Economics Dynamics Problems 209

# Economics Dynamics Problems 209 - Systems of ﬁrst-order...

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Unformatted text preview: Systems of ﬁrst-order differential equations In a two-equation 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)=(1-x(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)=1-y(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 three-dimensional 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 three-dimensional 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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