To infinity and some glimpses of beyond

Panayotis G. Kevrekidis, Constantinos I. Siettos and Yannis G. Kevrekidis ()
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Panayotis G. Kevrekidis: University of Massachusetts Amherst
Constantinos I. Siettos: National Technical University of Athens
Yannis G. Kevrekidis: Princeton University

Nature Communications, 2017, vol. 8, issue 1, 1-13

Abstract: Abstract When mathematical and computational dynamic models reach infinity in finite time, extending analysis and numerics beyond it becomes a notorious challenge. We suggest how, upon suitable transformations, it may become possible to go beyond infinity with the solution becoming again well behaved and the computations continuing normally. In our Ordinary Differential Equation examples the crossing of infinity occurs instantaneously. For Partial Differential Equations, the crossing of infinity may persist for finite time, necessitating the introduction of buffer zones, within which an appropriate transformation is adaptively identified. Along the path of our analysis, we present a regularization process via complexification and explore its impact on the dynamics; we also discuss a set of compactification transformations and their intuitive implications. This methodology could be useful toward a systematic approach to bypassing infinity and thus going beyond it in a broader range of evolution equation models.

Date: 2017
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DOI: 10.1038/s41467-017-01502-7

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