I understand the concept of non-linear equations. I was recently having a conversation with a colleague and he used the term "highly non-linear" equation.
This got me thinking, how do we classify which non-linear behavior is higher than another? How do we classify or quantify the levels of non-linearity in a physical system?
Answer
Linear equations are always an approximation of real physical processes, so it's safe to say every real system is non-linear to some extent. You still get to call many systems linear, if their behaviour can be described by linear equations with sufficient precision.
Likewise, if you call a system non-linear, it usually means it is substantially non-linear, so that linear equations are not sufficient to describe it. Often, you can still linearise those equation around a certain point to get a local linear solution. Despite being local, these local solution can be quite useful, especially if the "working point" can remain stable. For example, that's how most electronic components are modelled.
Finally, there are systems which have no useful working point (e.g., chaotic systems). If such systems are also non-linear, local linear solutions don't make much sense, as their state will quickly "wander off" to the point such solutions lose all precision. That's what the kind of systems I would call highly non-linear.
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