Fatal Attractor

In physics and economics, Equilibria are a lot like suitors in romantic comedies: having none is bad, exactly one is great, and having more than one makes everything very messy. The concept of equilibrium is widely used, for good reason. Finding equilibrium points in physical space, phase space, or a supply and demand curve is second nature to researchers because it satisfies two of the most basic desires: (1) That the problem be solvable and (2) that solution depend on a few variables as possible. Not overdetermined or underdetermined, just determined. The famous introductory physics example of a ball in a valley between two smooth hills. No matter where the ball starts, eventually the ball will be at the lowest point of the valley. This is the only stable equilibrium – the only point at which  the sum of all forces (here gravity) are not trying to make the ball move. In fact, if you perturb the ball, it will just fall back down to the same spot.

Think about how neat and tidy this is! No matter what the initial conditions or history of the ball, if we just “wait awhile” (for some value of awhile) we KNOW where the ball will be waiting for us. We collapsed all of the messy details of the hill and gravity and Newtonian mechanics by just looking for a point where the forces are balanced (formally, where the first derivative of the potential is zero), and to make sure it is stable – the bottom of a valley (stable equilibrium) and not precariously poised a the top of a peak (unstable equilibrium) – we can check that the second derivative of the potential is positive. The reason this trick of looking for equilibrium works is that having that extra condition – that the forces be balanced – allows us find the one magic point… Assuming there is just one. But what happens if there is not just one?

Imagine you are playing a Tic-Tac-Toe carnival game, which now offers nine possible “final states” for the ball to end in. Assuming it stays in the box at all, which pocket will the ball rest it? The ball can bounce around inside the box, so clearly the final state will depend very exquisitely on the initial conditions (how you tossed the ball), as well as the exact shape of the box. The the ball hits even a tiny knothole in the wood, it can radically alter the trajectory of the ball, and slightly different launch angles and velocities can similarly lead to highly unpredictable behavior. This is why the game is so hard, and why Casinos and use a Roulette wheel as the next best thing to a pseudo-random number generator, even though the flight of the ball and wheel are determined by Newton’s laws.  The Eudaemonic Pie is a book that recounts an attempt to break the bank by calculating where the ball was going to land based to how it was thrown using electronics that could be concealed under clothing. This was met with limited success, due to the complexity of the situation.  Even so, Casinos no longer accept bets once the croupier has released the ball.

In short. situations with multiple equilibriua have multiple solutions that depends on the initial conditions and history. That is to say, they are path dependent. In these diagrams, the graph shows the rate of change of the variable, and the arrows show the direction of change (right for positive, left for negative). The equilibrium points occur when the rate of change is zero, and where the graph crosses the x-axis. For example, if the rate of growth of a population depends linearly on the current population (A), it will increase forever. This is exponential growth. More realistically, if there is some constraint due to limited space or resources, then the rate of growth will be slow for small populations, and small, or even negative, for large populations (B). This is logistic growth. There will be two fixed points, either zero population, or at the “carry capacity.” As evident by the arrows, for any population above zero, the population will eventually reach the carrying capacity. So there are two equilibria, also called “fixed points,” but all nonzero values are in the “basin of attraction” of the carrying capacity.

Most interestingly, is the case when there is a reason that small populations are at a special disadvantage (C). Think about a species that hunts in packs, and if the population is too sparse to constitute a pack, none of them can eat. This is often called the “Allee Effect.” Now there are three fixed points, with one in the middle being a threshold value that if the population falls below, it is doomed to extinction. There are two possible outcomes, with points greater than the threshold value fated to eventually reach the carrying capacity, and populations that dip below attracted to zero. The fate of the population now rests on never going below this value, so it is history dependent.

Phase diagrams in two dimensions can have more elaborate basins of attraction:

Even in derministic systems, without any randomness, the basins of attraction that indicate which equilbirum point is finally reached can be extemely detailed, as the value careens in crazy directions as it is pulled by the different attractors. Here is a fractal created by plotting which solution Newton’s Method finds.

This is somewhat similar to the motion of a chaotic pendulum that swings over stationary magnets:

There is a branch of study called catastrophe theory that deals with situations in which fixed points can be created or destroyed by a small change in a continuous parameter.  The “catastrophes” can occur when a large qualitative change happens in response to a very tiny change. Fold bifurcations are one example, in which an Allee species may be devastated by a downturn in conditions, and will not recover even when conditions go back to what they were.

Figure 2

Catastrophes are a lot like phase transitions. Magnet hysteresis occurs because the number of magnetic domains that will be aligned for a given external field will depend on the initial state of the magnet.

In electronics, data can be stored in the magnetic domains of a hard drive disc or a flip-flop circuit, because of the bi-stability.

Some attractors might be considered “bad.” In materials science, there are sometimes metastable states that are local minima, but not the most stable (global minimum). Sometimes external energy is required to get the system of the trap of a local minimum, as in annealing metal. Similarly in people’s lives, living in a rut is comfortable but sub-optimal, and very hard to get out of.  In the Upside of Down the author writes that failure is sometimes needed to “kick” us out of a local minimum because of status quo/optimism/normalcy/diplomatic/sunk cost biases. She also talks about the dangers of unemployment scarring, in which not having a job makes getting a job harder. That makes joblessness “sticky,” in the sense that it becomes a fixed state that is very hard to get out of.

When I watch the X-Men movies, I sometimes see Prof X and Magneto as representing two different equilibria. Prof X thinks that humans and mutant can live in harmony, while Magneto is convinced that they cannot. As the Days of Future Past movie explores,  two fixed states are possible – harmony, or mutual aggression – and the final destination is very path dependent. A single incident can lead to escalation that ends in total war. From this point, detente is very difficult. On the other hand, if each side believes that the other can be trusted, there can be peace.

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Author: lnemzer

Assistant Professor Nova Southeastern University

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