LORENZ NEEDED A SIMPLE SET OF EQUATIONS
After discovering the extreme sensitivity to initial conditions exhibited by his original set of twelve equations, Lorenz wanted to study this phenomenon more carefully. However, Lorenz knew that he needed a smaller set of equations to study. Twelve simultaneous nonlinear equations were simply too cumbersome for the work he had in mind.
Lorenz borrowed a set of seven convec-tive fluid motion equations from a col-league and then eliminated four of the equations that he believed were of minor importance. He hoped that the extreme sensitivity to initial conditions that he had seen previously would be present in the remaining equations. It was.
Lorenz knew that the three equations did not model weather particularly well but predicting the weather was not his present goal. He was now focused on studying the sensitivity to initial conditions phenomenon.
THE LORENZ EQUATIONS
The three equations Lorenz chose are shown below.
dx/dt = s(y – x)
dy/dt = rx – y – xz
dz/dt = xy – bz
These differential equations are called “autonomous” meaning that the independent variable, time, explicitly appears only in the derivatives to the left of the equal signs and not in the terms on the right side of the equations.
The equations describe the convective motion of the atmosphere when it is heated by the ground below and cooled from above. However, describing what the variables x, y, and z in these equations actually represent is beyond the scope of this discussion and is not necessary for us appreciate the significance of Lorenz’ work.
What is important, though, is to know that Lorenz chose
s = 10 r = 28 and b = 8/3
as the values for the coefficients in his equations.
The Lorenz equations are coupled because more than one variable appears in each equation. They also are nonlinear due to the presence of the terms xz and xy in the second and third equations, respectively.
The study of equations such as these is called “Nonlinear Dynamics.” The term “dynamics” (we sometimes see the phrase “dynamical systems” used) refers to the fact that the variables change with time. The study of nonlinear dynamics is important because the vast majority of the phenomena in our world are nonlinear.
Nonlinear equations usually are quite difficult to solve. Fortunately, mathematicians have developed some special techniques to gain information about these systems.
One of the techniques consists of making a “phase space” plot of the variables. Simply stated, this means plotting values of one variable versus corresponding values of another variable. Time is not explicitly involved.
Initial values are assigned to each of the variables. These are used to closely approximate new values for the variables at an instant in time later. The process is repeated over and over and is called “iteration.”
The values of the variables are then plotted in phase space. The resulting paths defined by the plotted points are commonly referred to as “trajectories.”
Each point on a trajectory in phase space represents the values of the variables at a particular instant. Moving along a trajectory shows how a system’s variables change with time.
The trajectories do not occupy just any and all regions of phase space. Regardless of the starting point (initial conditions) the trajectories evolve with time toward a specific region in phase space called an “attractor.”
Attractors are phase space pictures of the possible steady state(s) for a system. The trajectory for a physically realizable (i.e., lossy) non-chaotic system asymptotically approaches its attractor as time increases.
When the response of a system of nonlinear equations is not chaotic, an attractor can only have one of several possible well-defined geometrical shapes.
The trajectories of chaotic nonlinear equations, however, evolve toward what is called a totally different type of attractor, called a “strange attractor.” These “strange attractors” are irregular and complex in shape. They can take a variety of forms.
A strange attractors is the result of a trajectory that never repeats itself. In addition, a chaotic trajectory line never crosses or touches itself (although it may come infinitesimally close to touching itself).
Strange attractors define the existence of chaos and will be produced by each of the chaotic circuits we will simulate and/or build. They are fascinating to see.
For a very good discussion of phase space (and chaos in general), see James Gleick’s excellent and easily readable book “Chaos – Making a New Science.” Another easy to follow discussion is given in the book entitled “Chaos Theory Tamed” by Garnett Williams.
Lorenz used a phase space approach, thereby unlocking much valuable information about his system of equations. Much more significantly, he discovered important, never before known information about chaos.
Because Lorenz’ system had three independent variables (x, y, and z), a complete phase space diagram would have required a three-dimensional plot. However, two-dimensional plots are much easier to make and to analyze, so that is the graphing method Lorenz used.
One of the two-dimensional phase space plots for his set of three equations is shown below.
(Click on the image to enlarge it.)
In this diagram, the value of z is plotted versus the value of x. Lorenz’ plots of x versus y and y versus z had a somewhat similar appearance. This is not surprising since they showed the same three-dimensional strange attractor, observed from a different direction.
The fact that the phase space plot resembled a butterfly caused the image to be referred to as the “Lorenz butterfly.” (The choice of this word “butterfly” had nothing to do with the “butterfly effect” discussed earlier.) Lorenz’ butterfly truly was a “strange attractor” and resulted because his equations were chaotic.
For a more complete description of Lorenz’ work, I recommend the book “The Essence of Chaos” by Edward Lorenz.
CHARACTERISTICS OF ALL CHAOTIC SYSTEMS
As the result of Lorenz’ work, together with that of other later researchers, several important characteristics of chaotic systems have been deduced.
1. Chaos is not random. Chaos is deter-ministic. That means, it is governed by mathematical equations.
2. All chaotic systems exhibit extreme sensitivity to initial conditions. Thus, long-range predictions of the behavior of a chaotic system are impossible. (Good short-range predictions often are possible, however.)
3. All chaotic systems are nonlinear (but not all nonlinear systems are cha-otic).
4. All chaotic systems that are described by a set of autonomous differential equations are at least three dimensional. (The number of dimensions is the number of variables required in the equations to describe the system.)
5. The trajectories traced out in phase space for a chaotic system never cross or touch. If they did, the values of the variables would then repeat themselves and future values would be predictable.
6. The rate at which two adjacent trajectories diverge depends on what are called the “Lyapunov exponents” of the chaotic system. The number of Lyapunov exponents is equal to the number of dimensions (independent variables) of the phase space.
Lyapunov exponents can be positive or negative but at least one of them must be positive if the system is chaotic. The larger the most positive exponent is, the more sensitive to initial conditions is the system and the faster the trajectories diverge. This results in a shorter time of predictability for the system’s behavior.
We Soon Will Solve the Lorenz Equations
On page 10 of this website, we will discuss a circuit that electronically “solves” the Lorenz equations and produce the Lorenz butterfly strange attractor . The circuit is amazingly simple and readily lends itself to both computer simulation and to physical construction.
After reading the discussion on page 10, you will be able to simulate this Lorenz circuit using LTspice or to actually construct the circuit using a few electronic components. (Why not plan to do both?) In any case, you will be able to produce the Lorenz butterfly yourself.
I know that you are getting anxious to work with chaotic circuits yourself. So before we talk more about the Lorenz butterfly, I want you to see how simple and easy to simulate or build chaotic circuits really can be.
Consequently, on page 5 of this website, we will discuss the first actual circuit that was specifically designed to demonstrate a chaotic response. It is an extremely simple circuit to simulate and/or build, as you will see on page 8.
IT’S TIME TO START!
We are now ready to begin talking about the actual simulation and building of chaotic circuits that can be accomplished by anyone who has the interest. The first circuit we will discuss is on the following page of this website.
The circuit is called “Chau’s Circuit.” It is a chaotic circuit which has been widely studied and is very well documented. It’s an excellent “starter” circuit.