X, = (1.5+0.28 e*) X,,-0.96 X,5+€, (13.47) 
where o7 = 0.025, namely €, is N[0, 0.025]. 
The eigenvalue A stays Ul? = 0.96 for 25° < Arg. (A) < 40°, and actually 
moves between Ap = 0.89 + 0.41i and A. = 0.75 + 0.63i when |X,_;| changes between 
0 and © as schematically shown in Fig. 13.2. The generated time history is shown over 
t = 1-100 in Fig. 13.3. To check the characteristics of this model expressed by Eq. 13.47, 
the model 
X; = (1.5 oP 0.28e7 -1) Xi) Im 0.96 Xi-2 + Qa: (13.48) 
was simulated,!© where a, is no longer random but is the form a; = sin{27 fit): th. where 
the frequency f(t) changes with time. 
4.00 
uit Ayal i Wi Uys ld i Wylie Hh 
00 100.00 200.00 300.00 400.00 500.00 600.00 700.00 800.00 900.00 1000.00 
o—4.00 0.00 
Fig. 13.3. Generated exponential AR model 
= (1.54 0.286%) X,, -0.96X-0+€;, €1: N[O, 0.025]. 
(From Ozaki.9) 
Figures 13.4 and 13.5 show these simulations. In Fig. 13.4, the frequency fiz) 
increases with time from 0.005 to 0.1, and in Fig. 13.5 the frequency f(t) decreases with 
time from 0.1 to 0.005. In Fig. 13.4, the amplitude suddenly becomes small at around 
f= 0.062; in Fig. 13.5, the amplitude changes abruptly at around f= 0.052 and actually 
demonstrates the jump phenomenon shown in Fig. 13.1. 
In the same way, by making the damping amplitude dependent, Ozaki,°** proposed 
the model for expressing the Van der Pol type oscillator as 
or, more generally, 
X= (91+ & *1) Xi-1+ (2+ me*1) X2+€1. (13.50) 
386 
