Are all systems really non-linear?

A few years ago I had a course on control and systems engineering. In one of the first chapters of the book we used, it was stated that "all physical systems are nonlinear". When discussing this in class, the professor said that he wasn't really convinced that this statement was true. If I remember it correctly, he made a case along the lines of "maybe all systems are linear, but we don't want to deal with or understand the additional dynamics".

It is a while ago, and my memory of his reasoning has faded a bit, but I've had a few more control courses since then and the idea still intrigues me. Especially since the "all physical systems are nonlinear"-statement is repeated in every book on systems theory I've come across, I would be interested in any insight into the validity of this statement.

• It is trivially true that all systems are nonlinear for a large enough range of operating conditions. For example temperatures can not go below absolute zero, and relativistic effects will occur at high velocities. More practically, any electrical device has practical limits on its operating currents and voltages, and many materials will change phase between solid/liquid/gas. Linear systems are a useful approximation to reality, but it is important to understand when the approximation breaks down in any real-world system. Dec 26 '20 at 13:35
• @alephzero That's an answer, and a good one - post it! Dec 26 '20 at 13:58

All quantum mechanical systems are linear, and all systems are quantum mechanical. So, all systems are linear, and nonlinearity appears only in the description of a system: in an inexact, approximate description.

• "all systems are quantum mechanical" ­— this is a bold claim. Quantum mechanics is also an approximation, and who knows what nonlinearities are there beyond it in the real nature. Dec 26 '20 at 22:30
• This answer is too simple because linearity depends on the variables being considered. QM is linear with respect to superpositions of the state vector, but necessarily with respect to other things. Dec 27 '20 at 17:33
• That's a good point: any linear system can appear nonlinear when described in terms of an unfortunate choice of variables. Dec 27 '20 at 18:22

"all physical systems are nonlinear"

I think this should not be understood as a hard rule, but as an aphorism that outlines one of the very important differences between theoretical physics and engineering: engineering is supposed to deliver solutions that work in practice, which means not fail catastrophically, which means there is a list of important gotchas, and this is one that you should really remember.

In the context of control engineering, "all systems are nonlinear" makes perfect sense as a rule of thumb, because if you think your system is linear, then your design will fail.

For example, none of your state variables like current, voltage, speed, etc will go to infinity because at some point something will explode or catch fire. What this means is that a system will tend to be linear for small variations of state variables, it will probably follow predictable equations when state variables are constrained inside a certain domain, and have abrupt and sometimes unpredictable transitions between different sets of control laws when certain boundaries are crossed.

For example, above a certain threshold of force, the tires on a vehicle will skid, and that changes the control equations completely. Likewise if some of the wheels lift off the road because you're turning too hard. In these two cases, what changes is not just the control equations, but also the goal and priorities the control system should aim for.

In other words, while it may be true that all quantum mechanical systems are linear, it most likely won't be the first thing to be concerned about if your aircraft stalls and falls towards the ground like a brick.

One of the fundamental differences between theoretical physics and engineering is class action lawsuits... which also are a nonlinear phenomenon, btw.

Note that nonlinearities don't just happen at large state space variable excursions, they can also manifest as small-signal thresholds. For example a motor won't turn at all if there is not enough current to create enough force to overcome friction in the bearing and gears.

So when they say "all physical systems are nonlinear" that's just a colorful way of saying you should always know the limits of your model, you should know when your approximations hold and when they don't, to make sure your design handles transitions between different modes of behavior gracefully and not catastrophically.

Most of the differential equations of motion for engineering systems have exact solutions when the equations are constituitively linear and have constant coefficients. In the real world this corresponds to systems without exotic components and for small deflections only, and that's the price you pay for being able to calculate values of voltages, pressures, forces, currents, flow rates and deflections to guide your design process: your design is an approximation which is accurate under a limited set of circumstances, which makes it good enough for most engineering work where you add safety margins that cover the cases where the approximation is poor.

When you have to model the behavior of systems with deflections large enough to invalidate things like small-angle approximations and where for example the compliances are not at all linear (springs made from air and rubber are the classic examples), then closed-form solutions either don't exist or are horribly complicated, and solving those equations of motion require numerical integration on a computer- still doable, and adequately accurate again within certain circumstances, and again sandbagged in practice with safety margins.

What all this means today is that nonlinearities are routinely managed even in very complex systems by throwing a great big cheap powerful computer at them, something that did not exist 45 years ago when I was taking control systems courses in grad school, and letting the machine do all the hard work while you sit back and sip a cup of coffee.

Even in the case where the problems in the old days were so hard that you had to solve them by constructing scale models of airplanes and blowing air at them in a wind tunnel, today the wind consists of ones and zeroes which are digitally (and fully nonlinearly) blown at a digital representation of an airplane, and the answers thus obtained are good to three decimal places of accuracy.

Are all systems really non-linear?

To give the most lazy answer: There is at least one non-linear system. All systems are inevitably coupled, as we cannot perfectly isolate them. Hence all systems are non-linear.

To be a bit more practical and closer to engineering applications: Linearity certainly brakes down when you subject a system to sufficiently high forces (or currents, etc.) to destroy the system.

When discussing this in class, the professor said that he wasn't really convinced that this statement was true. If I remember it correctly, he made a case along the lines of "maybe all systems are linear, but we don't want to deal with or understand the additional dynamics".

This doesn’t sound like your professor was making anything close to a sound argument, but rather saying something like: “There are purely linear systems, because I want them to be.” Well, reality doesn’t care very much about the wishes of your professor.

The answer is not straightforward, it is either linear or it is a combination of both linear and non-linear. Which one is true is still undetermined. It comes down to whether there are a finite number of physical laws that can describe and predict an infinite possible number of phenomenon with total accuracy and precision when combined. Without knowing all the laws, linear systems may seem to be non-linear. If there are a finite number of laws, and we know all of them, then all systems will in fact be linear. If there are an infinite number of physical laws then all systems will only be linear to a certain extent within certain domains and conditions, but will also have an element of non-linearity. In that case, the more laws we know, the more linear systems will seem to be, but there will always be an element of non-linearity, which could seem to decrease but it will never be 0

• a combination of both linear and non-linear – Well, one non-linearity makes something non-linear. — If there are a finite number of laws, and we know all of them, then all systems will in fact be linear. – Why? Do you assume all laws to be linear? Last time I checked, there were plenty of non-linear physical laws. Dec 28 '20 at 9:32