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Random dynamical systems: addressing uncertainty, nonlinearity and predictability

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Abstract

Nonlinearity and noise play a significant role in an enormous range of subjects across the entire spectrum of science and engineering. This paper considers several research topics that encompass the area of random dynamical systems (RDS). A general overview of the problems, the multidisciplinary methods required for their analysis, and relevant results achieved in RDS are given with particular emphasis on developments during the past 25 years. The first part of this paper focuses on developing methods to unravel complex interactions between noise and nonlinearities using a mix of multidisciplinary approaches from theory, modeling, and simulation. Practical applications of these research results are beginning to appear across the entire spectrum of mechanics; for example, vibration absorbers, panel flutter, variable speed machining processes, and mixing and transport phenomena in fluid mechanics. The second part of this paper focuses on developing new algorithms and tools for the collection, assimilation and harnessing of data by threading together ideas ranging from random dynamical systems to information theory. A new particle filtering algorithm that combines stochastic homogenization with filtering theory is presented. Importance sampling and control methods are then used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering.

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Acknowledgements

The author acknowledges the support of the AFOSR under Grant Numbers FA9550-12-1-0390 and FA9550-17-1-0001 and the National Science Foundation (NSF) under grant number CMMI 1030144. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF. The author acknowledges the careful reading of the manuscript and helpful comments by R. Beeson and H. C. Yeong. This paper contains results from several joint work with N. Lingala, J.H. Park, Z. Rapti, P. Singh, R. Sowers, H. C. Yeong of University of Illinois and P. Imkeller, and N. Perkowski of Humboldt-Universität zu Berlin.

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  1. Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 306 Talbot Laboratory, 104 S. Wright St, Urbana, IL, 61801, USA

    Navaratnam Sri Namachchivaya

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  1. Navaratnam Sri Namachchivaya
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Correspondence to Navaratnam Sri Namachchivaya.

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This is an expanded version of the keynote lecture presented by the author at the 8th European Nonlinear Dynamics Conference (ENOC 2014) in Vienna, Austria, on July 8, 2014, under the same title.

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Namachchivaya, N.S. Random dynamical systems: addressing uncertainty, nonlinearity and predictability. Meccanica 51, 2975–2995 (2016). https://doi.org/10.1007/s11012-016-0570-4

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