Economics Dynamics Problems 194

Economics Dynamics Problems 194 - 178 Economic Dynamics...

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178 Economic Dynamics (ii) All trajectories remain bounded but do not approach the critical point as t →∞ . This occurs when tr( A ) 2 < 4det( A ) and r = β i and s =− β i ( α = 0). (iii) At least one of the trajectories tends to inFnity as t . This occurs when (a) tr( A ) 2 > 4det( A ), r > 0 and s > 0or r < 0 and s > 0 (b) tr( A ) 2 < 4det( A ), r = α + β i , s = α β i and α> 0. 4.9 Stability/instability and its matrix specifcation Having outlined the methods of solution for linear systems of homogeneous au- tonomous equations, it is quite clear that the characteristic roots play an important part in these. Here we shall continue to pursue just the two-variable cases. ±or the system ˙ x = ax + by ˙ y = cx + dy where A = ± ab cd ² and A λ I = ± a λ b λ ² we have already shown that a unique critical point exists if A is nonsingular, i.e., det( A ) ±= 0 and that r , s = tr( A ) ² ³ tr( A ) 2 4det( A ) 2 (4.26) ±urthermore, if: (i) tr( A ) 2 > 4det( A ) the roots are real and distinct (ii) tr( A ) 2 = 4det( A ) the roots are real and equal (iii) tr( A ) 2 < 4det( A ) the roots are complex conjugate.
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