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12

Systems and Networks

Some Examples

The simplest bistable system is described by

StartFraction d u Over d t EndFraction equals u minus u cubed period^du

dt ^{= u −u3 .}

(12.27)

There are three stationary states, at u equals 0u = 0 (unstable; the Lyapunov number is +1)

and u equals plus or minus 1u = ±1 (both stable), for which the equation for small deviations is

StartFraction d Over d t EndFraction delta u equals minus 3 delta u ^d

dt ^{δu = −3δu}

(12.28)

and the Lyapunov numbers arenegative 3−3. This system can be considered as a memory box

with an information volume equal tolog Subscript 2log2(number of stable stationary states) = 1 bit.

A slightly more complex system is described by the two equations

StartLayout 1st Row d u 1 slash d t equals u 1 minus u 1 u 2 minus a u 1 squared 2nd Row d u 2 slash d t equals u 2 minus u 1 u 2 minus a u 2 squared EndLayout right brace period^{du1/dt = u1 −u1u2 −au2}

1

du2/dt = u2 −u1u2 −au²

2

)

.

(12.29)

The behaviour of such systems can be clearly and conveniently visualized using a

phase portrait (e.g., Fig. 12.3). To construct it, one starts with arbitrary points in the

left parenthesis u 1 comma u 2 right parenthesis(u1, u2) plane and uses the right-hand side of Eq. (12.29) to determine the increments.

The main isoclines (at whose intersections the stationary states are found) are given

by

StartLayout 1st Row d u 1 slash d t equals upper F 1 left parenthesis u 1 comma u 2 right parenthesis equals 0 2nd Row d u 2 slash d t equals upper F 2 left parenthesis u 1 comma u 2 right parenthesis equals 0 EndLayout right brace period^{du1/dt = F1(u1, u2) = 0}

du2/dt = F2(u1, u2) = 0

)

.

(12.30)

Total instability, in which every Lyapunov number is positive, results in dynamic

chaos. Intermediate systems have strange attractors (which can be thought of as

stationary states smeared out over a region of phase space rather than contracted to

a point), in which the chaotic régime occurs only in some portions of phase space.

Reception and Generation of Information

If the external conditions are such that in the preceding example (Eq. 12.29) the

starting conditions are not symmetrical, then the system will ineluctably arrive at one

of the stationary states, as fixed by the actual asymmetry in the starting conditions.

Hence, information is received.

On the other hand, if the starting conditions are symmetrical (the system starts out

on the separatrix), the subsequent evolution is not predetermined and the ultimate

choice of stationary state occurs by chance. Hence, information is generated. ²¹

21 Cf. the discussion in Chap. 6.