Equation 13.43 can be transformed into 
X,= (6, +2 X*_) Xi1t+o2 Xi2+&. (13.43’) 
This expression shows that the coefficient of X,;, the first autoregressive coefficient, 
changed from @, in Eq. 13.38 to(@,; +2 X2,) in Eq. 13.43’. This means the frequency, 
determined by the coefficient of X;_), became amplitude dependent. Although this looks 
fine, Ozaki and Oda found that the time series X,, defined by the nonlinear autoregressive 
model Eq. 13.4’ was not stable but could be diverged. Therefore, instead of Eq. 13.43, 
they proposed 
_V¥2 
X,= (git me) X-1+ 2 X-2+€:. (13.44) 
This is the exponential type of autoregressive model for a Duffing type oscillator. By ex— 
pressing it this way, we can make both roots A(0), A(0) of the instantaneous characteristic 
equation, when X,_; = 0, 
J? — ($1 +) -$2 = 0 (13.45) 
and also the roots A(), A(©) of the instantaneous characteristic equation when 
Xi] —> © 
17-6 -62=0 (13.46) 
lie inside the unit circle as shown in Fig. 13.2. Accordingly, with Eq. 13.44 as the model, 
we can express the amplitude—dependent and stable oscillation under white noise oscilla- 
tion excitation. 
Fig. 13.2. A(0), A(%) for stable Duffing type model. 
When 2, > 0, the model corresponds to a hard spring system and when 7 < 0 the 
model corresponds to a soft spring type oscillator. Haggan and Ozaki (1979)™ and 
Ozaki?’-* showed an example of this type as 
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