TOM HOLROYD – 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, the influence of the linear aspect of the equation is much larger numerically than the nonlinear aspect, and as a result in most cases, dominates the behavior of model and real world systems. But not always.
Now, the weight of the last statement easily missed: students are often taught not worry about the nonlinear H.O.T. and focus instead on the linear parts of the equation around fixed points.
But in real life, in the complementary nature, real dynamic structure~functions don’t necessarily ignore or discard the higher order nonlinearities.
This doesn’t mean that modelling physical systems via “local linearization of a nonlinear manifold” is uninformative. It is often informative and representative of modelled system around its fixed points. It is in fact a great method of modelling when focusing upon a system’s fixed points.
But in the complementary nature, systems are not ruled by their ‘fixed points’ alone. Real dynamic structure~functions include real nonlinearities, and these nonlinearities can dominate the action and run the show. There are growing number of scientist~philosophers today who feel that nonlineaty is actually more the rule than the exception.
But in educational and other real world scenarios of this methodology and thinking, linear analysis still pretty much runs the show. upon. So whose right? Is the linear more fundamental in nature than the nonlinear? The straight over the crooked? Or is it the other way around? Are nonlinearities more fundamental in nature than the linear, etc.?
They ‘squiggle’ of course, they are complementary aspects. So one isn’t eternally dominant over the other. The complementary nature of linear and nonlinear is officially communicated by the complementary pair, linear~nonlinear.
As with all complementary pairs, the squiggle sign (~) is used on one hand to symbolize the complementary nature of this pair of aspects, and the other that coordination dynamics is happening. The latter information is functional. It leads to a well established set of principles arbitrarily~specifically applicable to any field and level of observation or wonder. It leads to a well established, highly published scientific paradigm that specifies approaches to speculate upon, observe, test and learn about the system and level of interest.
If acted upon, new information will be discovered which in some cases will replace other information that guides a person’s moment-to-moment actions and that information will dissipate, will vanish away. That’s what “functional information” is like. It influences the ongoing dynamics of systems.
“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 communicates 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 with ‘squiggle sensibility way that in our models of nature, it is a good idea not to ignore the higher order terms, not to assume that the linear always runs the show.
THE EXCITING BUT REAL-WORLD EXAMPLE:
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.”
SQUIGGLE MESSAGE:
“linearity~nonlinearity: Depending on surrounding circumstances, i.e., where the system is located in the dynamic landscape of its ambient 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).”
