as i understand it, with infinite precision; the real numbers within some range, say -15 volts to +15 volts, have a bijection to infinite strings of bits (some infinitesimally small fraction of which are all zeroes after a finite count). with things like the logistic map you can amplify arbitrarily small differences into totally different system trajectories; usually when we plot bifurcation diagrams from the logistic map we do it in discrete time, but that is not necessary if you have enough continuous state variables (three is obviously sufficient but i think you can do it with two)

given these hypothetical abilities, you can of course simulate a two-counter machine, but a bigger question is whether you can compute anything a turing machine cannot; after all, in a sense you are doing an infinite amount of computation in every finite interval of time, so maybe you could do things like compute whether a turing machine will halt in finite time. so far the results seem to support the contrary hypothesis, that extending computation into continuous time and continuously variable quantities in this way does not actually grant you any additional computational power!

this is all very interesting but obviously not a useful description of analog computation devices that are actually physically realizable by any technology we can now imagine

It would not be very interesting. You will lose all the interesting properties of analog computer and it would be a poorly performing turing machine. Still it would have those loops and branches necessary, since you can always build digital on top of analog with some additional harness.