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Fig. 5.29 Zero isoclines of
the one-dimensional
reaction-diffusion equation
dx i =
dt ¼ fxðÞ:
Attractors of this system are stationary states that correspond to the intersections
of the zero isocline f ( x i ) ¼ 0 with the abscissa (Fig. 5.29 ). These states can be stable
or unstable. Three variants of the dynamics are possible for a system described by
the considered one-dimensional equation. Two of them apply to the case when the
system has one steady state, and the third one to a system with three states. A simple
analysis of the stability of the system shows that if f ( x ) takes only one zero value,
the steady state is stable. If three zero points correspond to f ( x ), then the two steady
states ( df ( x )/ dx
0) is unstable. In this
case the system is bi-stable, i.e., it can be in two different states.
Consider a network consisting of subsystems of this type, which are connected
by diffusion, mass transfer, or some other mechanism. Then, if all subsystems are
mono-stable, i.e., have one stable state, the network also has only one stable state,
which is homogeneous. If the subsystems have three states, the network can have up
to 2 N stable states. Each subsystem can in principle have two states, and the network
structure is given by different distributions of states of all subsystems.
Stationary structures in related reaction-diffusion systems have been studied
experimentally in recent years. In particular a system has been studied built on the
basis of 16 linearly related reactors with complete mixing. The dynamics of the
medium was determined by the well-known chlorite-iodate reaction:
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0) are stable, while the third one ( df ( x )/ dx
>
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