Integrating "Beyond Infinity" and Back

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Consider the simple ordinary differential equation with initial condition . The solution is obviously given by . It tends to infinity as tends to 1. We now consider a time-discrete approximation to this solution as provided by the asynchronous leapfrog method. Since the evolution step formula does not involve potentially undefined operations, any such discrete approximation is well defined for all its values. Of course, the approximated values grow dramatically and will soon transcend what can be represented even with Mathematica's arbitrary-precision numbers. Since the asynchronous leapfrog method is a reversible integration method, we should be able to go along each finite discrete trajectory back to its initial point. If the final point was "close to infinity", the computation needs to be done with a large number of digits in order to come back to its initial point. This is what the present Demonstration studies. Values are input by means of setter bars instead of sliders since updating the curve for higher precision and