Economics Dynamics Problems 190

Economics Dynamics Problems 190 - orientation, the critical...

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

View Full Document Right Arrow Icon
174 Economic Dynamics Figure 4.23. the general solution as x = [ c 1 e rt v + c 2 e rt v 2 + c 2 e rt t v ] = [( c 1 v + c 2 v 2 ) + c 2 t v ] e rt = u e rt Then u = ( c 1 v + c 2 v 2 ) + c 2 t v which is a vector equation of a straight line which passes through the point c 1 v + c 2 v 2 and is parallel to v . Two such points are illustrated in Fgure 4.23, one at point a ( c 2 > 0) and one at point b ( c 2 < 0). We shall not go further into the mathematics of such a node here. What we can do, however, is highlight the variety of solution paths by means of two numerical examples. The Frst, in Fgure 4.24, has the orientation of the trajectories as illus- trated in Fgure 4.23, while Fgure 4.25 has the reverse orientation. Whatever the
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: orientation, the critical point is again an improper node that is stable. Had r > , then the critical point would be an improper node that is unstable. Case 4 (Complex roots, α ±= and β > ) In this case we assume the roots λ = r and λ = s are complex conjugate and with r = α + β i and s = α − β i , and α ±= 0 and β > 0. Systems having such complex roots can be expressed ˙ x = α x + β y ˙ y = − β x + α y...
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