Aperiodic behavior

Figure 9-33a shows the predicted shear stress as a function of strain for the initial foam orientation depicted in Fig. 9-32. The stress grows continuously until at y = 1.15 a T1 reorganization occurs which brings the cell structure back to its starting state, and the stress jumps back to zero. Thereafter, the stress history repeats itself. Similar periodic stress patterns and stress jumps have been predicted for the three-dimensional tetrakaidecahedron foam model (Reinelt 1993). If the initial orientation is rotated through an angle of r/12 with respect to that shown in Fig. 9-32, the stress history also has jumps, but is aperiodic (see Fig. 9-33b). Aperiodic behavior is the norm, and periodic stress histories occur only for special initial orientations (Kraynik and Hansen 1986). These unsteady, discontinuous stress...

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. 

What is so incredible about chaos theory is that unstable aperiodic behavior can be found in mathematically simple systems. These simple mathematical systems display behavior so complex and unpredictable that it is acceptable to merit their descriptions as random. An interesting question arises concerning why chaos has just recently been noticed. If chaotic systems are so mandatory to our everyday life, how come mathematicians have not studied chaos theory earUer The answer can be given in one word computers. The calculations involved in studying chaos are repetitive, boring and number in the millions, and computers have always been used for endless repetition. 

Controlling chaos, even in the rather limited fashion described here, is a powerful and attractive notion. Showalter (1995) compares the process to the instinctive, apparently random motions of a clown as he makes precise corrections aimed at stabilizing his precarious perch on the seat of a unicycle. A small, but carefully chosen, variation in the parameters of a system can convert that system s aperiodic behavior not only into periodic behavior, but also into any of a large (in principle, infinite) number of possible choices. In view of the appearance of chaos in systems ranging from chemical reactions in the laboratory, to lasers, to flames, to water faucets, to hearts, to brains, our ability first to understand and then to control this ubiquitous phenomenon is likely to have major consequences. 

The Belousov-Zhabotinsky reaction shows oscillations of great variety and complexity it even exhibits chaos. In chaotic systems arbitrarily close initial conditions diverge exponentially the system exhibits aperiodic behavior. A... 

Figure 6 displays the Poincare maps for all experiments. Note that even the projections in canonical planes (see Figure 5) seem ordered in layers. That is, a toroidal structure can be seen form the Poincare surface. That is, small amplitude oscillations were detected in time series (see Figures 3 and 4) for all experiments. The t3rpical behavior of aperiodic (possibly chaotic) oscillations can be confirmed is one takes a look at the corresponding Poincare section... 

Fig. 13. Kinetic oscillations during the CO/O reaction on Pt(110) at I = 540 K, />0, = 7.5 x 10-5 torr, and for varying pm. (From Ref. 71.) (a) pco = 3.90 x 10 lorr constant behavior (fixed point), (b) pt0 = 3.K4 x I0"5 torr onset of harmonic oscillations with small amplitudes (Hopf bifurcation), (c)pco = 3.66 x 10 5 torr harmonic oscillation with increased amplitude, (d) pc0 = 3.61 x I0-5 torr first period doubling, (e) pc0 = 3.52 x 10 torr second period doubling, (f) pco = 3.42 x 10 5 torr aperiodic (chaotic) behavior.

A shortcoming of both models is that they do not capture the occurrence of complex periodic or aperiodic potential oscillations under current control, which were observed in many different electrolytes. Impressive studies of such complicated temporal motions during formic acid oxidation can e.g. be found in Refs. [118, 121], Schmidt et al. [131] suggest that the adsorption of anions, which leads to a competition for free surface sites not only between two species, formic acid and water, but between three species, formic acid, water and anions, can induce complex nonlinear dynamics. This conjecture is derived from differences in the oscillatory behavior found in perchloric and sulfuric acid for otherwise similar conditions. Complex motions were only observed in the presence of sulfuric acid. 

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. 

As pointed out, for 9 > 9C there exist infinitely many unstable steady states of period 1, 2, 4, 8,. .. and no stable steady states. This means that almost any initial condition leads to an aperiodic trajectory that looks random as in Figure 3.3 D, but actually the behavior is chaotic. In this figure, two chaotic orbits for 9 = 4 are coplotted. Only the initial conditions of the two trajectories differ... 

Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions. 

Aperiodic long-term behavior means that there are trajectories which do not settle down to fixed points, periodic orbits, or quasiperiodic orbits as t —> . For practical reasons, we should require that such trajectories are not too rare. For instance, we could insist that there be an open set of initial conditions leading to aperiodic trajectories, or perhaps that such trajectories should occur with nonzero probability, given a random initial condition. 

Solution No. Trajectories are repelled to infinity, and never return. So infinity acts like an attracting fixed point. Chaotic behavior should be aperiodic, and that excludes fixed points as well as periodic behavior. ... 

By our definition, the dynamics in Example 9.5.1 are not chaotic, because the long-term behavior is not aperiodic. On the other hand, the dynamics do exhibit sensitive dependence on initial conditions—if we had chosen a slightly different initial condition, the trajectory could easily have ended up at C instead of C Thus the system s behavior is unpredictable, at least for certain initial conditions. 

Figure 12.4.1 shows a time series measured by Roux ct al. (1983). At first glance the behavior looks periodic, but it really isn t—the amplitude is erratic. Roux et al. (1983) argued that this aperiodicity corresponds to chaotic motion on a strange attractor, and is not merely random behavior caused by imperfect experimental control. 

Even more exotic behavior has been found by those searching for it, but is exceedingly rare Oscillations may be periodic but contain waves of different amplitudes, or be entirely aperiodic. Only a glimpse of what may happen can be given here. 

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