Nonlinear analysis

The ubiquity of near-linear systems and the tools we have for analyses thereof can sometimes give the impression that nonlinear systems are exotic or even downright flamboyant. However, a great many systems¹ important for a mechanical engineer are frequently hopelessly nonlinear. Here are a some examples of such systems.

• A robot arm.

• Viscous fluid flow (usually modelled by the navier-stokes equations).

• Nonequilibrium thermodynamics.

• Anything that “fills up” or “saturates.”

• Nonlinear optics.

• Einstein’s field equations (gravitation in general relativity).

• Heat radiation and nonlinear heat conduction.

• Fracture mechanics.

• The 3-body problem.

Lest we think this is merely an inconvenience, we should keep in mind that it is actually the nonlinearity that makes many phenomena useful. For instance, the laser depends on the nonlinearity of its optics. Similarly, transistors and the digital circuits made thereby (including the microprocessor) wouldn’t function if their physics were linear.

In this chapter, we will see some ways to formulate, characterize, and simulate nonlinear systems. Purely analytic techniques are few for nonlinear systems. Most are beyond the scope of this text, but we describe a few, mostly in (lec:nonlinear-system-characteristics?). Simulation via numerical integration of nonlinear dynamical equations is the most accessible technique, so it is introduced.

We skip a discussion of linearization; of course, if this is an option, it is preferable. Instead, we focus on the nonlinearizable.

For a good introduction to nonlinear dynamics, see Strogatz and Dichter (2016). A more engineer-oriented introduction is Kolk and Lerman (1993).