Randomness and Chaotic Behavior

Finally, the heterogeneous dynamic picture of the gastrointestinal tract becomes even more complicated by the coexistence of either locally or centrally 

It can be concluded that the use of nonlinear dynamics in gastrointestinal absorption studies can provide a tool for  

---

The last section of this chapter is devoted to the regulatory aspects of oral drug absorption and in particular to the biopharmaceutics classification system and the relevant FDA guideline. At the very end of the chapter, we mention the difference between randomness and chaotic behavior as sources of the variability encountered in bioavailability and bioequivalence studies. 

Instead of accepting the random and chaotic behavior of classical reactors, one can attempt to design and build reactors that are characterized by regular spatial structures. Such structures may be designed in full detail up to the local surroimdings of the catalyst. This opportunity offers control of the local enviromnents, allowing simplification of the fluid mechanics to well-understood behavior, such as laminar flow. The engineer can then easily direct the interaction of transport phenomena and reaction. 

A billion cars and coimting, himdreds of millions of them with catalytic converters—this application is a landmark success of catalytic science and technology. Automobile catalytic converters are mostly monoliths— like ceramic honeycombs with porous catalyst layers on their inner wall surfaces. These monoliths are the most widely used structured reactors, the topic addressed by Moulijn, Kreutzer, Nijhuis, and Kapteijn. In contrast to the classical reactors containing discrete particles of catalyst and characterized by random and chaotic behavior, structured reactors are characterized by regular structures and predictable laminar flow. Structured reactors can be designed in full detail up to the local surroimdings of the... 

Microfluidics handles and analyzes fluids in structures of micrometer scale. At the microscale, different forces become dominant over those experienced in everyday life [161], Inertia means nothing on these small sizes the viscosity rears its head and becomes a very important player. The random and chaotic behavior of flows is reduced to much more smooth (laminar) flow in the smaller device. Typically, a fluid can be defined as a material that deforms continuously under shear stress. In other words, a fluid flows without three-dimensional structure. Three important parameters characterizing a fluid are its density, p, the pressure, P, and its viscosity, r. Since the pressure in a fluid is dependent only on the depth, pressure difference of a few pm to a few hundred pm in a microsystem can be neglected. However, any pressure difference induced externally at the openings of a microsystem is transmitted to every point in the fluid. Generally, the effects that become dominant in microfluidics include laminar flow, diffusion, fluidic resistance, surface area to volume ratio, and surface tension [162]. 

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

So, apart from the regular behavior, which is either steady-state, periodic, or quasi-periodic behavior (trajectory on a torus, Figure 3.2), some dynamic systems exhibit chaotic behavior, i.e., trajectories follow complicated aperiodic patterns that resemble randomness. Necessary but not sufficient conditions in order for chaotic behavior to take place in a system described by differential equations are that it must have dimension at least 3, and it must contain nonlinear terms. However, a system of three nonlinear differential equations need not exhibit chaotic behavior. This kind of behavior may not take place at all, and when it does, it usually occurs only for a specific range of the system s control parameters 9. 

Oscillations may involve periods with more than a single wave, and may be chaotic (with no recurring periods at all). Aperiodic behavior is called chaos, but is not random The seeming randomness results from the fact that minutes difference in starting conditions can lead to drastically different behavior (butterfly effect of meteorology). Even relatively simple hypothetical networks can produce chaotic behavior. However, while periodic oscillation are possible in networks with only two independent mathematical variables, chaos requires at least three. 

Although at first sight turbulence seems to be structureless and randomized, studies" of oscillographs like that in Fig. 3.3 show that this is not quite so. The randomness and unpredictability of the fluctuations, which are nonetheless constrained between definite limits, exemplify the behavior of certain mathematical chaotic nonlinear functions. Such functions, however, have not yet proved useful in quantitatively characterizing turbulence. This is done by statistical analysis of the frequency distributions. 

Figure I. Comparison of regular (i.e., nonchaotic deterministic), deterministic chaotic, and random behavior—nutdifiedfrom (52). The rate of divergence of neighboring trajectories for deterministic chaotic behavior is determined by the...

Gribbin, John. Deep Simplicity Brining Order to Chaos and Complexity. New York Random House, 2005. An examination of how chaos theory and related fields have changed scientific understanding of the universe. Provides many examples of complex systems found in nature and human culture. Mandelbrot, Benoit, and Richard L. Hudson. The Misbehavior of Markets A Fractal View of Financial Turbulence. New York Basic Books, 2006. Provides a detailed examination of fractal patterns and chaotic systems analysis in the theory of financial markets. Provides examples of how chaotic analysis can be used in economics and odier areas of human social behavior. Stewart, Ian. Does God Play Dice The New Mathematics of Chaos. 2d ed. 1997. Reprint. New York Hyperion, 2005. An evaluation of the role that order and chaos play in the universe through popular explanations of mathematic problems. Includes accessible descriptions of complex mathematical ideas underpinning chaos theory. 

Stewart, 1989). In many respects, the idea that systems with a deterministic dynamics can behave in ways that we normally associate with systems subject to random forces—that identical experiments on macroscopic systems can lead to very different results because of tiny, unavoidable differences in initial conditions— is truly revolutionary. Chaos has become a more central part of mathematics and physics than of chemistry, but it is now clear that chemical systems also exhibit chaotic behavior (Scott, 1991). In this section, we define chaos and some related concepts. We also give some examples of chaotic chemical systems. In the next section, we discuss the intriguing notion that understanding chaos may enable us to control chaotic behavior and tailor it to our needs. 

A stochastic description of explosion phenomena is set up, both for isothermal and for exothermic reaction mechanisms. Numerical simulations and analytic study of the master equation show the appearence of long tail and multiple humps in the probability distribution, which subsist for a certain period of time. During this interval the system displays chaotic behavior, reflecting the random character of the ignition process. An estimate of the onset time of transient bimodality is carried out in terms of the size of the system, the intrinsic parameters, and the initial condition. The implications of the results in combustion are discussed. 

Let us consider the case of a = 30 corresponding to a weakly developed chaotic attractor in the individual nephron. The coupling strength y = 0.06 and the delay time T2 in the second nephron is considered as a parameter. Three different chaotic states can be identified in Fig. 12.16. For the asynchronous behavior both of the rotation numbers ns and n f differ from 1 and change continuously with T2. In the synchronization region, the rotation numbers are precisely equal to 1. Here, two cases can be distinguished. To the left, the rotation numbers ns and n/ are both equal to unity and both the slow and the fast oscillations are synchronized. To the right (T2 > 14.2 s), while the slow mode of the chaotic oscillations remain locked, the fast mode drifts randomly. In this case the synchronization condition is fulfilled only for one of oscillatory modes, and we speak of partial synchronization. A detailed analysis of the experimental data series reveals precisely the same phenomena [31]. 

---