The Complementary Nature is Linear~Nonlinear

by Tom Holroyd and David A. Engstrøm

In school physics classrooms the world over, students are taught to use equations such as

dx/dt = f(x, p) + H.O.T.

to model nature mathematically.

In such equations,  f(x, p) is some linear function, and H.O.T. stands for Higher Order Terms, or in other words, stand for nonlinear terms of the mathematical model.  Quite often when applied to real world scenarios, analysis of this kind of equation focuses upon stability of solutions near x = 0, or ‘fixed points’. Since x^2, x^3, etc. get smaller as x approaches 0, the higher order terms become mathematically insignificant.

In other words (for the non math people), the influence of  the linear aspect of the equation is much larger numerically than the nonlinear aspect, and as a result in most cases, dominate the behavior of model and real world systems.

But not always…

Now, as a topic of some refresher physics class lecture, the weight of this last statement can and is often easily missed: “Okay,” says the student, “the prof. says not to worry about the nonlinear H.O.T. and focus on analyzing the linear part of the equation around the fixed points, then I won’t… and I will. Do it right and I will pass the exam with flying colors…”

But in real life, in nature, (in the complementary nature), real dynamic structure~functions don’t necessarily ignore or discard the higher order terms, these H.O.T., these nonlinearities.

Now, this doesn’t mean that modelling physical systems via the catchy sounding mathematical technique of  “local linearization of a nonlinear manifold” is useless. Indeed, it can be quite informative and illustrative of the behavior of the modelled system around its fixed points. That is to say, it is a great method of modelling when focusing upon fixed points.

But in real life, in nature (in the complementary nature), systems are not ruled by their ‘fixed points’ alone. Real dynamic structure~functions include real nonlinearities. Moreover, in many real life scenarios, these nonlinearities actually dominate the action and run the show. Here in 2009, there are growing number of scientist~philosophers who feel that nonlineaty is actually more the rule than the exception.

Hmm. But we just told you that in the usual learning scenario, and many, many real world applications of this methodology and thinking, the linear aspect is focused upon. So whose right? Is the linear more fundamental in nature than the nonlinear? Or is it the other way around? Is the nonlinear aspect more fundamental than the linear?

Before we get into the fun illustrative example, let’s remind you where you are reading this post:

That’s right, The Squiggle Sense blog…and

“The squiggle sense is a human sixth sense of the complementary nature.” The complementary nature is revealed to the squiggle sense as complementary pairs, like linear~nonlinear, where the squiggle symbol communcates the message that linear and nonlinear are inextricable, dynamic complementary aspects. You can’t have one without the other. One can dominate a given context, but neither aspect is most fundamental.

This being our modus operandi, this being the expression of our squiggle sense, we can say in a squiggle sensible way that in our models of nature, maybe it would be a good idea not to ignore the H.O.T.s, the higher order terms, not to assume that the linear always runs the show.

“Okay fine. So give us an exciting but real-world example. Make it very dramatic, so that we will remember the moral, the take home message better…?”

THE EXCITING BUT REAL-WORLD EXAMPLE:

ROGUE WAVES

This story is about waves. Ocean waves. Really, really BIG ocean waves…

For years, oceanographers and computer graphics experts have been modeling the surface of the ocean, with hope to understand its behavior better. This seems a worthwhile undertaking, considering 3/4 of our little planet is covered in ocean.  Historically speaking, such efforts  began with the simplest possible model, that is, assuming the surface behavior of the ocean can be approximated as the sum of lots of sine waves with random phases, frequencies, and amplitudes.  Such models are ‘nice and linear,’ no H.O.T.s to mess with, and lo-and-behold, a fair amount of actual ocean wave behavior is nicely mimicked, and we can all ooh and aah about how mathematics describes nature.

For years, mariners have told stories of giant waves that swallow boats. More often than not such reports were doubted.  One  reason for the doubt was that eyewitnesses are scarce, because they usually didn’t survive.

These days, wave data from oil rigs and satellites have shown that the Earth’s ocean surface is more interesting than was previously believed. Wave heights of 30 meters have been observed, and cannot be explained by the linear wave models of the ocean surface. So what gives?

It is known by most that ocean waves are caused by wind. What many don’t appreciate is that  ocean waves are also caused by ocean currents, and the winds on the other side of the ocean surface that we don’t see and tend to ignore.  Further, changes in temperature, salinity, geology, tides, the Earth’s rotation, and many other factors all contribute to underwater weather. (You have probably seen dust devils. Little spirals of air that swirl the leaves around. Small cyclones. These things happen underwater, too. And energy can be transferred from one wave to the next (have you ever looked at convection patterns?) These numerous factors all contribute to the H.O.T.s of the system, its nonlinearities, which can also have a drastic effect upon ocean waves (and upon the mathematical solutions of  model equations) .

For example, ocean waves can grow unusually large, for example, when underwater currents flow in opposition to the wind.

THE TAKE HOME MESSAGE:

“Nature is not constrained by the linearity of our models. Nonlinearity exists, and can be significant.”

THE TAKE HOME SQUIGGLE MESSAGE:

“linearity~nonlinearity: Depending on surrounding circumstances, i.e., where the system is located in the space of its parameters, behavioral change may be smooth and linear or abrupt and nonlinear. Nonlinearity is a requirement for multistability and its biological manifestation, multifunctionality. (TCN p. 222).”