Random dynamical systems: addressing uncertainty, nonlinearity and predictability

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.