A dynamical system is a system of variables that undergo changes over time. A familiar example is the motion of a ball free-falling in the earth’s gravitational field; the ball’s acceleration, velocity, and position all vary over time as the ball drops. System dynamics is the analysis of dynamical systems via mathematical formulas. Techniques used in analyzing dynamical systems of equations include, among others, mathematical analysis (e.g. calculus) computer simulation, time-series plots, Fourier analysis, and phase portraits.

In tracking studies, we typically display a time-series plot showing how perception, error, output, and disturbance variables changed over time during the experimental run. But this is not the only way we could display the results. In the mass-spring-damper simulations I referred to previously, by pressing a radio button the user could switch the graphical display from a time-series view to a phase diagram (also called a phase portrait). The phase portrait shows the system spiraling down toward the position where velocity is zero. The spiral occurs because the damper is draining energy from the system. This is one example of how a phase portrait can help the viewer to understand the behavior of the system over time.

Vensim is an example of a computer application that makes it relatively easy to simulate dynamical systems and analyze their behavior, and contains some mathematical procedures for finding “best-fitting” system parameters. Bill Powers used Vensim to fit system parameters of a control system to data and found that it did a better job than the “ecoli” search method he used to fit parameters to data.

In discussions of reorganization using the ecoli method, Bill Powers has talked about the “landscape” of parameters that the method must explore in order to find a set of parameters that will provide a good fit to the data in the sense of minimizing the squared error (least squares criterion). You can visualize this landscape as containing hills and valleys, with the optimal least-squares solution finding the deepest valley. But if the parameter landscape is complex, there will be many valleys, some deep, some shallow, some wide, some narrow. Depending on where on the landscape the search for optimal parameters begins, the search may lead to a relatively shallow valley in which the search becomes “stuck.”

Now here’s the kicker: The parameter landscape is nothing more than a phase portrait and the valleys are what are termed “attractors.” The landscape does nothing more than describe the behavior of the system (with respect to the size of error) as it moves through parameter space. Note that attractors are not forces that “attract” the system to a lower error state, they simply describe how ecoli reorganization will proceed from given starting points, given the data and the specific control system model whose parameters are being fitted. Phase portraits *describe* a dynamical system’s behavior. They do not propose a mechanism to explain this behavior. That mechanism is supplied by the system of variables that is put in motion to *produce* the behavior being diagramed.

It can now be seen that the above statement is incorrect. Control systems are just a particular kind of dynamical system; consequently, they cannot be “quite different than those that produce the goal-directness of dynamical systems.”

I think what you are worrying about is that dynamical systems include not only control systems, but other systems that may *appear* to produce goal-directedness. A nice example of this is the mass-spring-damper system of my simulation. The phase portrait of this system spirals down to a point attractor, giving the appearance that the system had the goal of arriving there. But this is merely an equilibrium system, not a control system, spiraling down to a minimum-energy state. Heylighen does a nice job distinguishing an equilibrium system from a control system, noting that control systems use energy from a source external to the system to resist the entropy-increasing effects of disturbances:

“For example, a ball that rolls down into a bowl will come to rest in the lowest point at the bottom of the bowl, having dissipated its energy of motion through friction. It does not matter where in the bowl (initial state) the ball started its trajectory: the end point will always be the same equilibrium state of minimal potential energy. This type of equifinality is not what we would intuitively see as goal-directedness, though.”

“This problem can be avoided by demanding that the final state would be a steady state “far-from-equilibrium”, i.e. a state that requires active intervention and a continuing flow of energy to reach and maintain. Indeed, life is a far-from equilibrium condition, and organisms cannot survive without a constant source of energy to keep their metabolism going.”

I like Heylighen’s discussion of phase portraits because he shows how useful they can be toward understanding a system’s behavior. For example, a control system will remain stable only so long as disturbances are not strong enough to overwhelm the system’s ability to compensate for them via its output. Cases in which disturbances can overwhelm the system are shown in the phase portrait as regions outside of the system’s attractor basin. Furthermore, a broad basin indicates a wide region of possible disturbances that the system will successfully compensate for. (See the paper for further insights into the information that can be extracted by viewing phase portraits.)

I strongly recommend reading Heylighen’s (2022) paper to those who are interested in expanding their knowledge of the philosophical problems that for millennia impeded progress in understanding the goal-directed nature of human and animal behavior, of the history of attempted solutions to the problem, of some current solutions, and of the use of phase diagrams in the analysis of dynamical systems.