Loop Polarity, Loop Dominance, and the Concept of Dominant Polarity

There is a conspicuous gap in the literature about feedback and circular causality between intuitive statements about shifts in loop dominance and precise statements about how to define and detect such important nonlinear phenomena. This paper provides a consistent, rigorous, and useful set of definitions of loop polarities, dominant polarity, and shift in loop dominance, and illustrates their application in a range of system dynamics models. Consistent with the usual definitions, the polarity of a first-order feedback loop involving a level x and a single inflow x is defined to be the sign of dx/dx. Loop polarity is shown to depend upon the sign of parameters not usually considered to be part of the loop itself. The definition of loop polarity is then extended to multi-loop first order systems. All positive loops with gain less than one, such as economic multipliers, are shown to be multi-loop systems with dominant negative polarity. The shifts in loop dominance that occur in nonlinear system arise naturally as changes in the sign of dominant polarity. The concepts developed in the paper are then applied to simple higher-order nonlinear feedback systems. The final application to a bifurcating system suggests that all bifurcations in continuous systems can be understood as consequences of shifts in loop dominance at equilibrium points. Loop Polarity, Loop Dominance, and the Concept of Dominant Polarity George P. Richardson The Rockefeller College of Public Affairs and Policy University at Albany – State University of New York Albany, NY 12222 Introduction Underlying the formal, quantitative methods of system dynamics is the goal of understanding how the feedback structure of a system contributes to its dynamic behavior. Understandings are captured and communicated in terms of stocks and flows, the polarities of feedback loops interconnecting them, and shifts in the significance or dominance of various loops. However, there is a conspicuous gap in our literature between intuitive statements about shifts in loop dominance and precise statements about how we define and detect such important nonlinear phenomena. This investigation is an attempt to bridge that gap. In the effort to construct formal definitions of shifts in loop dominance, it became clear that our common definitions of loop polarities were not sufficiently precise. There is an underlying unease in our own field and in the cybernetics literature that we do not really know what a positive loop is. Ashby, for example, was bothered by the convergent behavior of the discrete positive loop xt+1 = (1/2) yt, yt+1 = (1/2) xt. He used its apparently contradictory goal-seeking behavior to support his claim of the "inadequacy" of feedback as a tool for understanding complex dynamic systems (Ashby 1956, p. 81). To avoid such anomalies, some define a loop to be positive if it gives "divergent behavior." Graham (1977) finds problems with that characterization and suggests instead that a loop be called positive if its open-loop steady state gain is greater than one. Richmond delightfully exposed our confusions by describing a well-meaning professor trying to explain to a concerned student: "Positive loops are ... er, well, they give rise to exponential growth ... or collapse ... but only under certain conditions ... under other conditions they behave like negative feedback loops..." He concluded that the nicest way out of the confusion is to define a positive loop to be a goal-seeking loop whose goal continually "runs off in the direction of the search" (Richmond 1980). Some, of course, ignore all the subtleties and obtain loop polarities simply by counting negative links (Richardson and Pugh 1981). We begin then with a tighter, more formal definition of the polarity of a feedback loop. Our focus, however, is on the concept of loop dominance and the phenomenon of shifts in loop dominance in multi-loop nonlinear systems. Rigorous Definition of Loop Polarity We shall base our definition of loop polarity on the assumption that every dynamically significant feedback loop in a system contains at least one level (accumulation or integration).1 The development will be in terms of continuous systems. A similar development holds for feedback processes couched in discrete terms, provided the principle of "an accumulation in every loop" is maintained. Consider a single feedback loop involving a single level x and an inflow rate x = dx/dt.2 Define the polarity of the feedback loop linking the inflow rate x and the level x to be sign( dx dx ) = sign dx/dt dx This formal definition is consistent with our more intuitive characterizations: "dx" can be thought of as "a small change in x" which is traced around the loop until it results in "a small change dx" in the inflow rate x = dx/dt. If the change in the rate, dx , is in the same direction as the change in the level, dx, then they have the same sign. Since x here is an inflow rate and thus is added to the level, the loop reinforces the initial change and is therefore a positive loop. In such a case, sign(dx/dx) is also positive, so the formal definition is consistent with the intuitive one. If the resulting change in the inflow rate is in the opposite direction to the change dx, then sign(dx/dx) is negative and the polarity of the loop is negative by both our intuitive and formal definitions. The formal definition is equivalent to defining the polarity of a first-order feedback loop to be the sign of the slope of its rate-versus-level curve.3 To extend the definition to feedback loops in which x is an outflow rate, we merely have to agree to attach a negative sign to the expression for x if it represents an outflow. Then the definition above holds for all loops involving a single level x and a single inflow, outflow, or net rate x. The first few examples that follow are very familiar; they are intended to establish some confidence in this formal definition of loop polarity before we use it to derive some less familiar results. Example (1): Exponential growth or decay. Let x = bx, where b is a constant. Then the polarity of the feedback loop is sign( dx dx ) = sign d(bx) dx = sign(b) which is positive if b is positive and negative if b is negative. The result makes intuitive sense, as may be seen by interpreting x as a net rate such as net population growth. If births exceed deaths, the coefficient b is positive and the loop produces exponential population growth. Similarly, if deaths exceed births, b is negative and the loop exhibits exponential decay behavior. The usual case is b > 0, and that prompts us to call all such first-order net-rate formulations positive loops. However, the polarity of such a loop in fact depends on a parameter whose sign is set by environmental conditions outside the loop. Without knowledge of the sign of b, the polarity of the loop represented by x = bx is undetermined.4 Example (2): Exponential adjustment to a goal. Let x = (x* x) T , where x* and T are constants. Loop polarity = sign( dx dx ) = sign (x* x)/T dx = sign -1 T which is negative if the time constant T is positive, and positive if T is negative. In applications of this structure, as in exponential smoothing, the time constant T is always positive, so the loop is always negative. When x* = 0, this formulation reduces to example (1) with b = -1/T < 0: again, a negative loop by both formal and intuitive definitions. In each of these cases, the formal definition of loop polarity behaves appropriately but yields no new insights. Cases involving more than one loop provide more interesting testing ground. Multi-Loop Structures: Loop Dominance The formal definition of loop polarity leads to a precise concept of loop dominance in simple systems. Consider a first-order system containing several feedback loops and the level variable x. Let x represent the net increase in x. Define the dominant polarity of the first-order system to be Loop Dominance Page 4