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Unformatted text preview: Systems of ﬁrstorder 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 wellknown 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 secondorder differential equation can be reduced to a system of ﬁrstorder 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.
 Fall '11
 Dr.Gwartney
 Economics

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