Chaotic Modeling

Mathematical models are great things, but they have their limitations.  Take medicine, for example.

Now  I love medicine.  But one of the odder aspects of the medical profession is that, despite existing in one form or another for thousands of years, it has only recently been trying to formulate its own job in positive terms.  Ask yourself.  What exactly do doctors, as professionals, want?  What is their goal?

Health.

Wonderful!  And what is health?

The absence of disease.

I’m making it into a joke, but the surprising truth is that medicine has never had a scientific definition for this most desirable state.  Lately, though, a part of the medical profession has been trying to establish criteria for exactly what the word “health” means.  That, in turn, has led them to re-examine what “normal” means.  And this, in turn, has led them to ask for help from the dread world of mathematics and something called System Dynamics.

Their problem, you see, is that we are really messy.  It would be nice if every healthy individual were the same.  Then we could just say, “You are healthy if your temperature is 98.6, your heartbeat is 80 beats per minute, you blood pressure is 120/80,…”

It would be nice and it would also be easy to model.

Take temperature.  What is a “normal” human temperature?  Most of us would answer, I suspect, that it is 98.6 (carefully saying 98 point 6).  That is what we were taught.

In fact, the “normal” temperature varies from individual to individual over a several degree range.  Not only that, but for any individual it not only varies with age (higher when young to lower with old), it even varies substantially throughout the day.

And that is the way it is with every system and indicator in the body: Each of them will vary around some idiosyncratic “normal” value.  Pick any one you want — heart beat, blood pressure, glucose level, blood oxygen — each of them finds its own special average value, its own special exclusion range, and, most importantly, its own special response pattern.

Back to body temperature.  As the ambient temperature goes up and down, as we eat and exercise, energy is drawn from the hot gut to heat the blood and then radiated through the skin with perspiration to help cool it back down again.  It is a highly dynamic and complex system.

Now to the mathematics.  If we want to model body temperature as a dynamic system, then the simplest math model would be what is known as a mono-stable function.  That is the type used to model the thermostat on your wall.  It has one set value and it likes it.  Imagine a marble in a bowl.  At rest, the ball is sitting at the bottom of the bowl (98.6 degrees).  Perturb it to the right (say, with a lower ambient temperature).  The further it goes from center, the steeper an incline it faces and the stronger the corrective force to drive it back down to the bottom of the bowl.  Perturb it to the left (say with a higher ambient temperature) and the same thing happens, only with the opposite correction.

In any dynamic system a condition towards which the system tends to evolve or towards which it returns if perturbed is called an attractor.  Health, then, is modeled as that condition in which all our systems are successfully tending towards their own attractors and their attractors are good ones.

I added that last bit because, just to complicate our modeling, our bodies sometimes exhibit a fondness for what one might call “bad attractors.”

Take, for instance, blood pressure.  Ignoring the individual variations, we say a healthy adult should have blood pressure readings of 120/80 as an ideal.  As we go through our day, sitting and rising, running and walking, eating and sleeping, our blood pressure should, while taking excursions from those values, return to them as the “attractors” of the systems that regulate blood pressure.

However, for those with high blood pressure, something different occurs.  Some true physiological shift happens and the body suddenly finds 150/95, say, a more attractive target.  For those people, as they go through their day, sitting and rising, running and walking, eating and sleeping, their blood pressure takes excursions but stubbornly returns to 150/95.  They can take medications or herbs or what have you to move the target back to 120/80, but as soon as they remove these external elements, their bodies return to 150/95.

The only way to truly fix the problem is to try to reverse the initial shift and return the attractors to 120/80.  Leaving the attractors high and then artificially forcing the results low is to work against the body’s dynamic systems.  It beats dying of a stroke, but it is a dumb way to mimic “health.”  Unfortunately, in most cases we don’t really know what caused the initial shift so we don’t know how to fix it.

That is one kind of problem.  Another occurs when human beings exhibit what in system dynamics would be called bi-stability.  That is, instead of one possible attractor, there are two.  To visualize this, imagine a deep valley with a hill in the middle of it, creating two shallower vales on either side.

Now, if you put a marble in one of the vales, it will be happy there.  Perturb it up the hill and it will want to roll back down into its vale.  Push it up the valley wall and it will also tend to roll back down into its vale.  But, if you give it a really hard push, it can roll all the way up the hill and down into the second vale.

Where it is equally happy and equally stable.

Now imagine that you are happily living in that wonderful condition we call “health.”  Then, one day, your insides start acting up.  You can’t seem to digest your food properly.  Everything you eat goes right on through with dizzying (and painful) speed.

