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WM Keck Science

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We demonstrate that many physical systems possess an often overlooked property known as a partial-ordering structure. The detection and analysis of this special geometric property can be crucial for understanding a system's dynamical behavior. We review here the fundamental dynamical features common to all such systems, and describe how the partial ordering imposes interesting restrictions on their possible behavior. We show, for instance, that though such systems are capable of displaying highly complex and even chaotic behaviors, most of their experimentally observable behaviors will be simple. Partial orderings are illustrated with examples drawn from many branches of physics, including solid state physics, fluids, and chemical systems. We also describe the consequences of partial orderings on some simple nonlinear models, and prove, for example, that for general two-dimensional mappings with the partial-ordering property, period 3 implies chaos, in analogy with the well-known result of Li and York [Am. Math. Mon. 82, 985 (1975)] for (ordinary) one-dimensional mappings.


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