(1) LINEAR THRESHOLD AR MODEL (2) NONLINEAR THRESHOLD AR MODEL (8) EXPONENTIAL AR MODEL 
Fig. 13.9. Schematical expression of behaviors of characteristic roots 
by Ozaki.99 
Ozaki! investigated the stability and existence of limit cycles and their relation to the 
form of the characteristic equation. For example, he showed that a nonlinear threshold 
model, 
oe 0.8 Xe stIGin for IX,_4| =>1.0 
(0.8 + 1.3 X2,-1.3Xt)) Xei+€:, for X,4! < 1.0 (13.60) 
has characteristic roots that behave as shown in Fig. 13.10. This model also has three 
stable singular points and gives a process that fluctuates around one of these points and 
jumps from one stable singular point to another, depending on the white input, as shown 
in Fig. 13.11. 
With these discussions and examples, Ozaki showed the greater generality of his 
exponential models. 
X; 
Fig. 13.10. Behavior of characteristic roots of a 
nonlinear threshold AR model. 
(From Ozaki.99) 
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