Economics Dynamics Problems 197

# Economics Dynamics Problems 197 - Systems of ﬁrst-order...

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Unformatted text preview: Systems of ﬁrst-order differential equations 181 be stable. This means that if there is only one limit cycle satisfying the theorem, it must be stable. Example 4.15 The following well-known example has a limit cycle composed of the unit circle (see Boyce and DiPrima 1997, pp. 523–7): x = y + x − x(x2 + y2 ) y = −x + y − y(x2 + y2 ) Utilising the VisualDSolve package within Mathematica, we can show the limit cycle and two trajectories: one starting at point (0.5,0.5) and the other at point (1.5,1.5). The input instructions are: ’ In[2]:= PhasePlot [{x [t] == y[t] + x[t] - x[t] (x[t]ˆ2 + y[t]ˆ2), y’ [t] == -x[t] + y[t] - y[t] (x[t]ˆ2 + y[t]ˆ2)}, {x[t], y[t]}, {t, 0, 10}, {x, -2, 2}, {y, -2, 2}, InitialValues -> {{0.5, 0.5}, {1.5, 1.5}}, ShowInitialValues -> True, FlowField -> False, FieldLength -> 1.5, FieldMeshSize -> 25, WindowShade -> White, FieldColor -> Black, Nullclines -> True, PlotStyle -> AbsoluteThickness [1.2], InitialPointStyle -> AbsolutePointSize [3], ShowEquilibria -> True, DirectionArrow -> True, AspectRatio -> 1, AxesLabel -> {x, y}, PlotLabel -> ‘ ‘Unit Limit Cycle”]; which produces ﬁgure 4.29 showing a unit limit cycle. Example 4.16 (Van der Pol equation) The Van der Pol equation is a good example illustrating an asymptotically stable limit cycle. It also illustrates that a second-order differential equation can be reduced to a system of ﬁrst-order differential equations that are more convenient for Figure 4.29. ...
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## This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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