94Economic DynamicsIn deriving the stability of a periodic point we require, then, to compute [fk(y)] ,and to do this we utilise the chain rule[fk(y)]=f(y∗1)f(y∗2). . .f(y∗n)wherey∗1,y∗2, . . . ,y∗kare thek-periodic points. For example, ify∗1andy∗2are twoperiodic points off2(y), then[f2(y)]=f(y∗1)f(y∗2)and is asymptotically stable iff(y∗1)f(y∗2)<1All other stability theorems hold in a similar fashion.Although it is fairly easy to determine the stability/instability of linear dynamicsystems, this is not true for nonlinear systems. In particular, such systems cancreate complex cycle phenomena. To illustrate, and no more than illustrate, themorecomplexnatureofsystemsthatarisefromnonlinearity,considerthefollowingquadratic equationyt+1=ayt−by2tFirst we need to establish any fixed points. It is readily established that two fixedpoints arise sincey∗=ay∗−by∗2=ay∗1−by∗awhich gives two fixed pointsy∗=0andy∗=a−1bThe situation is illustrated in figure 3.5, where the quadratic is denoted by the graph
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Equilibrium point, Nonlinear system, Stability theory, Economic Dynamics, y∗, two