Intermittent chaos

So far we have concentrated on the particular parameter values <7 = 10, b =, r = 28, as in Lorenz (1963). What happens if we change the parameters It s like a walk through the jungle—one can find exotic limit cycles tied in knots, pairs of limit cycles linked through each other, intermittent chaos, noisy periodicity, as well as strange attractors (Sparrow 1982, Jackson 1990). You should do some exploring on your own, perhaps starting with some of the exercises. 

If this were the only context in which CML models were used, their utility would be severely limited. For values y beyond the stability limit, the Euler method fails and one obtains solutions that fail to represent the solutions of the reaction-diffusion equation. However, it is precisely the rich pattern formation observed in CML models beyond the stability limit that has attracted researchers to study these models in great detail. Coupled map models show spatiotemporal intermittency, chaos, clustering, and a wide range of pattern formation processes." Many of these complicated phenomena can be studied in detail using CML models because of their simplicity and, if there are generic aspects to the phenomena, for example, certain scaling properties, then these could be carried over to real systems in other parameter regimes. The CML models have been used to study chemical pattern formation in bistable, excitable, and oscillatory media." ... 

We recall from our earlier discussion of chaos in one-dimensional continuous systems (see section 4.1) that period-doubling is not the only mechanism by which chaos can be generated. Another frequently occurring route to chaos is intermittency. But while intermittency in low dimensional dynamical systems appears to be constrained to purely temporal behavior [pomeau80], CMLs exhibit a spatio-temporal intermittency in which laminar eddies are intermixed with turbulent regions in a complex pattern in space-time. 

Many examples of stick slip involve a rather different type of motion that can lead to intermittency and chaos [17a,39,51,53,181]. Instead of jumping... 

We should not be surprised to see ghosts—they always occur near saddle-node bifurcations (Sections 4.3 and 8.1) and indeed, a tangent bifurcation is just a saddle-node bifurcation by another name. But the new wrinkle is that the orbit returns to the ghostly 3-cycle repeatedly, with intermittent bouts of chaos between visits. Accordingly, this phenomenon is known as intermittency (Pomeau and Manneville 1980). 

In experimental systems, intermittency appears as nearly periodic motion interrupted by occasional irregular bursts. The time between bursts is statistically distributed, much like a random variable, even though the system is completely deterministic. As the control parameter is moved farther away from the periodic window, the bursts become more frequent until the system is fully chaotic. This progression is known as the intermittency route to chaos. 

Figure 10.4.5 shows an experimental example of the intermittency route to chaos in a laser. 

The problems encountered in mathematical modeling of tumble/growth agglomeration do not relate to the theories, formulas, and possibilities to solve the ever more complicated equations. With modem computing possibilities, a whole series of assumptions can be introduced into the model equations and responses to certain imaginary process conditions can be predicted. However, the real system often produces unexpected results intermittently or even consistently without offering a clear indication of why such deviations occur. Introduction of new mathematical methods, such as, for example, fuzzy logic or chaos theory, produce more complicated model equations and closer to life results but still are not able to serve as unequivocal bases for control schemes. 

The next problem to consider is how ehaotie attractors evolve from the steady state or oseillatory behaviour of chemical systems. There are, effeetively, an infinite number of routes to chaos [25]. However, only some of these have been examined carefully. In the simplest models they depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the eontrol parameter space allow at least a partial observation of these well known routes. For chemical systems we deseribe period doubling, mixed-mode oscillations, intermittency, and the quasi-periodic route to chaos. 

Figure C3.6.6 The figure shows the C2 coordinate, for < 0, of the family of trajectories intersecting the ( 2, 3) Poineare surfaee at = 8.5 as a function of bifurcation parameter k 2- As the ordinate k 2 decreases, the first subharmonie easeade is visible between k 2 0.1, the value of the first subharmonic bifurcation to k 2 0.083, the subharmonie limit of the first cascade. Periodic orbits that arise by the tangent bifurcation mechanism associated with type-I intermittency (see the text for references) can also be seen for values of k 2 smaller than this subharmonie limit. The left side of the figure ends at k 2 = 0.072, the value corresponding to the chaotic attractor shown in figure C3.6.1(a). Other regions of chaos can also be seen.

In addition to the period-doubling route to chaos there are other routes that are chemically important mixed-mode oscillations (MMOs), intermittency and quasi-periodicity. Their signature is easily recognized in chemical experiments, so that they were seen early in the history of chemical chaos. 

In recent years, some studies on maintenance of chaos and control of chaos had been undertaken. The former has recently been experimentally demonstrated in a magnetomechanical system demonstrating intermittency. There is interest in such studies in view of the likelihood that Pathological destruction of chaotic behaviour possibly due to some underlying disease may be implicated in heart failure and brain seizers . 

Three different routes to chaos—period-doubling, periodic-chaotic sequences, and intermittency—have been observed in the BZ reaction. Figure 8.15 shows time series for three periodic states of a period-doubling sequence, while Figure 8.16 illustrates both periodic and chaotic states from a periodic-chaotic sequence, and Figure 8.17 gives an example of intermittency. 

Chaos, intermittency and hysteresis in the dynamic model of a polymerization reactor. Chaos, Soliton Fractals, 1, 295-315. 

Quasiperiodicity in Antiferro-Like Structures and Spatial Intermittency in Coupled Logistic Lattice—Towards a Prelude of a Field-Theory of Chaos. 

Belousov-Zhabotinsky reaction when conducted in a well-mixed medium [91, 92, 95]. The so-called chemical chaos and its homoclinic nature was shown to arise from the nonlinear complexity of the chemical kinetics. The remarkable feature of the homoclinic intermittent bursting illustrated in Figures 9 to 12, is that it results from the interaction of the diffusion process with a chemical reaction which itself would proceed in a stationary manner if diffusion was negligible. 

C.7. 85, Consider the bifurcations of the symmetric cycle as a evolves from positive to negative values. Can it undergo a period-doubling bifurcation saddle-node one Exploit the symmetry of the problem. For the map (C.7.3), find the analytical expression for the principal bifurcation curves. Does the saddle-node bifurcation here precede the appearance of the Lorenz attractor (i.e. can chaos emerge through the intermittence ) Vary A from positive to negative values. Examine the piece-wise linear map with A > 1, and determine the critical value of A, after which the Lorenz attractor emerges. ... 

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