# Non-linear properties in linear systems

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Common sense would suggest that all observed properties in a linear system would likely be linear. The response of a pendulum to an impulse can be computed, and for multiple impulses, the individual solutions can be superimposed to obtain the resultant solution. Qualitatively however, non-linear behavior can be readily observed. e.g. two impulses with the proper timing can (instead of creating twice the amplitude) bring the system to a stop. Non-linear properties can also be observed in optical interference patterns even though Maxwell's equations are completely linear. The energy patterns of individual sources do not superimpose. Thereby, non-linear features are also readily observed in optical systems that are governed by linear equations. This puzzle seems to have a simple but not-so-obvious solution that "not all of the equations governing the properties of these systems are linear." In particular, the energy equations are in general quadratic in the linear parameters so that system properties involving the energy are not required to exhibit linear behavior. In the case of the pendulum with impulses 180 degrees out of phase, the sinusoids cancel in a linear fashion, but the system's energy response is non-linear. In the case of the optical interference pattern, we actually have energy and photon conservation even though the energy patterns of two sources do not superimpose. Instead, the electric fields of the two sources superimpose in a perfectly linear fashion. Am posting this in case anyone else may have been stumped by this puzzle. Perhaps the solution is obvious to many, but some viewers may find it of interest.

Qualitatively however, non-linear behavior can be readily observed. e.g. two impulses with the proper timing can (instead of creating twice the amplitude) bring the system to a stop.
This is not an example of nonlinear behavior.

Non-linear properties can also be observed in optical interference patterns even though Maxwell's equations are completely linear.
This is only true if you are dealing with non linear media. For linear media superposition holds.

Thereby, non-linear features are also readily observed in optical systems that are governed by linear equations.
No, in non linear media the equations are non linear.

This puzzle seems to have a simple but not-so-obvious solution that "not all of the equations governing the properties of these systems are linear." In particular, the energy equations are in general quadratic in the linear parameters so that system properties involving the energy are not required to exhibit linear behavior.
Yes. You can always take any linear equation and write a non linear equation using the same terms.

Your conclusion is correct (you can write nonlinear equations for a linear system), but your examples are not correct (the examples you mentioned are linear).

This is not an example of nonlinear behavior.

This is only true if you are dealing with non linear media. For linear media superposition holds.

No, in non linear media the equations are non linear.

Yes. You can always take any linear equation and write a non linear equation using the same terms.

Your conclusion is correct (you can write nonlinear equations for a linear system), but your examples are not correct (the examples you mentioned are linear).
Yes, the systems are linear, but the non-linear behavior is in the behavior of the energy properties. For example, the spatial energy patterns of each separate source do not superimpose in a linear fashion to form the spatial energy pattern for two mutually coherent sources, even though the system (with Maxwell's equations) is completely linear in the electric and magnetic field parameters and is thereby designated as "totally linear". The energy equations of the electromagnetic system, which also govern the behavior of the system, are U=E^2/(8*pi)+B^2/(8*pi), and are non-linear in the E and B fields. Thereby some features of a non-linear variety (e.g. interference patterns where the energy patterns don't superimpose) can be expected to be observed in electromagnetic systems.

Yes, the systems are linear, but the non-linear behavior is in the behavior of the energy properties.
Sure, energy is not linear, but these specific points you make are not about energy and are not examples of nonlinearity:
two impulses with the proper timing can (instead of creating twice the amplitude) bring the system to a stop.
. Non-linear properties can also be observed in optical interference patterns
As I said above, your conclusion is correct but some of your supporting examples were not. I am not refuting your conclusion but just telling you that some of your examples don't support it. Your argument will be stronger if you stick to examples that support the conclusion.

Sure, energy is not linear, but these specific points you make are not about energy and are not examples of nonlinearity:
As I said above, your conclusion is correct but some of your supporting examples were not. I am not refuting your conclusion but just telling you that some of your examples don't support it. Your argument will be stronger if you stick to examples that support the conclusion.
With the pendulum, (or a mass on a spring), two impulses (energy) "input" can result in zero (energy) "output". That again indicates non-linear in the sense that the output isn't proportional to the input. In some ways, it's very much a "triviality". A more mathematical example is with Parseval's theorem, which is second order in the parameters and holds for any completely linear system for which Fourier transforms apply. The fact that second order equations appear in the fundamental equations governing the system would indicate that system properties of a non-linear nature could be observed in these linear systems. The discussion probably has limited appeal for you. It's somewhat trivial in a way, but I find it of some interest.

A follow-on: I can easily give a couple other examples: An electronic feedback amplifier with a phase delay where the timing of the input pulses determines how much appears at the output. Another example is a transmission line with a segment that has a change in impedance. The timing of input r-f signals will determine how much gets through (in an energy sense) because of multiple reflections (analogous to the Fabry-Perot effect in optics). On this one, it is not unlikely that the reader will read the post and discount it, but I am sticking with it as worthwhile physics. The topic may seem trivial, but I pondered it off and on considerably with no satisfactory answer until I realized that there was a non-linear (in the linear parameters) energy principle and/or equation at work in each of these systems.