What definition can be given to potential and actual bifurcation points without resorting to post-arithmetic concepts of mathematics?
What is their meaning and significance?
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Greatly simplifying, a bifurcation is when, with a small and smooth change in the values of the parameters of a dynamic system, a “qualitative” change in the behavior of the system suddenly occurs. Bifurcation point – values of parameters at which this change occurs. For example, if you heat a layer of oil in a frying pan, then when a certain value of the temperature difference between the bottom and top of the liquid is reached, a new mechanism of thermal conductivity arises – the liquid molecules begin to form special structures (the so-called Benard cells/Rayleigh filters), which transfer heat to the surface more efficiently. If the bifurcation has already occurred, then the point is actual, if it is probable, then it is potential.
Colloquially, the expression “bifurcation” is sometimes used as, for example, a synonym for the moment from which events can develop according to one of two scenarios.
If I understand your question correctly, it's something like this. Of course, a mathematician will explain much better.