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Economics Dynamics Problems 128

Economics Dynamics Problems 128 - tude Identical real roots...

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112 Economic Dynamics Figure 3.13. characteristic root and | r | > | s | , then y n = r n c 1 + c 2 s r n Since | s / r | < 1, then ( s / r ) n 0 as n → ∞ . Therefore, lim n →∞ y n = lim n →∞ c 1 r n There are six different situations that can arise depending on the value of r . (1) r > 1 , then the sequence { c 1 r n } diverges to infinity and the system is unstable (2) r = 1, then the sequence { c 1 r n } is a constant sequence (3) 0 r < 1 , then the sequence { c 1 r n } is monotonically decreasing to zero and the system is stable (4) 1 < r 0 , then the sequence { c 1 r n } is oscillating around zero and con- verging on zero, so the system is stable (5) r = − 1 , then the sequence { c 1 r n } is oscillating between two values (6) r < 1 , then the sequence { c 1 r n } is oscillating but increasing in magni-
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Unformatted text preview: tude. Identical real roots If the roots are real and equal, i.e., r = s , then the solution becomes y n = ( c 1 + c 2 ) r n = c 3 r n But if c 3 r n is a solution, then so is c 4 nr n (see Chiang 1992, p. 580 or Goldberg 1961, p. 136 and exercise 14), hence the general solution when the roots are equal is given by y n = c 3 r n + c 4 nr n We can now solve for c 3 and c 4 given the two initial conditions y and y 1 y = c 3 r + c 4 (0) r = c 3 y 1 = c 3 r + c 4 (1) r = ( c 3 + c 4 ) r...
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