Complex systems locally stable states

The simplest manifestation of self-organization in a reacting system is the occurrence of bistability, that is, the coexistence of two locally stable homogeneous states. In all electrical models, bistable behavior results from the interaction of an N-shaped stationary polarization curve with a sufficiently large ohmic resistor in the external circuit. These two features also represent the backbone for all more complex forms of self-organization where, owing to exactly these two properties of the system, the double-layer potential takes on the role of the autocatalytic variable. 

Figure 1.1. Prototypical potential energy surface of a simple system (a) and of a complex system (b). In a simple, low-dimensional system, dynamical bottlenecks for transitions between long-lived stable states most often coincide with saddle points on the potential energy surface. Locating these stationary points reveals the reaction mechanism. In a typical complex system, the potential energy surface is rugged and has countless local minima and saddle points. Nevertheless, there can be well-defined long-lived stable states and rare transitions between them. Such transitions can occur via a multitude of different transition pathways.

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

In the case of the fast binary reaction we could eliminate the reaction term from the reaction-diffusion-advection equation. But in general this is not possible. In this chapter we consider another class of chemical and biological activity for which some explicit analysis is still feasible. We consider the case in which the local-reaction dynamics has a unique stable steady state at every point in space. If this steady state concentration was the same everywhere, then it would be a trivial spatially uniform solution of the full reaction-diffusion-advection problem. However, when the local chemical equilibrium is not uniform in space, due to an imposed external inhomogeneity, the competition between the chemical and transport dynamics may lead to a complex spatial structure of the concentration field. As we will see in this chapter, for this class of chemical or biological systems the dominant processes that determine the main characteristics of the solutions are the advection and the reaction dynamics, while diffusion does not play a major role in the large Peclet number limit considered here. Thus diffusion can be neglected in a first approximation. 

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