Routes to chaos

It is clear from the phase portrait of the chaotic attractor that all the periodic orbits that have lost their stability throughout the bifurcation sequence are visited during the course of time a chaotic trajectory winds its way 

All the essential features of the dynamics of the Rdssler system are contained in and recoverable from the Poincare map derived using the analysis 

The logistic map, referred to previously, is a now famous, but simple, quadratic function that displays the same period-doubling cascade into chaos exhibited by the Rossler system (and by quite a few experimental examples, as well). A map is an equation that gives an iterative rule for generating a sequence of points x, Xj, X3. given an initial value Xq. The logistic map is given by 

Another route to chaos that is important in chemical systems involves a torus attractor which arises via bifurcation from a limit cycle attractor. Again chaos is found to be associated with periodic behavior and to arise from it through a sequence of transformations and associated bifurcations of a periodic state of the system. The specific sequence is different in this case, however, and somewhat more complex. 

As mentioned in a previous section, a limit cycle can, at times, undergo a Hopf bifurcation. This would be revealed in a stability analysis of the cross-sectional point in the Poincare section of the limit cycle. A Hopf bifurcation would correspond to an associated pair of eigenvalues whose real part passes from negative to positive while all other eigenvalues remain negative. In physical terms, a Hopf bifurcation means a second frequency becomes available to the system, and this is reflected in the disappearance of the limit cycle attraaor and the appearance of a torus attractor. (The limit cycle still exists actually, but the bifurcation renders it unstable so that all trajectories are repelled from it.) 

How do periodic oscillations get transformed to chaotic oscillations This happens when the magnitude of bifurcation parameter is changed. In the case of chemical reactions in CSTR, bifurcation is the flow rate. Another question is In what steps periodic oscillations are converted to chaos  

There are three main routes to chaos in dissipative dynamical systems as given below  

These are associated with periodic doubling or regular oscillation intercepted by occasional bursts of noise. The features of three routes are summarized below  

Feigenbaum route Infinite cascade of periodic Chemical reaction 

Manneville-Pomeau Intermittent transition Chemical reaction Benard experiment 

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

At the th period doubling the period of the oscillation is In the limit —> co we arrive at the strange attractor where the time variation of the concentrations is no longer periodic. This is the period-doubling route to chaos. 

In addition to tire period-doubling route to chaos tliere are otlier routes tliat are chemically important mixed-mode oscillations (MMOs), intennittency and quasi-periodicity. Their signature is easily recognized in chemical experiments, so tliat tliey were seen early in the history of chemical chaos. 

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

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. 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

As a summary for the second study case, the bubble streams interactions induce the presence of complex oscillatory phenomenon [51]. For instance, since the bubble rise almost-linearly into the plume, the presence of the central plume induce an almost-periodic behavior. We shown that the number of fundamental frequencies (which are in some sense induced by the modes of the bubble streams) increases when the superficial gas velocity increases yielding a route to chaos (periodic- quasiperiodic-chaotic behavior). 

To achieve the instability of homogeneous broadened line lasers, a satisfaction of much more difficult conditions is required large gain and the so-called bad-cavity properties. This special regime for damping constants and mode intensity is fulfilled in the far-infrared lasers [36]. In 1985 Weiss et al. [37,38] experimentally found a period doubling route to chaos in the NH3 laser. Further experimental investigation of chaotic dynamics in such lasers was reported later [39]. 

The CO2 lasers were also investigated in connection with chaotic behavior, and here we mention the most important papers in the field. The chaotic behavior associated with a transverse mode structure in a cw CO2 laser was observed in 1985 [40]. In the CO2 laser with elastooptically modulated cavity length, a period doubling route to chaos was also found [41]. 

Chaos was also investigated in solid-state lasers, and the important role of a pump nonuniformity leading to a chaotic lasing was pointed out [42]. A modulation of pump of a solid-state NdPsOn laser leads to period doubling route to chaos [43]. The same phenomenon was observed in the case of laser diodes with modulated currents [44,45]. Also a chaotic dynamics of outputs in Nd YAG lasers was also discovered [46 18]. In semiconductor lasers a period doubling route to chaos was found experimentally and theoretically in 1993 [49]. 

The case of a frequency mismatch between laser pumps and cavity modes was investigated [83], and for the first time, chaos in SHG was found. When the pump intensity is increased, we observe a period doubling route to chaos for Ai = 2 = 1. Now, for/i = 5.5, Eq. (3) give aperiodic solutions and we have a chaotic evolution in intensities (Fig. 5a) and a chaotic attractor in phase plane (Imaj, Reai) (Fig. 5b). 

Another very successful apphcation of nonlinear dynamics to the heart is through mathematical modehng. An example in which a simple model based on coupled oscillators describes the dynamics of agonist induced vasomotion is in the work of de Brouwer et al. [586], where the route to chaos in the presence of verapamil, a class IV antiarrhythmic drug, is studied. 

DeBrouwer, S., Edwards, D., and Griffith, T., Simplification of the quasi-periodic route to chaos in agonist-induced vasomotion by iterative circle maps, American Journal of Physiology, Vol. 274, No. 4(2), 1998, pp. H1315-1326. 

A particularly important question is the exact nature of the transition to chaos at r 0 (r 1/2, respectively). For the box model discussed in this section, the transition to chaos is sudden (Bleher et al. (1990)). This means that at least some trajectories in box C are chaotic for any r with 0 < r < 1/2. Box C is regular only in two cases (i) for r = 0 (in which case C is identical to R), and (ii) for r = 1/2 (apart from the chaotic disconnected regions discussed above). No doubt this route to chaos is important, but rather abrupt. Other systems show more slowly developing, and thus more interesting, routes to chaos as a control parameter is varied. A particularly important route to chaos, the period doubling route to chaos is discussed in the following section. 

The period doubling route to chaos is best illustrated with the help of the logistic map... 

It is universal for a large class of period doubling scenarios. Physical examples of this route to chaos include the driven pendulum (Baker and Gollub (1990)) and ion traps (Blumel (1995b)). 

C.G.Steimnetz and R.Larter, The Quasiperiodic Route to Chaos in a Model of the Peroxidase-Oxidase Reaction, Journal of Chemical Physics, 94, 1388-1396(1991). 

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. 

Figure 10.6.2 shows thatthe qualitative dynamics of the two maps are identical. They both undergo period-doubling routes to chaos, followed by periodic windows interwoven with chaotic bands. Even more remarkably, the periodic windows occur in the same order, and with the same relative sizes. For instance, the period-3 window is the largest in both cases, and the next largest windows preceding it are period-5 and period-6. 

This orbit diagram allows us to keep track of the bifurcations in the Rdssler system. We see the period-doubling route to chaos and the large period-3 window— all our old friends are here. 

First we introduce some notation. Let /(x, r) denote a unimodal map that undergoes a period-doubling route to chaos as r increases, and suppose that is the maximum of f. Let denote the value of r at which a 2 -cycle is born, and let R denote the value of r at which the 2 -cycle is superstable. 

Let f x,p) be any unimodal map that undergoes a period-doubling route to chaos. Suppose that the variables are defined such that the period-2 cycle is born at X = 0 when /i = 0. Then for both x and jU close to 0, the map is approximated by... 

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