To Infinity and Beyond: Some ODE and PDE Case Studies

When mathematical/computational problems “reach” infinity, extending analysisand/or numerical computation beyond it becomes a notorious challenge. We suggestthat, upon suitable singular transformations (that can in principle be computationallydetected on the fly) it becomes possible to “go beyond infinity” to the other side, withthe solution becoming again well behaved and the computations continuing normally.In our lumped, Ordinary Differential Equation (ODE) examples this “infinity crossing”can happen instantaneously; at the spatially distributed, Partial Differential Equation(PDE) level the crossing of infinity may even persist for finite time, necessitating theintroduction of conceptual (and computational) buffer zones in which an appropriatesingular transformation is continuously (locally) detected and performed. Theseobservations (and associated tools) could set the stage for a systematic approach tobypassing infinity (and thus going beyond it) in a broader range of evolutionequations; they also hold the promise of meaningfully and seamlessly performing therelevant computations. Along the path of our analysis, we present a regularizationprocess via complexification and explore its impact on the dynamics; we also discuss aset of compactification transformations and their intuitive implications.