Economics Dynamics Problems 128

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
112 Economic Dynamics Figure 3.13. characteristic root and | r | > | s | , then y n = r n ± c 1 + c 2 ² s r ³ n ´ Since | s / r | < 1, then ( s / r ) n 0as 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 inFnity 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-
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

Ask a homework question - tutors are online