Economics Dynamics Problems 196

Economics Dynamics Problems 196 - 180 Economic Dynamics...

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180 Economic Dynamics Table 4.1 Stability properties of linear systems Matrix and eigenvalues Type of point Type of stability tr( A ) < 0 , det( A ) > 0 , tr( A ) 2 > 4det( A ) Improper node Asymptotically stable r < s < 0 tr( A ) > 0 , det( A ) > 0 , tr( A ) 2 > 4det( A ) Improper node Unstable r > s > 0 det( A ) < 0 Saddle point Unstable saddle r > 0 , s < 0or r < 0 , s > 0 tr( A ) < 0 , det( A ) > 0 , tr( A ) 2 = 4det( A ) Star node or proper node Stable r = s < 0 tr( A ) > 0 , det( A ) > 0 , tr( A ) 2 = 4det( A ) Star node or proper node Unstable r = s > 0 tr( A ) < 0 , det( A ) > 0 , tr( A ) 2 < 4det( A ) Spiral node Asymptotically stable r = α + β i , s = α β i ,α< 0 tr( A ) > 0 , det( A ) > 0 , tr( A ) 2 < 4det( A ) Spiral node Unstable r = α + β i , s = α β i ,α> 0 tr( A ) = 0, det( A ) > 0 Centre Stable r = β i , s =− β i in predatory–prey population models. If a system has closed orbits that other trajectories neither approach nor diverge from, then the closed orbits are said to be stable. Geometrically, we have a series of concentric orbits, each one denoting
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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.

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