Economics Dynamics Problems 197

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

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.

Ask a homework question - tutors are online