Chaos - James Gleick [65]
A little realism, in the form of friction, changes the picture. We do not need the equations of motion to know the destiny of a pendulum subject to friction. Every orbit must eventually end up at the same place, the center: position 0, velocity 0. This central fixed point “attracts” the orbits. Instead of looping around forever, they spiral inward. The friction dissipates the system’s energy, and in phase space the dissipation shows itself as a pull toward the center, from the outer regions of high energy to the inner regions of low energy. The attractor—the simplest kind possible—is like a pinpoint magnet embedded in a rubber sheet.
One advantage of thinking of states as points in space is that it makes change easier to watch. A system whose variables change continuously up or down becomes a moving point, like a fly moving around a room. If some combinations of variables never occur, then a scientist can simply imagine that part of the room as out of bounds. The fly never goes there. If a system behaves periodically, coming around to the same state again and again, then the fly moves in a loop, passing through the same position in phase space again and again. Phase-space portraits of physical systems exposed patterns of motion that were invisible otherwise, as an infrared landscape photograph can reveal patterns and details that exist just beyond the reach of perception. When a scientist looked at a phase portrait, he could use his imagination to think back to the system itself. This loop corresponds to that periodicity. This twist corresponds to that change. This empty void corresponds to that physical impossibility.
Even in two dimensions, phase-space portraits had many surprises in store, and even desktop computers could easily demonstrate some of them, turning equations into colorful moving trajectories. Some physicists began making movies and videotapes to show their colleagues, and some mathematicians in California published books with a series of green, blue, and red cartoon-style drawings—“chaos comics,” some of their colleagues said, with just a touch of malice. Two dimensions did not begin to cover the kinds of systems that physicists needed to study. They had to show more variables than two, and that meant more dimensions. Every piece of a dynamical system that can move independently is another variable, another degree of freedom. Every degree of freedom requires another dimension in phase space, to make sure that a single point contains enough information to determine the state of the system uniquely. The simple equations Robert May studied were one-dimensional—a single number was enough, a number that might stand for temperature or population, and that number defined the position of a point on a one-dimensional line. Lorenz’s stripped-down system of fluid convection was three-dimensional, not because the fluid moved through three dimensions, but because it took three distinct numbers to nail down the state of the fluid at any instant.
Spaces of four, five, or more dimensions tax the visual imagination of even the most agile topologist. But complex systems have many independent variables. Mathematicians had to accept the fact that systems with infinitely many degrees of freedom—untrammeled nature expressing itself in a turbulent waterfall or an unpredictable brain—required a phase space of infinite dimensions. But who could handle such a thing? It was a hydra, merciless and uncontrollable, and it was Landau’s image for turbulence: infinite modes, infinite degrees of freedom, infinite dimensions.
Velocity is zero as the pendulum starts its swing. Position is a negative number, the distance to the left of the center.
The two numbers specify a single point in two-dimensional phase space.
Velocity reaches its maximum as the pendulum’s position passes through zero.
Velocity declines again to zero, and then becomes negative to represent leftward motion.
ANOTHER WAY TO SEE A PENDULUM. One