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Unformatted text preview: Systems of ﬁrstorder differential equations
In a twoequation system, the programme will produce a direction ﬁeld plot by
default if the system is a set of autonomous equations. Since we require only the
solution curves, then we include an option that indicates no arrows.
To illustrate the points just made, consider the Van der Pol model, equation (4.28), a simple set of instructions to produce a Maple plot similar to ﬁgure 4.30
is
phaseportrait(
[D(x)(t)=y(t), D(y)(t)=(1x(t)^2)*y(t)x(t)],
[x(t),y(t)], t=0..10,
[ [x(0)=0.5,y(0)=0.5],[x(0)=0.5,y(0)=4] ],
stepsize=.05
linecolour=blue,
arrows=none,
thickness=1); Producing more solution curves in Maple is just a simple case of specifying more
initial conditions. For instance, a Maple version of ﬁgure 4.36(b) can be produced
with the following instructions:
with(DEtools):
phaseportrait(
[D(x)(t)=1y(t),D(y)(t)=x(t)^2+y(t)^2],
[x(t),y(t)], t=0..3,
[[x(0)=2,y(0)=1],[x(0)=1.75,y(0)=1],
[x(0)=1.5,y(0)=1],[x(0)=1,y(0)=0],
[x(0)=1,y(0)=1], [x(0)=0.5,y(0)=1],
[x(0)=0,y(0)=1], [x(0)=1.25,y(0)=0],
[x(0)=0.5,y(0)=1, [x(0)=1,y(0)=1]],
x=2..2, y=1..3,
stepsize=.05,
linecolour=blue,
arrows=none,
thickness=1); Trajectories for threedimensional plots are also possible with Maple. Consider
once again the Lorenz curve, given in equation (4.34), with parameter values
σ = 10, r = 28 and b = 8/3. We can construct a threedimensional trajectory from
the initial point (x0, y0, z0) = (5, 0, 0) using the following input instructions:
with(DEtools):
DEplot3d(
[diff(x(t),t)=10*(y(t)x(t)),
diff(y(t),t)=28*x(t)y(t)x(t)*z(t),
diff(z(t),t)=x(t)*y(t)(8/3)*z(t)],
[x(t),y(t),z(t)], t=0..30,
[[x(0)=5,y(0)=0,z(0)=0]],
stepsize=.01,
linecolour=BLACK,
thickness=1); 193 ...
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 Fall '11
 Dr.Gwartney
 Economics

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