a hyper-jump through phase space.
By visualizing change as a journey through a space it becomes easier to recognize and discuss the patterns of continuous improvement. It also becomes easier to see which shapes are important, and which ones are not.
Attractors and Convergence
OK, now it gets a bit more mathematical. Just hold on tight, and I’ll try and steer you through this challenging landscape. Trust me; the scenery will be worth it.
When complex systems change, the journeys they take through their vast phase space typically fall in one of a few categories. Consider the example of the famous Game of Life. Regardless of the initial state of the game, after a number of steps the system ends up in a stable situation, in all but a few cases. The stable situation at the end is either one stationary state (a “still life”), or it is an everlasting cycle of a small number of states. We say that the stable situation is an attractor for all other states that lead into it. And the collection of all trajectories that lead to an attractor is called its basin of attraction (see Figure 2). Since each system usually follows trajectories that lead into attractors, the attractors lure the system into small regions of its entire phase space. Despite the vast range of possible states of the system, it finally settles into one of just an orderly few.
Attractors (A), basins of attraction (B), and disturbances (S)
Are you still with me? Good. Let’s make it a bit more concrete with the example of seaweed…
Theoretically there are 2^1000 possible versions of seaweed DNA. That’s a lot, actually. It’s quite a bit more than the number of atoms in the universe. However, the number of real observable forms of seaweed is extremely small, because all other forms are unstable and, within a few generations, would either die out or change and end up in one of the few stable forms. It doesn’t matter that an uncountable number of forms of seaweed is theoretically possible. In practice, the environment forces seaweed to end up in one of a small number of forms that are actually feasible for that environment.
Some scientists think that convergence , which is the fact that biological solutions like eyes and wings have been “invented” several times independently, is a good example of the concept of attractors [Lewin 1999:73]. In biological morphology there is an attractor of “things that have four legs,” and an attractor of “things with two wings,” etc. Five legs and one wing are valid forms, but they are not stable. (Except perhaps in the vicinity of an unstable nuclear power plant.)
And so I believe that, in order to make a software project work well in its environment, we must make sure that what works well is also stable. Because projects will converge on stable forms, but that doesn’t mean those forms also work well…
Stability and Disturbances
There are three kinds of attractors in complex systems [Gleick 1987:269]:
- A fixed point attractor keeps a system in one specific state. An organizational hierarchy could be a good example of a fixed point attractor. Almost all organizations end up in that structure, and then they stay there forever [Waldrop 1992:169];
- A limit cycle is an attractor where a system repeatedly goes through the same sequence of states. One example is the cycle of forming, storming, norming, performing, and adjourning, a well-known group development model [Arrow 2000:152];
- A chaotic or “strange” attractor is a trajectory that refuses to end up in any of