**Love, High School, and****Learning Dynamics Early**

Ted Carron

### "When I think back on all the crap I learned in high school, it's a wonder I can think at all."

- Paul Simon

Dynamics, especially nonlinear dynamics, offers a more useful and humane view of causality than the one we learn in high school – useful not only in economics, but also in the way we interpret all complexity.

Whatever else education does, it expands the pool of metaphors available to us when trying to understand the world. The ‘facts’ and ‘skills’ we learn fade away, usually as we walk out of the examination hall, but the underlying ideas persist, available as metaphors to process the world. This applies quite as much to the study of mathematics as it does to that of history and science, and we are unlikely to encounter mathematical concepts anywhere outside a maths class.

One of the most ubiquitous ideas, implicit in many subjects, is what Harvard’s Graduate School of Education refers to as ‘linear causality’: the idea that everything that happens does so as a result of some preceding event. This is obviously an important and useful idea. But there are any number of areas (e.g., relationships, ecology, climate science and, of course, the economy) where simple linear causality fails to give us a good handle on things. Worse than that, it may mislead us into poor decisions. Yet unless we are lucky, we leave high school having studied only these systems where there is a fixed, unchanging (albeit sometimes complicated) relationship between the variables.

I believe that a general appreciation of the more complex causal types found in dynamic systems would help.

If the student is lucky, he or she might encounter some more complex causal patterns in high school – equilibrium or positive and negative feedback, for example. (Popular culture may give us a degree of familiarity with these, but formal study – not necessarily mathematical – could make them more accessible.) These are at the outer edge of a group of relationships found in a study of dynamics. Other nonlinear dynamic concepts/metaphors are local equilibria, parameter drift, aperiodicity, catastrophe, emergence, phase change. I can’t elaborate on all those ideas here, obviously, but I do want to at least contrast the mathematics of static analysis with that of dynamics, to show how fundamentally different the two are, and what a different view of causality dynamics gives rise to.

Dynamics is the mathematics of change – as opposed to state. In a static system the value of one variable is rigidly linked via a formula to the values of the other variables. Such a description is regarded as static even if one of those variables is time. In a dynamic system, by contrast, it is the RATE OF CHANGE of each variable – not its value – which is determined.

A charming – though definitely tongue-in-cheek – demonstration of how dynamics can throw light on something otherwise incomprehensible comes from Strogatz’ dynamic analysis of the relationships between lovers (Strogatz 1988).

In his model there are two lovers, Romeo and Juliet. The model suggests that each has a level of affection for the other which changes over time at a rate determined by their fixed nature AND the level of affection in which each is currently held by the other. In pseudo-mathematical notation the model looks like this.

Notice that the actual level of affection is not calculated directly, only its rate of change. To determine the actual level of affection at any instant, you have to add up all the continuous changes over the period leading up to that instant.

Analysis or simulation of this model shows that a number of very different outcomes can emerge depending on the values of a,b,c,d. Below these are expressed in the form of diagrams called phase diagrams. Each represents the lover’s behaviours under different selected values of a,b,c,d. They indicate how the variables change from any particular initial state. The horizontal axis represents Romeo’s affection for Juliet and the vertical axis represents Juliet’s affection for Romeo. The position and direction of the lines and arrowheads (trajectories) indicate the direction that J and R will move/change.

**Diagram ‘A’** is relatively simple. It suggests that if the lovers’ affections will ‘orbit’ tragically, forever, first Romeo loving an indifferent Juliet, then vice versa.

**Diagram 'B'** is a little more complex. The black dot in the center (0,0) of the diagram represents something called a ‘global stable fixed point’. All arrows lead eventually to this point. No matter where the lovers’ affections start, their natures will eventually lead them to this point of mutual indifference from which they will never emerge.

**Diagram 'C'** is the most complex and interesting of all. The hollow point in the middle of this diagram is called a ‘saddle node’, which can be thought of as a kind of directional ‘repeller’. But notice that there are two ‘places’ to which lovers might ultimately be impelled, and they depend on their starting position. If they start in the northeast half above the diagonal (J+R>0), their love will reinforce itself and they will end up with infinite love for each other. If they start in the southeast diagonal half (J+R<0), then they will inevitably end up hating each other to the same infinite extent.

The fates of Romeo and Juliet are an emergent consequence of their natures and not the result of any identifiable linear cause. Under linear causality they would both ‘know’ that someone was to blame and that it was not themselves, so it must have been the other. Viewing the consequences as an emergent result of both their natures leads them to the less accusatory – and more humane – understanding, perhaps smoothing the waters between them.

The ‘Lovers’ model is a type of dynamic model called linear. Nonlinear dynamic models show even more complex behaviours where linear causality is even less appropriate. One example comes from Steve Keen’s nonlinear model of the economy. I won’t go into the maths, or even the principles, of the model here, but the diagram below displays a single trajectory from a Steve’s model (Keen 2009). The behaviour of this model is formally described as ‘aperiodic’. Aperiodic systems are distinctive because, although they are completely deterministic – and despite a superficial cyclic appearance – they never repeat themselves, Aperiodicity is another complex causality that occurs frequently in the real world.

Many emergent behaviours are encompassed by dynamic reasoning and complex causality. Once you are turned on to these patterns, you start seeing them everywhere. It allows a more sophisticated view, and often apparent randomness is clarified. You don’t have to get into the maths of dynamics to appreciate that these seemingly arbitrary (random) behaviours may, and usually do, have relatively simple underlying deterministic causes, and that they can be understood and managed – if not controlled.

________________________________________

Strogatz, S.H. (1988), “Love affairs and differential equations,” Math Magazine 61,35.

Keen, Stephen (2009) “Household Debt: The Final Stage in an Artificially Extended Ponzi Bubble,” The Australian Economic Review, vol. 42, no. 3, pp. 347–57

copyright 2014 Ted Carron

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