System Science Methods for Nonlinear Model Development

Contributing author Inang Birol

Department of Chemical Engineering Northwestern University Evanston, IL

One of the nature's greatest mysteries is the reason why she is understandable. Yet, she is understandable, and the language she speaks is mathematics. The history of science is full of breakthroughs when the mathematics of a certain type of behavior is understood. The power of that understanding lies in the fact that, stemming from it, one can build a model for the phenomenon, which enables the prediction of the outcome of experiments that are yet to be performed.

Now, it is a part of scientific folklore [195], how Lorenz [350] realized the phenomenon that was later given the name deterministic chaos; how it stayed unnoticed on the pages of the Journal of Atmospheric Sciences for a period of time only to be rediscovered by other scientists; and how it unfolded a new scientific approach. In fact, the existence of chaotic dynamics has been known to mathematicians since the turn of the century. The birth of the field is commonly attributed to the work of Poincare [473]. Subsequently, the pioneering studies of Birkhoff [55], Cartwright [89], Littlewood [344], Smale [551], Kolmogorov [284] and others built the mathematical foundations of nonlinear science. Still, it was not until the wide utilization of digital computers in late-1970s for scientific studies, that the field made its impact on sciences and engineering. It has been demonstrated that chaos is relevant to problems in fields as diverse as chemistry, fluid mechanics, biology, ecology, electronics and astrophysics. Now that it has been shown to manifest itself almost anywhere scientists look, the focus is shifted from cataloging chaos, to actually learning to live with it. In this chapter, we are going to introduce basic definitions in nonlinear system theory, and present methods that use these ideas to analyze chaotic experimental time series data, and develop models.

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