Chaos in deterministic systems: Strange attractors, turbulence, and applications in chemical engineering

Abstract

It is now well established that seemingly innocuous dynamical systems, dissipative or not, can produce complicated phase trajectories and, eventually, chaos. There is now an enormous amount of mathematics and physics literature on this subject, little of which has permeated into the chemical engineering community at large. This article is divided into two parts: the first one considers dissipative systems, the second Hamiltonian systems. One of the major achievements of the research in these areas has been the recognition that both difference and differential models share a group of universal features that go quite beyond the formal details of the models. For example, one of the fingerprints of many dissipative systems is the Feigenbaum cascade of period-doubling. The limit of the cascade is a chaotic situation resembling turbulence. Feigenbaum's analysis provides“universal” scaling laws and statistics which characterize“turbulent motions” arising via the period-doubling route. A sound mathematical framework characterizes the behavior of Hamiltonian systems. Chaos arises due to transversal intersections of stable and unstable manifolds belonging to hyperbolic points and prevented from becoming widespread by surviving invariant tori whose behavior is controlled by the Kolmogorov-Arnold-Moser theorem. In this article, emphasis is placed on differential models since they arise naturally in chemical engineering applications. An interesting example, in the context of fluid mechanics, is provided by mixing problems. The particle trajectory of every fluid particle in every fluid mechanical problem is represented by a dynamical system of three ODEs, autonomous if the flow field is steady, nonautonomous if it is not. In two dimensions, the problem can be framed in terms of Hamiltonian mechanics. An analysis of the particle trajectories (flows) provides a rational way to create chaotic (or good) mixing. Other applications can be drawn from fluid mechanics, transport processes, and reaction engineering. It is important to recognize that these new techniques provide additional ways of interpreting experimental data and they suggest new experiments for discriminating between possible mechanisms and routes to complex or turbulent behavior.