The Rossler Attractor



n 1.The confused unorganized state existing before the creation of distinct forms. 2:complete disorder syn confusion, jumble, snarl, muddle, disarray.

The new science of chaos theory has sparked the imaginations of many modern scientists from a variety of disciplines. Members from the 'hard sciences like math and physics to the 'soft disciplines of psychology and sociology come to chaos in search of answers. Current areas of inquiry which utilize Ideas from chaos are planetary climate prediction, neural network models, data compression, turbulence, and economics to name a few. The common thread throughout these topics is a tendency to be complex. As an example, think of the complexity of interactions between something as seemingly simple as water flowing through a faucet into your kitchen sink. If you gradually increase the amount of water coming out of the faucet you will notice that the state of the water changes from an orderly continuous stream into a gushing and splashing flow. As this happens the ability of physics to say what is happening within the flow decreases dramatically. This is where the model of chaos steps in. Chaos theory attempts to find order within a seemingly unordered universe.

It is no wonder that this new science has arrived in such close conjunction with high speed computers. Computers are in fact indespensible to the study of such complex systems. This is because the amount of data necessary to realize so called chaotic patterns is immense. Computers also lend easily to methods such as iterating a process over and over until a pattern can be realized. A human performing such a process by hand is a maddening thought!

Some of the first research in this area was done in the 1960's at the University of California, Santa Cruz by a group of students who would stay late in the new computer labs to get computer time to run experiments. These computers were true plug and play machines! But it was Edward N. Lorenz who discovered the first strange attractor on a "supercomputer" in the 1962 at M.I.T. In fact, the Rossler attractor, which we will be exploring shortly is based upon the Lorenz attractor.

The Lorenz Attractor



The basic idea behind an attractor is that a dynamic system will tend toward certain states as time goes on. The simplest form of an attractor is the point attractor. Consider a normal pendulum, it doesn't matter where you release it from, it will always come to rest in the same position, perpindicular to the ground. This state is the attractor for the system.
To further the concept, think of the same pendulum. Now place three magnets as the vertices of an equilateral triangle with the pendulum at rest in the center. Each time you swing the pendulum it will come to rest either in the center or favoring one of the three magnets. The image shown below displays the attractor basin for this system. Each of the heart shaped regions in the center corresponds to the final state of the pendulum coming to rest favoring a particular magnet. The black areas correspond to the borde r between attractor basins.

Now consider a pendulum which is not subjected to friction. In this situation the pendulum will be attracted to different cycle states depending on the amount of energy in the system. This system has an infinite number of final states.


The Strange Attractor

The truth of the matter is this, there is not yet a formally accepted definition of a strange attractor. Strange attractors tend to arise in dissipative dynamical systems, such as the first pendulum example given above. Dissipative simply means that the system loses energy as time goes on. The Collins Reference Dictionary of mathematics states the definition of a strange attractor as ...such that its Hausdorf Dimension is non-integral, or else dependant on initial conditions... Obviously this is not a complete definition, but it does give us a sort of intuitive way of thinking about a strange attractor.


Measuring Dimension

The Hausdorf Dimension is a means of measuring the dimension of a mathematical object. For instance, the dimension of a point is 0, a line resides in 1 dimensional space, a plane 2, and of course our friend 3-space in which we live. So by definition a strange attractor is an object which is neither a point, a line, or a plane!. For example, the dimension of the Rossler Attractor is estimated at between 2.01 and 2.02. To understand what this means, think about this, the equations which describe the Rossler attractor will describe a curve or line in 3 dimensional space for periodic solutions. But when you have a chaotic solution, which is never periodic, that is, it never visits a point which it has previously visited, then the path of the Rossler attractor as a whole (time to infinity) becomes more than a collection of lines, and slightly more than a collection of planes. The key point is that for a non-periodic solution, the attractor never retraces a previously traveled path. This non-integer dimension is also what qualifies strange attractors as a fractal. A fractal is defined to be any object with a non-integer Hausdorf Dimension.


The Rossler equations

The Rossler attractor is defined by a set of three Differential equations

Where the coefficients a, b, and c are adjustable constants. As you can see the xz term causes this to be a non-linear set of equations. Which is a major cause of the chaos exhibited in this system. To investigate this attractor, think of the equations as describing a path in space. For certain choices of the constants, we can see periodic behavior.

For instance, set a = 0.2 ; b = 0.2 and c = 8 and you get the following path.

But for other choices you will not get a periodic solution. One way to look at the overall behavioir of an attractor is to construct a final state diagram, or a Feigenbaum diagram. To do this, choose one parameter which you wish to vary and set the other s to constants. As you vary the one parameter, mark what is the final state of the system on the vertical axis. Shown below is a final state diagram for a = 0.2 ; b = 0.2 and 2.5 < c < 10. Notice the relationship between c = 8 on the Feigenbaum diagram be low and the periodic solution above.

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