You go to your doctor.  He diagnoses you with Irritable Bowel Syndrome (IBS).  He tries everything he knows to no avail.  He sends you to specialists.  They try everything up to and including steroids.  Some seem to help for a while (often with horrendous side effects), but eventually you always return to that miserable condition they casually call IBS.  It is as if your body were fighting the cure.  As if it had found a new attractor and was using all of its resources to maintain it.

For some of those who come down with some form of Irritable Bowel Syndrome, there comes a magic moment when the body, for yet another unknown reason, switches back.  Food begins to be properly digested and that magical condition we call “health” returns.  But it can be fragile.  Another unfathomable switch and, Wham, they are back in misery again.

In our model, we say they have become bi-stable.

All of which brings me to weather forecasting.

Weather systems are dynamic, too.  They are so dynamic that they are modeled by a special family of mathematics we call Chaotic.  Chaotic math has a couple of very interesting properties.

First, they have attractors, too, but they are called strange attractors.  They are strange because the systems don’t tend to drive towards them and then land there until something drives them off (like our marble).  Instead, they continually (and apparently randomly) oscillate around these values without ever displaying any particular attraction for the values themselves.

An example might be temperature.  We might know that the average temperature in a given area in summer might be 75.  Year after tedious year, the average comes out to almost exactly 75.  Yet for any given day and time, the odds of the temperature actually being 75 are pretty slim.  Instead, it varies all day and from day to day following no particular pattern yet producing a consistent average.

That is the way a strange attractor works.  If you look closely, monitor from moment to moment, there is no obvious pattern.  All seems to be random.  Step back a bit and you discover some order imposed on the chaos.

The other interesting property of chaotic math is called The Butterfly Effect.

Let’s suppose you want the temperature in your neighborhood to change.  So you go outside and wave your arms, hoping to start a breeze.

Common sense tells you that this is hopeless.  Whatever microscopic change you might create will obviously be damped out by the sheer mass of the air, not to mention all the currents and gradients it has.

But Chaotic Mathematics has a charming quirk.  Every once in a while it can display what is known as An Extreme Sensitivity To Initial Conditions.

What this means is that sometimes some minor initial input is effectively amplified by the system.  At least in theory, a butterfly flapping its wings at precisely the right moment might cause a hurricane tomorrow in Tokyo.

This may seem especially bizarre, but real weather happens to share this oddity, which explains why all our wonderful weather models and supercomputers still can’t accurately predict tomorrow’s weather.  Real weather is constantly being grossly affected by small (and often unseen) inputs that make jokes out of our models’ tidy forecasts.

But this mathematics has another funny quirk.  It is multi-stable.

It works like this.  Imagine you have some giant system, like the weather, with lots of energy sources and sinks and lots of perturbations.  In detail, it is a mess, but in broad it displays patterns of considerable long term stability.  We call this our Climate.

Imagine, now, that you change some long term variable like the amount of radiation the atmosphere absorbs.  The mean temperature goes up a fraction.   Assuming that is not one of those abnormally sensitive initial conditions, the system jogs itself around a bit and finds a slightly varied pattern of stability.  It is still the same basic climate, but shifted a bit.

You change the absorption some more.  The climate shifts some more.  Parts that were arid are now getting some rainfall.  Fertile plains are starting to dry up.  As the process continues, the system accommodates a greater and greater change by shifting into new patterns of stability on the same basic configuration.

We call this global warning and our models forecast big enough changes ahead to have us worried about our future.

But chaotic systems are not that linear.  If you take a chaotic system and push it far enough, eventually it does something very interesting.

First it does exactly what our global warming models do: it shifts its patterns a bit to accommodate the changed inputs.  Push it more and it shifts more.  Keep on pushing and it will keep on shifting until a magic moment occurs when it cannot simply adjust its pattern anymore.

All of a sudden there is a complete collapse into what looks like totally random turmoil.  But if you push it a bit more it will suddenly stabilize into a completely new pattern of stability.  It has gained a completely new set of attractors.  It bears precious little resemblance to the old system, but it demonstrates considerable stability in its own right.  Minor changes give minor results and the overall condition is stable.

If you now push this new stable condition too far you might end up in still a third stable condition.  And so on.

Why is that important?  Because we have no idea how to model those events.  All of our global warming models just keep on shifting their basic patterns without ever becoming unstable.  But there are a number of climatologists who believe that the earth’s climate, if pushed far enough, can abruptly shift states with no warning.  In fact, they think it has happened before, fairly recently.

We call it the Ice Age.

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