Info

rdG

— -

Vi +

—- 1

— v.

+

-—Vi

3,vi

Byh '

1 9vi ' 1

'dy2 "

where «11 =B F/'d\\ ,n 12 — 9 F/dy 2, and so on. The stability property of the steady-state point (v 1, y2) in a linear system of differential equations such as this is determined directly from Ihe coefficient matrix. The coefficient matrix for equations (24.29) and (24.30) is

3yi ih'2

The following theorem states the relevance of this matrix.

(24.311

Theorem 24.7 If the determinant of the coefficient matrix in (24.31) is nonzero, the qualitative behavior of the trajectories of die nonlinear system in definition 24.5 in the neighborhood of its steady-state point (j,. y2) is the same as that of the trajectories of the linear homogeneous system consisting of (24.29) and (24.30). except if the linear system is acenter. In that case the nonlinear system could be acenterora focus.

Theorem 24.7 implies that the local stability properly of a nonlinear differential equation system can be determined directly from the coefficient matrix of its linearized form. Thus, the stability results summarized in table 24.1 can be used to determine the qualitative behavior of the trajectories of a nonlinear system around its steady slate. For example, if the determinant of A in equation (24.31) is negative. Ihe steady state of the nonlinear system is a saddle point, so the trajectories display the properties of a saddle point, at least locally. The qualifier thai the results are locally valid is a direct consequence of the fact that the linearization of the nonlinear system is only valid in the local neighborhood of the steady state.

Example 24.16 Determine the behavior of the trajectories of the following nonlineai differential equation system in the neighborhood of the steady state:

_VZ = Y| - C V2 where a and <■ are positive constants and 0 < h < I. Solution

The steady state is found by setting y, = 0 = y2. This gives y:(uc - by: ) =0

The solutions are y2 = 0 and

The solutions for >'i are y, = ry2. We shall focus our attention on die strictly positive values of the two steady states. The coefficient matrix of the first-order linear approximation to the nonlinear system is

The determinant of the coefficient matrix is -ac + (/»—! )by2~2 and is negative because 0 < b < 1 and a, c > 0, and because y2 > 0. Thus we know immediately that the strictly positive steady state is a saddle-point equilibrium. We therefore conclude that the behavior of the nonlinear system is that of a saddle-point equilibrium in the neighborhood of the steady-state point.

Determining the global behavior of a nonlinear system can be a much more difficult task. However, for many problems encountered in economics, and all of the problems encountered in this book, the nonlinear differential equation systems are sufficiently well behaved that the global behavior of the nonlinear system can be determined using phase diagram analysis in conjunction with theorem 24.7. ■

Construct the phase diagram for the nonlinear differential equation system in example 24.16.

The procedure is the same as lor a linear system except that qualitative graphing techniques may have to be substituted for explicit graphing. Begin by analyzing the motion of yi.

Selling v, =0 gives

Example 24.17 A Nonlinear Phase Diagram

Solution

as the yi isocline. To determine the shape of this equation, it is helpful to determine its slope and intercepts if possible. As yt -»• 0. y2 —» cc because the exponent is negative (0 < b < 1). In addition, as y, —► oo. y> —«- 0. Thus the graph is asymptotic to both axes. The slope is dn 1 (a V1-«/»-1» ^ = <()

Figure 24.12 shows the yi isocline labeled yt = 0.

Figure 24.12 Phase diagram I'nr example 24. <>

The motion of vi at points off the isocline is determined as follows. We calculate

Asa result vi is increasing at points to the right of the isocline. Since the differential equation is monotonically increasing in Vi, we know that yj must continue to be increasing everywhere to the right of the isocline. By similar reasoning, y, < Otothe left of the isocline. The appropriate horizontal arrows are marked on figure 24.12.

Now determine the motion of yi. Since v2 is a linear differential equation, this is relatively straightforward: the y; isocline is the line = yi/c and yj is decreasing above and increasing below the isocline.

Figure 24-. 12 shows ihe phase diagram. Wc have determined that the stead) state is a saddle-point equilibrium, so we know that in the neighborhood of the steady state there is a saddle path and that all other trajectories diverge. The phase diagram analysis shows that globally, the behavior of trajectories is consistent with die presence of a saddle point. I

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