Attractors

            In chaos theory, attractors are states towards which a system may evolve when starting from certain initial conditions. Attractors can be unique states, called fixed point attractors. But they can also be a whole range of states in the case of periodic or quasi-periodic attractors. Sometimes the specific condition or system state at, or near, an attractor is entirely unpredictable. Such attractors are called chaotic, and their general surrounding region is called a basin. Within a basin, a dynamic system is under far-from-equilibrium conditions.

            The magnetic attraction of the orbit of a pendulum for a fixed point, for example, is caused by a special attractor known as the fixed-point attractor. These are stable equilibrium points where dynamic systems will tend to come to rest. According to Çambel (1993), "Dynamic systems are attracted to attractors the way fireflies are attracted to light" (p. 59).

            An easy way to think of the fixed-point attractor is to consider a ping-pong ball and the surface of the sea. We can drop the ball over the sea where it will fall until it contacts the surface. We can also hold the ball under the water and then let go, where the ball will float upward toward the surface. Either way, the surface of the sea acts as an attractor for the ball and no matter where the ball is released, it will always wind up on the surface. But once the ball reaches the surface, it will be buffeted by winds and currents in unpredictable ways because the dynamics of the surface of the sea are very complex (Cohen & Stewart, 1994).

            The fixed-point attractor is just one of four known types. Other types include the limit cycle, tori, and chaotic attractors. A fixed-point attractor draws systems to a single point in phase space. A limit-cycle attractor attracts systems to a cyclic path in phase space (a range of final resting points). The tori attractor, plotted in phase space, looks like an innertube or doughnut.

            The configuration of attractors in phase space can help determine if a system is conservative (maintains energy) or dissipative (energy must be supplied from outside the system) and can also help determine if a system is chaotic. They serve as the geometric counterpart to the thermodynamic entropy function (Çambel, 1993).

            Chaotic attractors are found in conditions of turbulence. They attract complex systems from order into disorder. David Ruelle, who first called the chaotic attractor for turbulence "strange," found that this attractor pulled complex systems into a space of fractional dimension, where they became caught somewhere between a two-dimensional plane and a three-dimensional solid (Briggs & Peat, 1989; Ruelle & Takens, 1971).

            To appreciate this situation, consider a piece of paper, which is essentially a two-dimensional object. Crumple the paper. As it is compressed, the two-dimensional sheet of paper will approach a three-dimensional object.

            Probably the best-known method for measuring attractors is the Lyapounov (sometimes spelled Ljapunouv or Lyapunouv or Lyapounouv) function (or number, or exponent, depending on how we are using it) after the Russian scientist Alexander Michailowitsch Ljapunouv (1857-1918). The Lyapounov exponent in a system provides a way of measuring the conflicting effects of stretching, contracting, and folding in the phase space of our attractor. They give a picture of all the properties of a system that leads to stability or instability” (Gleick, 1987, p. 253).

            The Lyapunouv exponent “quantifies the average growth of infinitesimally small errors in the initial point” (Peitgen, Jurgens, & Saupe, 1992, p. 516). These small errors cause instability when continuously fed back into a system.

            There are several Lyapunouv exponents, one for each dimension that a system operates within. Later we will show a two-dimensional phase space for the ego.

            The first exponent is usually the largest and measures chaoticity or the degree of chaos present. The second measures the speed at which trajectories are pulled into (or repulsed from) a chaotic attractor. Another way of looking at this is that the first exponent indicates the average amplification of system errors while the second indicates the average area change in phase space.

            The archetypes of Jung’s collective unconscious are equivalent to chaotic attractors in psychic phase space. Attractors and Lyapounuv numbers associated with the psyche are discussed later.