I like to know your idea about is there any third system after linear, none linear classification.

Could we consider more universe dimension than 4 dimensions (time, XYZ) like 11 dimensions of  string theories?

Dose this class is closed and could not extend (logical rules in this definition)?

If yes, how could extend logic? what is the first option for extending the logic for this purpose?

Is stochastic math equation is as one candidate?

Thabks for your attention.

  • $\begingroup$ Any more detail needed? $\endgroup$ – Laurent Duval May 23 at 15:36

Mathematically, linearity is a binary property: a system is either linear or not.

Realistically (and with some careful definition of terms, mathematically) a nonlinear system can have degrees of nonlinearity. In all cases the system is still nonlinear, but in some cases it can be treated as linear. Here is an incomplete listing of terms that I have seen or used:

  • Operated in the linear region. This means that if you're careful about the inputs the system receives its behavior is linear, and when you do your analysis you can treat it as such. There may be mechanical stops or numerical issues that are nonlinear behavior, but as long as the system states never exceed certain boundaries, these nonlinearities are never active.
  • Mildly nonlinear. This means that for the purposes of design and analysis, the deviation of the actual system from linear behavior is small enough that it doesn't matter. This is such a common case in circuit design and control systems design that we often don't think of the system as "nonlinear" per se.
  • Quantization noise. (If someone has a suggestion for a name for the general case, I'm listening.) Quantization is a nonlinear phenomenon, but we can deal with it by treating it as random noise (or, in some cases, bounded but antagonistic noise) that is added in "magically" and design our systems around that treatment.
  • Strongly nonlinear. Some systems just have nonlinearities that cannot be ignored. A relay is an example of this -- it's output is either on or off; if you try to linearize it you find that it has zero gain outside of the switching point, and essentially infinite gain at the switching point. There are other examples, of various degrees of tractability.
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  • $\begingroup$ Thanks Tim, that was intresting for me. $\endgroup$ – modern Dec 24 '19 at 6:39

Let me remind that properties (like linearity, time invariance) are not made for fun: they both model useful processes or systems AND they can be used to derive interesting properties, algorithms, etc.

From linearity combined with time-invariance, you get... convolution, Fourier, etc. When systems are more complicated, people have however tried to model them, and build tools from these models. Of course, you can go into higher dimensions, and find systems that are not fully linear, but are linear in some variables, and not in some others. So you can find bilinear (some time-frequency transforms, e.g. Wigner-Ville) or trilinear systems (some tensor models). I have not used those beyond the fourth dimension. Some systems, using complex values, are more subtle, like sesquilinear systems, linear in one variable, conjugate linear in the other:

$$\mathcal{S}(ax,by) = a\overline{b}\mathcal{S}(x,y)$$

As you mentioned logic, one can also play on the scalar number system. Rings, quaternions, etc. can provide exotic tastes.

As said by @TimWescott, some systems are approximately linear in some regions. So beyond linear, there are affine systems, polynomial ones (related to Volterra filters), systems that are linear under a monotonic transform (like homomorphic filters, that are linear under a logarithm).

Beyond those more of less natural linear extensions, there are morphological filters, order statistics and stack filters, etc.

The above gradations provide interesting alternative to "fully nonlinear" systems, that can be very wild and maybe not so practical to use directly.

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