Economics Dynamics Problems 105

Economics Dynamics Problems 105 - Discrete dynamic...

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Discrete dynamic systems 89 (i) The equilibrium point y is stable if given ε> 0 there exists δ> 0 such that ± ± y 0 y ± ± implies ± ± f n ( y 0 ) y ± ± for all n > 0. If y is not stable then it is unstable . (ii) The equilibrium point y is a repelling fxed point if there exists ε> 0 such that 0 < ± ± y 0 y ± ± implies ± ± f ( y 0 ) y ± ± > ± ± y 0 y ± ± (iii) The point y is an asymptotically stable (attracting) equilibrium point 2 if it is stable and there exists η> 0 such that ± ± y 0 y ± ± implies lim t →∞ y t = y If η =∞ then y is globally asymptotically stable . All these are illustrated in Fgure 3.2(a)–(e). In utilising these concepts we employ the following theorem (Elaydi 1996, section 1.4). THEOREM 3.1 Let y be an equilibrium point of the dynamical system y t + 1 = f ( y t ) where f is continuously differentiable at y . Then (i) if ± ± f ± ( y ) ± ± < 1 then y is an asymptotically stable (attracting) Fxed point (ii) if ± ± f ± ( y ) ± ± > 1 then y is unstable and is a repelling Fxed point (iii) if ± ± f ± ( y ) ± ± = 1 and (a) if f ±± ( y ) ²= 0 , then y is unstable (b) if f ±± ( y ) = 0 and f ±±± ( y ) > 0 , then y is unstable (c) if f ±± ( y )
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