Figure 6-8
Trajectory of the pendulum in phase space.
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the entire system. As the system changes, the point traces out a trajectory in phase space—a closed loop in our example. When the system is not a simple pendulum but much more complicated, it will have many more variables, but the technique is still the same. Each variable is represented by a coordinate in a different dimension in phase space. If there are sixteen variables, we will have a sixteen-dimensional space. A single point in that space will describe the state of the entire system completely, because this single point has sixteen coordinates, each corresponding to one of the system’s sixteen variables.
Of course, we cannot visualize a phase space with sixteen dimensions; this is why it is called an abstract mathematical space. Mathematicians don’t seem to have any problems with such abstractions. They are just as comfortable in spaces that cannot be visualized. At any rate, as the system changes, the point representing its state in phase space will move around in that space, tracing out a trajectory. Different initial states of the system correspond to different starting points in phase space and will, in general, give rise to different trajectories.
Velocity
Figure 6-9
Phase space trajectory of a pendulum with friction.
Strange Attractors
Now let us return to our pendulum and notice that it was an idealized pendulum without friction, swinging back and forth in
perpetual motion. This is a typical example of classical physics, where friction is generally neglected. A real pendulum will always have some friction that will slow it down so that, eventually, it will come to a halt. In the two-dimensional phase space this motion is represented by a curve spiraling inward toward the center, as shown in figure 6-9. This trajectory is called an “attractor,” because mathematicians say, metaphorically, that the fixed point at the center of the coordinate system “attracts” the trajectory. The metaphor has been extended to include closed loops, such as the one representing the frictionless pendulum. A closed-loop trajectory is called a “periodic attractor,” whereas the trajectory spiraling inward is called a “point attractor.”
Over the past twenty years the phase-space technique has been used to explore a wide variety of complex systems. In case after case scientists and mathematicians would set up nonlinear equations, solve them numerically, and have computers trace out the solutions as trajectories in phase space. To their great surprise these researchers discovered that there is a very limited number of different attractors. Their shapes can be classified topologically, and the general dynamic properties of a system can be deduced from the shape of its attractor.
There are three basic types of attractors: point attractors, corresponding to systems reaching a stable equilibrium; periodic attractors, corresponding to periodic oscillations; and so-called strange attractors, corresponding to chaotic systems. A typical example of a system with a strange attractor is the “chaotic pendulum,” studied first by the Japanese mathematician Yoshisuke Ueda in the late 1970s. It is a nonlinear electronic circuit with an external drive, which is relatively simple but produces extraordinarily complex behavior. 16 Each swing of this chaotic oscillator is unique. The system never repeats itself, so that each cycle covers a new region of phase space. However, in spite of the seemingly erratic motion, the points in phase space are not randomly distributed. Together they form a complex, highly organized pattern—a strange attractor, which now bears Ueda’s name.
The Ueda attractor is a trajectory in a two-dimensional phase space that generates patterns that almost repeat themselves, but