Economics Dynamics Problems 173

Economics Dynamics Problems 173 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 157 Then systems 4.4 and 4.5 are simply specific examples of the homogeneous linear system x = Ax ˙ (4.13) while system 4.6 is a specific example of the nonhomogeneous linear system x = Ax + b ˙ (4.14) For linear homogeneous systems, if the determinant of A is not zero, then the ∗ ∗ only solution, the only fixed point, is x∗ = 0, i.e., (x1 = 0 and x2 = 0). On the other hand, for nonhomogeneous linear systems, the equilibrium can be found, so long as A is nonsingular, from 0 = Ax∗ + b x∗ = −A−1 b (4.15) When considering the issue of stability/instability it is useful to note that linear nonhomogeneous systems can always be reduced to linear homogeneous systems in terms of deviations from equilibrium if an equilibrium exists. For x = Ax + b ˙ 0 = Ax∗ + b subtracting we immediately have in deviation form x = A(x − x∗ ) ˙ (4.16) ∗ ∗∗ (x1 , x2 ). which is homogeneous in terms of deviations from the fixed point x = There will be no loss of generality, therefore, if we concentrate on linear homogeneous systems. The matrix A is of particular importance in dealing with stability and instability. Two important properties of such a square matrix are its trace, denoted tr(A), and its determinant, denoted det(A), where4 tr(A) = a11 + a22 det(A) = a11 a21 a12 = a11 a22 − a12 a21 a22 (4.17) It should be noted that both the trace and the determinant are scalars. The matrix A is nonsingular if det(A) = 0. There is another property of the matrix A that arises for special linear systems. Consider the following general linear system y = Ax This can be viewed as a transformation of the vector x into the vector y. But suppose that x is transformed into a multiple of itself, i.e., y = λx, where λ is a scalar of proportionality. Then Ax = λx or (A − λI)x = 0 4 See Chiang (1984) or any book on linear algebra. (4.18) ...
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