Therefore from 
le a 
wan 0, 2[(a —Geg)X* + (Wh — W2,)xk + NxE(x,X,))° = 0 (10.8) 
OQeg 
and 
dle*| > i 
art 0, 2[(@ —Geq)x,X + (W§ — Weg)x* — NxB(x, x, = 0. (10.9) 
eq 
If this process x(r) is stationary, 
[xx] = 0. (10.10) 
Therefore from Eqs. 10.8 and 10.9 
eq =a +e, x, 1/L7I, (10.11) 
w2, = 0) +L 8G, 4, O1/L7 1. (10.12) 
If we assume ergodicity, the time average can be replaced with the ensemble aver- 
age, and therefore 
Geq= atn . Elig(x,%))/EL", (10.11°) 
Weq = H+ . Elx, g(x, x)]/Elx"]. (10.12’) 
This shows how Qeg, We, can be estimated from the original expression Eq. 10.4. When 
the form of 77 - g[x, x,t] is given, the equivalent linearized values Gg, Weg can easily be 
obtained by Eqs. 10.11’ and 10.12’. 
For example, for a Duffing type oscillation, 
Ft aitws{x+ ex = fi), (10.13) 
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