178Economic Dynamics(ii) All trajectories remain bounded but do not approach the criticalpoint ast→∞. This occurs when tr(A)2<4det(A) andr=βiands=−βi(α=0).(iii) At least one of the trajectories tends to inFnity ast. This occurswhen(a) tr(A)2>4det(A),r>0 ands>0orr<0 ands>0(b) tr(A)2<4det(A),r=α+βi,s=α−βiandα>0.4.9 Stability/instability and its matrix specifcationHaving outlined the methods of solution for linear systems of homogeneous au-tonomous equations, it is quite clear that the characteristic roots play an importantpart in these. Here we shall continue to pursue just the two-variable cases.±or the system˙x=ax+by˙y=cx+dywhereA=±abcd²andA−λI=±a−λb−λ²we have already shown that a unique critical point exists ifAis nonsingular, i.e.,det(A)±=0 and thatr,s=tr(A)²³tr(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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