Stable attractor

Fig. 4.5 Sets of stable attractors for the first six critical values of a. Note the self-similarity between the boxed subpattern for oe and the entire pattern for 04 appearing two lines above. A Cantor-set-like fractal pattern appears in the limit an-+oo-...

Sincic and Bailey (1977) relaxed the assumption of only one stable attractor for a given set of operating conditions and showed examples of some possible exotic responses in a CSTR with periodically forced coolant temperature. They also probed the way in which multiple steady states or sustained oscillations in the dynamics of the unforced system affect its response to periodic forcing. Several theoretical and experimental papers have since extended these ideas (Hamer and Cormack, 1978 Cutlip, 1979 Stephanopoulos et al., 1979 Hegedus et al., 1980 Abdul-Kareem et al., 1980 Bennett, 1982 Goodman et al., 1981, 1982 Cutlip et al., 1983 Taylor and Geiseler, 1986 Mankin and Hudson, 1984 Kevrekidis et al., 1984). 

Thus in the oscillator death state (Mirollo and Strogatz, 1990), the stirring completely inhibits the chemical oscillations, transforming the unstable steady state of the local dynamics into a stable attractor of the reaction-advection-diffusion system. This arises from the competition between the non-uniform frequencies that lead to the dispersion of the phase of the oscillations, while mixing tends to homogenize the system and bring it back to the unstable steady state. Therefore this behavior is only possible in oscillatory media with spatially non-uniform frequencies. The phase diagram of the non-uniform oscillatory system in the plane of the two main control parameters (v, 5) is shown in Fig. 8.4. 

Synthesis involves a complex set of processes and complex systems often show chaos, in particular in autocatalysis. Chaotic processes diverge two systems that follow the same process but have an unnoticed difference in start concentrations follow completely different paths. Chaos is not recognized if the system is locked into what is called a stable attractor or a limit cycle. Then it seems to be simply causal and is usually easy to model. When the system is in a strange attractor it seems to run amok. Chaos in synthesis can be recognized by several indicators ... 

Whichever qualitative features are examined, the apparently rich chemical literature reduces to a few simple classes. It is difficult to judge the reason why so few dynamic classes have appeared so far. Perhaps there are certain (still unanalyzed) features of kinetic equations that lead to simple dynamics. Another possibility is that chemists have tended to study only a small subclass of chemical kinetics and have ignored the rest. For example, a type of dynamical behavior that it is hard to imagine not existing is one in which there are two stable attractors. A limit cycle oscillation and a stable steady state where transitions can occur between the two as a result of large perturbations of concentrations. An example has been previously given in which this type of... 

Unique Stable Attractor X Unstable Steady State o Average on Periodic Attractor Average on Chaotic Attractor A Psuedo Maximum for Qiaotic Attractor... 

We choose the same variables for the chemical as for the electrochemical system. The reactions at the electrodes are sufficiently fast that the measured potential is the equilibrium potential fluctuations in that potential and in the imposed current are neglected. This is analogous to neglecting fluctuations in concentrations of species in equilibrium with mass reservoirs. For systems for which equilibrium is the only stable attractor, the chemical potential of each chemical species, say that of A2, is... 

A first approximation to the description of the full process is to assume that, by some mechanism independent of the fluctuations, the system remains uniform in space. In this "zero-dimensional" description, considerable attention was focussed on the decay from an initial unstable state, and on the passage times between simultaneously stable states [25-29]. Two characteristic scales emerge from this analysis. For the evolution around the unstable state, the time needed for the probability distribution to forget the initial condition and begin to develop peaks toward the stable attractors is... 

This crude model does not take into account the possibility of correlated noise events. Even if d < fo transitions are still possible if a series of noise events act before relaxation to the stable attractors is complete. This effect is clearly manifest in ifneq if only direct transitions are taken into account ifneq is predicted to be zero if d < To, but simulations show Kneni f n = 5.0) = 0.09 0.01 for d = 0.4. The simple model must be refined to describe such correlated effects. 

Our results suggest that the above dynamics can be viewed as an evolution in a stochastic potential whose qualitative aspect depends on time at the beginning it is similar to the deterministic potential, but subsequently it deforms (the deformation depending on the volume and initial conditions) and develops a second minimum. This minimum is responsible for the transient "stabilization" of the maximum of P(X,t) before the inflexion point. As the tunneling towards the other minimum on the stable attractor goes on, the first minimum disapears and the asymptotic form of the stochastic potential, determining the stationary properties of P(X,t), reduces again to the deterministic one. This phenomenon of "phase transition in time" is somewhat reminiscent of spinodal decomposition. 

Experiment 6. Lastly, yet another type of solution emerges when but minor parameter changes are made. These lead, however, to a structural change in the pattern of singular points. With the parameters of Fig. 5.9 a, for example, there occur five singular points two of these are stable attractors for a phase transition, upon which the formerly quasi-periodic motion breaks down and quickly ends in a stationary solution (such as long term temporary over- or ui erem-ployment equilibra [5.21]. Here, the stationary solution is reached with d 0.5 and X -0.8 (Fig. 5.9b). 

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