X-—c(1-X2)X + aX =n(0), (13.36) 
the system will produce a perturbed limit cycle. 
Ozaki? !-93.96.97 and Haggan and Ozaki™?> proposed a new type of nonlinear model 
called an exponential autoregressive model through the following considerations. In Eq. 
13.26, when g, and g> are constants, the system becomes a purely linear oscillator and the 
equation of motion is expressed as 
X+cX+aX=n(t). (13.37) 
This expression can be inverted into an ARMA(2.1) model, as was explained in some 
detail in Section 6.3.2 of Part II, as 
X,= PiX-1 + P2X1-2 + O1€-1 + €2- (13.38) 
Here n(t) is a continuous Gaussian white noise and €,; is a discrete Gaussian white noise. 
The oscillation expressed by Eq. 13.37 is governed by its characteristic equation, as dis- 
cussed in Section 6.3.2, 
w+cu+a=0 (13.39) 
and by its roots (or eigenvalues) 
c : 
pone iva—(c?/4). (13.40) 
We know this model X, diverges when c < 0 and converges when c > 0. On the other 
hand, the model expressed by Eq. 13.38 diverges when the roots of its characteristic 
equation 
2-12-92 =0 (13.41) 
are outside the unit circle and converges to a stable process when two roots of Eq. 13.41 
are inside the unit circle, as described in some detail in Sections 5.2.4 and 5.2.3 and in 
Fig. 5.16. 
From Eq. 13.41, when @? < 4@, i.e., when the roots are unequal and complex 
(conjugate to each other) zone [ II ] in Fig. 5.16, 
1 
z= fhe 5 iy¥—402-?, (13.42) 
since IzI* = @2, when—@2 > 1 org <—1, this model diverges and when 1 >—¢2 > 0 or 
0>2 >-1, this model becomes a stable process. 
From the discussion of Green’s functions of ARMA(2.1) in Section 5.2.4 or of the 
autocorrelation of AR(2) in Section 5.2.3, we know that damping is determined by 
/—@2, i.e., by p2, and the frequency is dependent on@). 
In the analysis of a ship’s nonlinear rolling that comes from the nonlinear restoring 
moment (the Duffing type), Ozaki and Oda” first fitted a nonlinear model of the type 
X,= 1 Xi1+@2 X-2+H Xeyte. (13.43) 
384 
