My understanding is that they both are simulation of the same dynamics of different kinds or assumptions and errors. You can make electric circuits that share a solution trajectory of the same topology with an adequate mechanical system. The only difference is the value of the coefficients with in the representing differential equation. From a metaphysics point of view, electric circuit is what it is and definitely not a mechanical system. So there could be an argument claiming that the two are different things, phenomenon. Meanwhile, if your looking this whether both "are simulating" the same dynamics, then I would answer yes, it is simulating, simulation doesn't require perfect reality, just a best effort to make it close to reality. Numerical simulation relies on the righteousness of the dynamical equation and the numerical error the arithmetic holds within. This is not reality as well as the electrical circuits imitating mechanical systems. I wouldn't give credit to justify any kind of equality between any of the above given three types without each of them amended with extra adequate assumptions. Sure, electrical circuit absolutely constitutes a physical experiment, but not a simulation of a different kind of system without adequate assumptions.

I'm afraid I cannot see the merit of simulating dynamics with electrical circuits these days. Computers are still costly probably for developing countries, but if one can afford one, than what's the point of using a circuit board where you have very limited flexibility in adjusting the coefficients? You may save time against using simulation codes with very small step time. But I don't buy that merit over the flexibility that Numerical method gives us. If you could explain why such question is arising, it may help me think more in your position.

Thank you.

Yes Miguel, certainly, I subscribe to your opinion. If the components (particularly AO) of an analog computer work in their linear domain of their characteristics, and the passive components have tight tolerances, then the model reproduces the real phenomenon enough accurately. Moreover, the simulation is performed in real time and the numerical errors (inherent in numerical simulations) are totally missing, so that the analog modeling is more accurate than a numerical one. Do not forget: any model suffers of wrong input data because the parameters describing the real system are not known accurately.

Let me give my opinion based on realization theory. Suppose one wants to "realize" a certain dynamical system, possibly nonlinear. A modern way of doing it is to write the corresponding equations (which will be structurally and parametrically uncertain) and to simulate it on a digital computer. Other option, that dates back (at least) to the 1950s is to build a physical system that "realizes" the given equations. The physical system can be mechanical, electrical, electronic etc. Because of the ease of construction and, perhaps, accuracy, realization is best achieved using electrical/electronic components. That is the main reason why analog computers are (or use to be) electrical/electronic. Now, once a circuit OR a mechanical OR any other physical system has been built to realize a set of equations it is clear that such a realization will also have structural and parametric uncertainties. So, my opinion is that any physical system can be considered as a realization of a theoretical set of equations and in each case there are uncertainties. The point is that usually with electronics the uncertainties are less, thats all. Of course, in practice we move in the other direction: we start from a system (electronic or not), we postulate a set of equations for it (in the case of white-box modeling) and then estimated parameters. If the system is electronic, we stand a better chance to postulate equations that are closer to the true dynamics. In this case, again, it is easier to figure out what equations describe an electronic circuit. If the modeling is black-box (where the equations and not only the parameters must be estimated also), then the differences are even more subtle. In short, an electronic circuit is just as physical an experiment as any other, BUT with less uncertainties involved.

Prof. Aguirre answers this question quite nicely. I'd like to add a couple of anecdotes from my experience where this question has come up. Both are cases where I believe electronic circuits were misused as physical experiments.

I recently read a paper where the physical system of interest had both spatial and temporal dynamics. The authors introduced a system of coupled oscillators as a model. Such a model is obviously not completely accurate as it treats space discretely rather than as a continuum. Be that as it may, such a model is often useful insofar as it reproduces the phenomena of interest and is more tractable than a partial differential equation. In this paper I was reading, the authors proposed to demonstrate the validity of their coupled oscillator model by comparing it to an experiment. However, the experimental system they examined was not the original spatial-temporal system of interest, but rather was an analog computer-type implementation of their coupled oscillator model. Therefore, it did not include the "unmodeled" features of the supposed physical system of interest. In fact, it contained other unmodeled features that are specific to electronics and are not relevant to the original system.

Another paper I read involved some type of low dimensional chaotic dynamics. First, the authors presented some straightforward numerical solutions of the differential equations obtained with a desktop computer. The authors then suggested the system of interest was well suited for electronic implementation. Unfortunately, the specific implementation they described was a numerical integration of the system equations on an FPGA. That's just another digital computer.

So I would say that an electronic circuit can constitute a physical experiment, but there are enough cases in the literature where circuits have been badly used to justify some skepticism.

In my opinion, the answer is no. Unless the modelled dynamical system is the electronic circuit itself. The model of a physical system is inevitably an approximation of that system. If you implement its model, you are realizing an approximation of it, does not matter if it is in software or with electronics. On the other hand, if you implement that model by an electronic circuit, as the latter is also physical system, you will might be introducing different types of unknowns, which makes the realization another step away from the original physical system (unless, of course, the physical system is the electronic circuit itself)

Following from the principle of parsimony, just the essential mapping/correspondence between the experimental/simulation system and the target system is needed. As when a holometer setup has been proposed and used as a crucial experiment, to lend support for the principle of cosmological holography. The usual caveat applies: that the margins of uncertainty is sufficiently iterated and noted (e.g. artefacts connected to the experimental setup, not withstanding  the uncertainty principle).

http://theory.fnal.gov/jetp/talks/Lanza_Fermilab_GW_talk_20150327.pdf

http://www.fnal.gov/directorate/program_planning/Nov2009PACPublic/holometer-proposal-2009.pdf

http://www.huffingtonpost.com/2014/08/29/space-hologram-experiment-two-dimensions_n_5726262.html