Dynamical Systems

What the are and how they work...

Dynamical systems are collections of interrelated things that change over time in accordance with a fixed rule, normally expressed as a set of one or more mathematical expressions.

In our description of a system in general we used a pendulum as an example of a dynamical system. Forgetting for the moment the rest of the clock system, imagine that we pull the pendulum weight to one side slightly and release it. We would expect the pendulum to swing in decreasing arcs until it came to rest hanging straight down again. An undriven pendulum, one which is not pushed by some external force, will always exhibit this behavior due to the friction at the pivot and through the air. It will always come to rest with its state variables, its position, θ, measured by the angle from the vertical and the rate of change of that position, θ', called angular velocity, both zero.

If we create a graph with one state variable on the horizontal and the other running vertically, the state of the system at any time is a single point on that graph. Such a graph is called a phase space map of the system. For systems like the pendulum it is customary to plot the position horizontally and angular velocity vertically.

As time passes and the pendulum swings, its state point traces out a curve in the system's phase space. This curve is called a phase space portrait of the system. For an undriven pendulum the phase space portrait is a spiral winding in to the point (0,0) where both position and angular velocity are zero. Such a point, where a dynamical system comes to rest is called a point attractor. because the system seeks out that point.

It is possible that a system may be attracted to objects in phase space other than a point. There are cyclic attractors which may take the form of any finite number of points visited in sequence by the system or an infinite number of points lying on a curve like a circle or ellipse. There are also chaotic attractors where the system is attracted to a region of phase space that is not so simply defined, being fractal in nature.

Getting back to the clock system example, the elevated weights and the clock gears serve to push the pendulum, makeing it a driven rather than undriven pendulum. The driven pendulum attractor is an ellipse in phase space which will be of fixed size as long as the elevated weights continue to supply energy to the pendulum as they fall.