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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 < 0 or 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 a closed trajectory. In answering the question: ‘When do limit cycles occur?’ we draw on the Poincar´e–Bendixson theorem . This theorem is concerned with a bounded re-
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