Chaotic state

A chaotic state of the Belousov-Zhabotinskii reaction may be generated in a flow reactor. Concentrations of the reagents were found to vary randomly for the following reagent concentrations in the influent streams at 39.6°C, see Table 6.4. The changes in Ce(IV) concentration occurring during 

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Attractors can be simple time-independent states (points in F), limit cycles (simple closed loops in F) corresponding to oscillatory variations of tire chemical concentrations with a single amplitude, or chaotic states (complicated trajectories in F) corresponding to aperiodic variations of tire chemical concentrations. To illustrate... 

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

Assuming that the gas molecules within the gap are in chaotic state, then X can be written as... 

According to Stuart A. Kauffman (1991) there is no generally accepted definition for the term complexity . However, there is consensus on certain properties of complex systems. One of these is deterministic chaos, which we have already mentioned. An ordered, non-linear dynamic system can undergo conversion to a chaotic state when slight, hardly noticeable perturbations act on it. Even very small differences in the initial conditions of complex systems can lead to great differences in the development of the system. Thus, the theory of complex systems no longer uses the well-known cause and effect principle. 

Figure 4 Distributions for separations between the nearest distances nearest NPs, saddle points, NPs with the same (++) and opposite winding numbers (+-) in a chaotic Sinai billiard. The radial distribution of nearest distances for completely random points (26) is shown by the dashed curve in (a). The corresponding distribution for the Berry model function for a chaotic state (2) and random superposition of 16 eigen functions for a rectangular box with the same size and energy are shown by dots and thin curves, respectively.

Improvement Item Decontamination logs were not accurate due to chaotic state at the scene. Gross decontamination performed but quickly overwhelmed Fire Department did not have adequate resources to conduct decontamination activities for mass casualty situation. 

In equilibrium, impurities or vacancies wiU be distributed uniformly. Similarly, in the case of two gases, as above, once a thorough mixture has been formed on both sides of the partition, the diffusion process is complete. Also at that stage, the entropy of the system has reached its maximum value because the information regarding the whereabouts of the two gases has been minimized. In general, it should be remembered that entropy of a system is a measure of the information available about that system. Thus, the constant increase of entropy in the universe, it is argued, should lead eventually to an absolutely chaotic state in which absolutely no information is available. 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

For t < 0.294 hr and t > 0.305 hr the oscillations are simple RO and QHO, respectively. If t is increased from the lower limit in this range, an alternating sequence of periodic and chaotic regimes is revealed. Each periodic regime consists of a single RO and a number of QHO which increases by one from each such periodic regime to the next (Fig. 6a and c). The chaotic states that separate the periodic ones consist of single... 

In Fig. 6 the Br time series for one sequence of alternating states are shown (Fig. 6a-c) together with the corresponding power spectra (Fig. 6d-f). Figure 6a illustrates the first complex periodic state (one RO, one QHO) which appears as t is increased from 0.294 hr. The second complex periodic state (one RO, two QHO) is shown in Fig. 6c, and the intervening chaotic state in Fig. 6b. Each periodic state is characterized by a power... 

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

Entropy is often defined as an increase in disorder. A way of understanding entropy is to think in terms of the increase in entropy as an increase in the chaotic state (disorganized, untidy, or hectic state) of the system. Then, the greater the number of arrangements in a system, the greater the entropy. 

Secondly, the chaotic state of a resonant system is an infinite collection of unstable periodic motions (see Figure 14.9). The basic idea of the controlling chaos is to switch a chaotic system by a small perturbation among many different periodic orbits. Ott, Grebogy and Yorke [10] developed a scheme in which a chaotic system can be forced to follow one particular unstable periodic orbit. The problem is to calculate the perturbation that will shift the system towards the desired periodic orbit. This process is similar to balancing a marble on a saddle. To keep the marble from rolling off, one needs to move the saddle quickly from side to side. And... 

It is hard to say why the study of sedimentary carbonate geochemistry started the 1960 s with such vigor and ended the decade in such a chaotic manner. The tumultuous nature of the times, sudden infusion of previously undreamed of levels of funding, and rapid addition of new investigators to the field certainly were all part of it. However, it also seems probable that frustration with the unexpected complexities of the chemical behavior of carbonates and failure to arrive at solutions of basic problems also contributed substantially to the chaotic state of the field as the 1960 s ended. These were the wild adolescent years of the field. 

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]. 

Transitions Between Periodic and Chaotic States in a Continuous Stirred Reactor... 

The Belousov-Zhabotinskii reaction in an isothermal CSTR can undergo a series of transitions among periodic and chaotic states. One segment of this series of transitions is investigated in detail. Liapunov characteristic exponents are calculated for both the periodic and chaotic regions. In addition, the effect of external disturbances on the periodic behavior is investigated with the aid of a mathematical model. 

In the early investigations of hydrodynamic instability [9], [190], it was presumed that the instability evolves to a chaotic state characteristic of turbulence. Thus self-turbulization of premixed flames was attributed to hydrodynamic instability (analogous, in a sense, to the development of turbulence in shear flows). This viewpoint must be revised if the instability evolves to stable nonplanar structures, as suggested above. From numerical experiments with equations describing the self-evolution of flame surfaces in the limit of small values of the density change across the flame, it has been inferred [152], [198]-[200] that the hydrodynamic instability evolves... 

Hysteresis between a fixed point and a strange attractor) Consider the Lorenz equations with <7 = 10 and b = 8/3. Suppose that we slowly turn the r knob up and down. Specifically, let r = 24.4 -h sin or, where ft) is small compared to typical orbital frequencies on the attractor. Numerically integrate the equations, and plot the solutions in whatever way seems most revealing. You should see a striking hysteresis effect between an equilibrium and a chaotic state. 

The U-sequence has been found in experiments on the Belousov—Zhabotinsky chemical reaction. Simoyi et al. (1982) studied the reaction in a continuously stirred flow reactor and found a regime in which periodic and chaotic states alternate as the flow rate is increased. Within the experimental resolution, the periodic states occurred in the exact order predicted by the U-sequence. See Section 12.4 for more details of these experiments. 

Furthermore, the map is unimodal, like the logistic map. This suggests that the chaotic state shown in Figure 12.4.1 may be reached by a period-doubling scenario. Indeed such period-doublings were found experimentally (Coffman et al. 1987), as shown in Figure 12.4.4. 

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