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Economics Dynamics Problems 101

# Economics Dynamics Problems 101 - y t 1 − y t = y t Given...

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CHAPTER 3 Discrete dynamic systems 3.1 Classifying discrete dynamic systems Adiscretedynamicsystemisasequenceofnumbers, y t ,thataredefinedrecursively, i.e., there is a rule relating each number in the sequence to previous numbers in the sequence; we denote such a sequence { y t } . A first-order discrete dynamic system is a sequence of numbers y t for t = 0 , 1 , 2 . . . such that each number after the first is related to the previous num- ber by the relationship y t + 1 = f ( y t ) (3.1) We shall refer to (3.1) as a recursive equation . The sequence of numbers given by the relationship y t + 1 y t + 1 y t = g ( y t ) (3.2) we shall refer to as a first-order difference equation . 1 Examples are (i) y t + 1 = 2 + y t implies y t + 1 y t = 2 (ii) y t + 1 = 2 y t implies
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Unformatted text preview: y t + 1 − y t = y t Given the discrete dynamic system y t + 1 = f ( y t ), then if f ( y t ) is linear, the system is said to be linear ; if f ( y t ) is nonlinear then the system is said to be nonlinear . Examples (i) y t + 1 = 2 + 3 y t linear (ii) y t + 2 − 2 y t + 1 − 3 y t = 5 linear (iii) y t + 1 = 3 . 2 y t (1 − y t ) nonlinear (iv) y t + 1 = ry t ln( k / y t ) nonlinear Consider the general discrete dynamic system y t + 1 = f ( t , y t ) (3.3) ±or example y t + 1 = th ( y t ) 1 Often equation (3.1) and equation (3.2) are each referred to as a difference equation....
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