In engineering, great use is made of linear mathematical models. If you read popular books on chaos theory or new-age physics, you learn that linear models are oversimplified and bad, and are equated with uncreative straight-line thinking. I'd like to both touch on the element of validity in the criticism and also explain why linear models are valuable and often the right choice.

A mathematical model is a representation of how something is going to behave, most often how things are going to change over time. Mathematical models are always abstractions that ignore some details. A mechanical engineer can model the strength of steel bar using only a few numbers read from a handbook. Metallurgists who create new metal alloys need a more fine-grained understanding: what aspects of the crystalline structure create aspects of stiffness or susceptiblity to cracking. They have their own mathematical models. Various kinds of physicists work on even smaller-scale understandings of matter with their own models.

In some sense, the finest scale models such as quantum physics are better models because they can accurately predict the behavior of matter across a wider range of conditions. However, there are several good reasons why civil engineers don't design bridges with quantum physics:

- Much simpler models get the job done better. Simpler models are easier to use, and perhaps more important, can lead to an intuitive understanding of interactions in the design.
- The concerns at the different levels are so different that quantum physics doesn't have much to say about the best shape of metal beams.
- A simple model will have a limited range of situations in which it is acceptably accurate, but we may know (or be able to show) that simplifying assumptions are valid. When the steel bar melts, its mechanical properties change greatly. However, the engineer is not concerned with how the beam will behave once it melts, because at that point it is useless. More relevant is the the decision of whether to add insulation to prevent the beam from melting in a fire.

First of all, a linear model does not say that everything is a straight
line. *Linear* refers to the form of the differential equations, not the
predicted behavior of the system. Linear models can exhibit quite complex
behavior when they have multiple inputs and outputs, and even the simplest
models predict wave-like oscillating motion and
exponential increasing or fading
away, and combinations of these.

What *linear* does mean is that if the system is stable, then the
output is proportional to the input. If you increase the input 50%, the
output will increase 50%. If you ask somone a harmless question, and they
scream at you, then it is a reasonable metaphor to say they "went nonlinear"
because your input created a disproportionate output.

A linear system may also be unstable, which means that even in the absence of input, the output rapidly explodes toward infinity. When someone blows their top, you could as well attribute this to unstable (but linear) behavior. In real unstable systems, some nonlinearity will intervene preventing the output from growing forever. However, if your plan is to keep the output from blowing up by carefully controlling the input, then you can use a linear model and ignore the nonlinearity that would have resulted without your intervention.

Then computers cast some light in the nonlinear bushes. With computers, we
could look at the behavior of nonlinear systems, and found that even simple
nonlinear systems showed interestingly complex behavior. Systems with no
randomness built into them produce seemingly random fluctuating outputs.
This unpredictable behavior of nonlinear systems was called *chaos*, and
people worked on trying to understand the behavior of these systems.

From this work, we now understand more about how nonlinear systems (like the
earth's weather) can exhibit unpredictable behavior. We have also learned more
about the role of nonlinearity in living and *self-organizing* systems.
It turns out that a wide range of stable systems from cells in the body to
entire societies operate on the edge of chaos. There's enough unpredictable
behavior to keep things from getting stuck in a rut, but on average things are
pretty linear.

Although in many areas of science linear models do a good job of predicting the things that are predictable, there is a special reason for the relevance in engineering. Engineers don't want to make things that are unpredictable. If we can constrain the problem definition so that everything is linear, then we are happy. If something goes nonlinear, that is a problem to be fixed by keeping well away from there.

*Last update 10 February 2004*

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