicos produces a very nice oscilloscope plot, not shown, thanks to that green square (scope) function that performs the output plot. Optionally, Scicos can create a file that can be read in by another program, as was done for the above plot. Scicos has numerous palettes loaded with all sorts of nifty functions, including x-y plots, which are needed for other chaos studies.
Mathematica has available notebooks written by researchers to allow studying chaos. They simplify the process for creating bifurcation diagrams, phase space plots, statistical time-series calculations, etc. Any of these things can be programmed with other mathematics software, of course. It's just a matter of how much time you're willing to wait before getting started doing actual science.
Analog Simulation One of the first Heathkits I built, and the one that actually worked, was the model EC-1 analog computer. It was probably around 1961. I can remember carrying the kit back home from the store. Naturally, I couldn't wait to start building it. Months later, when I finally finished, there was a gorgeous instrument boasting nine operational amplifiers (op amps) with three initial condition supplies. After patching in the demo programs (patching is how you program most analog computers) to simulate a bouncing ball, a fired projectile, and such, the unit kind of just sat there. The EC-1 was really a very sophisticated instrument. This sophistication was not so much in the electronics end but, rather, in its problem-solving capabilities. Analog computers thrive on solving differential equations, which are the equations governing (or, rather, describing) natural phenomena, such as wave motion, atmospheric turbulence, electronic circuits, biological rhythms, chaotic phenomena, etc. If one is not familiar with differential equations, and I wasn't, analog computers are just nice things to look at. But once you study differential equations, you will find that worlds open, curtains part, salaries skyrocket, and people usually treat you better. The EC-1 used vacuum tubes for the op amps. The tube amplifiers had gains of 1000 (a measure of accuracy—the higher the better) and were very prone to drift. Drifting meant frequent amplifier balancing. Conveniently, the amplifiers were capable of outputting +/- 60 volts. I no longer have the Heathkit computer as I gave it away years ago, before eBay was born. To this day, I still walk around with a "kick me" sign on my back! Analog computers are practically synonymous with operational amplifiers. Today, op amps have little drift, very high open and closed loop gains, low power requirements, and they don't weigh a ton! However, analog computers seem to have taken a back seat to the ubiquitous desktop digital computer. Whereas digital computers can solve problems to any specified precision, they do so discretely in computed chunks. As fast as digital computers may be, it takes time to solve differential equations. With an analog computer, once a problem is "patched in" the solution is revealed. It's also great fun modifying system parameters in real time. While digital computers in theory can achieve precision to a specified number of decimal places, analog computers use voltages to achieve precision to within the level of system noise and beyond! Analog computers are fantastic for simulating natural phenomena if you know the differential equations governing the natural. There are now plans in the works that may allow a successful bridging of the two types of computers in ways previously undreamt of! Although hybrid computers (part analog and part digital) have been around for a long time, special chips are being developed that may allow digital computation to mesh with the virtually instantaneous computational speed of analog computers. Incidentally, analog computers played a part in other instrumentation produced by Heath (in its Malmstadt-Enke line), including a dropping mercury polarograph. (I actually purchased the polarograph, but the dropping mercury electrode was so precariously constructed that I never had the nerve to fill it with mercury!) At least one other company, McKee-Pedersen, employed analog computer circuitry in its line of laboratory instrumentation for chemical analysis. Sadly, both Heath and McKee-Pedersen are now just footnotes to history.
Analog computers employ feedback for proper operation, and they can operate nonlinearly, meaning the output is not linearly related to the input. Feedback and nonlinearity are also characteristic features of chaotic phenomena. And so, analog computers and op amps are ideal simulators for chaos. Until recently, one company still making the intrepid analog computer was Comdyna. I acquired a used system on the Internet. At right is a picture taken of the unit in operation, along with an X-Y monitor overlay of the computer's output. The unit's eight amplifiers, two of which are just sign inverters, are easily patched and set up to perform simulations.
In order to work with breadboarded circuits where externally wired integrated circuits (i.e., multipliers, A/D converters, and sensors) can be readily incorporated into op amp circuits, I assembled a number of Tektronix modules. Basically, there are two TM 506 power modules sitting one on the other with a group of six AM 501 operational amplifiers in the lowest row. The AM501 incorporates dual 741 op amps. The second row up includes a DC 503A universal counter/timer, an FG 503 function generator, an MR 501 X-Y monitor (cute as a bug), a DM 502 digital multimeter, and two PS 503 dual-power supplies. Finally, at the very top is a model 604 X-Y monitor. To its left is a homemade interface box that allows adjusting the x- and y- voltages being fed to the 604.
Chua's Circuit Invented by Leon Chua back in 1983, one version of the basic circuit is shown below both as a schematic and as its breadboard realization. Over the years, there have been many modifications and application-specific variations made to the original circuit. The circuit was designed from first principles by Chua, and it reveals a remarkable broad spectrum of chaotic behavior. Later, when we get to actual experiments, we will look at some of the results in phase space output by the electronics. In the photos below, the first depicts a Chua circuit drawn and simulated by Electronics Workbench v5.12 software. Here, the overlaid simulated scope output shows the chaotic signal generated with the specified component values. The two pictures that follow show an honest-to-goodness, real-life breadboarded circuit along with an equally real oscilloscope output. Both the simulated and breadboarded circuits use components of nearly identical values. I was pleasantly surprised (actually, shocked) that electronic design software could simulate the chaotic state produced in Chua's circuit. It is probably unrealistic to expect this kind of compliancy across-the-board in chaos simulations. Note the nonlinear components highlighted in the schematic. As mentioned above, nonlinearity is a requirement for chaos. Though not sufficient in and of itself to guarantee chaos, nonlinearity must be present in the feedback loop. Note also the inductor at the lower left. While capacitors are available in a large range of values, inductors are not. Also, for many circuits the inductor must have a very low internal resistance. I specified a 6.5 ohm internal resistance, here shown as a resistor in series with the coil. Usually, inductors are fixed in value. However, by using a small but powerful magnet, it is possible to vary the inductance of a coil if it has been wound on a ferrite bobbin. (The inductance can be reduced and restored depending on the magnet's motion.) I use just such a magnet while exploring circuits that appear reluctant to enter a chaotic state. By approaching and withdrawing the magnet from the coil, I can usually find the sweet spot at which chaos appears. This assumes the design is fairly close to correct, of course. If you plan to do a lot of chaotic circuit experiments with real inductors, by all means try the magnet trick while tuning.



Next, a look at some laser-based tools for exploring chaos.

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