Abstract
In many real-life systems, the state at the next moment of time is uniquely (and continuously) determined by the current state. In mathematical terms, such systems are called deterministic dynamical systems. In the analysis of such system, continuity is usually understood in the usual mathematical sense. However, as many formal definitions, the mathematical definition of continuity does not always adequately capture the commonsense notion of continuity: that small changes in the input should lead to small changes in the output. In this paper, we provide a natural fuzzy-based formalization of this intuitive notion, and analyze how the requirement of commonsense continuity affects the properties of dynamical systems. Specifically, we show that for such systems, the set of stationary states is closed and convex, and that the only such systems for which we can both effectively predict the future and effectively reconstruct the past are linear systems.
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Acknowledgments
This work was supported by:
\(\bullet \) the National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science), and HRD-1834620 and HRD-2034030 (CAHSI Includes),
\(\bullet \) the AT &T Fellowship in Information Technology,
\(\bullet \) the program of the development of the Scientific-Educational Mathematical Center of Volga Federal District No. 075-02-2020-1478, and
\(\bullet \) grant from the Hungarian National Research, Development and Innovation Office (NRDI).
The authors are thankful to the anonymous referees for valuable suggestions.
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Kosheleva, O., Kreinovich, V. (2023). Commonsense-Continuous Dynamical Systems – Stationary States, Prediction, and Reconstruction of the Past: Fuzzy-Based Analysis. In: Dick, S., Kreinovich, V., Lingras, P. (eds) Applications of Fuzzy Techniques. NAFIPS 2022. Lecture Notes in Networks and Systems, vol 500. Springer, Cham. https://doi.org/10.1007/978-3-031-16038-7_11
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DOI: https://doi.org/10.1007/978-3-031-16038-7_11
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