12.3 Synergetics

151

One expects that the evolution of a system is completely described by itsnn equa-

tions of the type (12.21), together with the starting and boundary conditions. Suppose

that a stationary state has been reached, at which all of the derivatives are zero, and

described by the variables u overbar Subscript 1 Baseline comma ellipsis comma u overbar Subscript n Baseline¯u1, . . . , ¯un, at which all the functions upper F Subscript iFi are zero. Small

deviations delta u Subscript iδui may nevertheless occur and can be described by a system of linear

differential equations

StartFraction d Over d t EndFraction delta u Subscript i Baseline equals sigma summation Underscript j Overscript n Endscripts a Subscript i j Baseline delta u Subscript j Baseline comma ^d

dt ^{δui =}

n

Σ

j

ai jδu j ,

(12.23)

where the coefficients a Subscript i jai j are defined by

a Subscript i j Baseline equals StartFraction partial differential upper F Subscript i Baseline Over partial differential u Subscript i Baseline EndFraction vertical bar Subscript u Sub Subscript i Subscript equals u overbar Sub Subscript i Subscript Baseline periodai j = ^∂Fi

∂ui

||||

ui=¯ui

.

(12.24)

The solutions of Eq. (12.23) are of the form

delta u Subscript j Baseline left parenthesis t right parenthesis equals sigma summation Underscript j Overscript n Endscripts epsilon Subscript i j Baseline e Superscript lamda Super Subscript i Superscript t Baseline commaδu j(t) =

n

Σ

j

εi je^λit ,

(12.25)

where theepsilon Subscript i jεi j are coefficients proportional to the starting deviations [viz.epsilon equals delta u left parenthesis 0 right parenthesisε = δu(0)].

Thelamdaλs are called the Lyapunov numbers, which can, in general, be complex numbers,

the eigenvalues of the system; they are the solutions of the algebraic equations

det StartAbsoluteValue a Subscript i j Baseline minus delta Subscript i j Baseline lamda Subscript j Baseline EndAbsoluteValue equals 0 comma det|ai j −δi jλ j| = 0 ,

(12.26)

wheredelta Subscript i jδi j is Kronecker’s delta. ²⁰ We emphasize that the Lyapunov numbers are purely

characteristic of the system; that is, they are not dependent on the starting conditions

or other external parameters—provided the external influences remain small.

If all of the Lyapunov numbers are negative, the system is stable—the small

deviations decrease in time. On the other hand, if at least one Lyapunov number

is positive (or, in the case of a time-dependent Lyapunov number, if the real part

becomes positive as time increases), the system is unstable, the deviations increase

in time, and this is what gives rise to “unexpected” phenomena. If none are positive,

but there are some zero or pure imaginary ones, then the stationary state is neutral.

20delta Subscript i j Baseline equals 0δi j = 0 wheni not equals ji /= j and 1 wheni equals ji = j.