Numerical instability and dynamical systems

European Journal for Philosophy of Science 11 (2):1-21 (2021)
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Abstract

In philosophical studies regarding mathematical models of dynamical systems, instability due to sensitive dependence on initial conditions, on the one side, and instability due to sensitive dependence on model structure, on the other, have by now been extensively discussed. Yet there is a third kind of instability, which by contrast has thus far been rather overlooked, that is also a challenge for model predictions about dynamical systems. This is the numerical instability due to the employment of numerical methods involving a discretization process, where discretization is required to solve the differential equations of dynamical systems on a computer. We argue that the criteria for numerical stability, as usually provided by numerical analysis textbooks, are insufficient, and, after mentioning the promising development of backward analysis, we discuss to what extent, in practice, numerical instability can be controlled or avoided.

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Julie Jebeile
University of Bern
Vincent Ardourel
Centre National de la Recherche Scientifique

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Philosophy and Climate Science.Eric Winsberg - 2018 - Cambridge: Cambridge University Press.
Explaining Chaos.Peter Smith - 1998 - Cambridge University Press.
What Are the New Implications of Chaos for Unpredictability?Charlotte Werndl - 2009 - British Journal for the Philosophy of Science 60 (1):195-220.

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