Mathematics, Science, and Dynamical Systems: An Introduction
In this short introduction, the section “Mathematics, Science, and Dynamical Systems” of the Handbook of the Mathematics of the Arts and Sciences is summarized.
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Abstract In this short introduction, the section “Mathematics, Science, and Dynamical Systems” of the Handbook of the Mathematics of the Arts and Sciences is summarized.
Keywords Dynamical systems · Differential equations · Difference equations
Mathematics is usually considered to be an abstract and general science for problemsolving and method development. It contains development, use, and analysis of methods. In some cases, the context in which an original problem or a class of problems was solved presents opportunities for axiomatization, and a mathematical theory can develop out of it (e.g., theory of differential and partial differential equations, etc.). In other words, the abstraction level is high enough to allow an analysis of the problems outside their original context (pure mathematics). This approach allows for focusing on the problem structure, isolation of key dependencies, and analysis of the generality of the methods. In other cases, the coupling to the original context is explicit and requires a solid understanding of both mathematical methods and the particular application in question (applied mathematics). In rare instances, theoretical mathematics like Minkowskian space-
T. Lindström () Department of Mathematics, Linnaeus University, Växjö, Sweden e-mail: [email protected] B. Sriraman () Department of Mathematical Sciences, University of Montana, Missoula, MT, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences, https://doi.org/10.1007/978-3-319-70658-0_143-1
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T. Lindström and B. Sriraman
time geometry, or Birkhoff’s ergodic theorem precipitate subsequent advances in the study of dynamical systems. A dynamical system is a model designed for predicting the future given a current state. In most cases, the model is a differential equation, but it can also be an equation based on recursion, i.e., using difference equations, which result in discrete dynamical systems. Differential equations have a number of advantages in this context. On an infinite decimal time-scale, the various contributions tend to operate independently and the various contributions can simply be added. This makes it possible to build models that, in principle, do the prediction for very extensive setups of processes. The process of integration is also better understood than summation in general, and this gives another advantage to the differential equation approach. The laws of nature can in most cases be expressed in terms of a set of differential equations. This means that the changes in the abundances of certain quantities are given in terms of functions of these abundances. The modeling itself is the easy part in this case, but the process of integrating these equations for formulating the solutions brings us in many cases to the unknown. Even if some simplifications of the reality can be completely known, taking an additional aspect into account might change the situation entirely. In other words, the complexity ca
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