Economics Dynamics Problems 207

# Economics Dynamics Problems 207 - Systems of ﬁrst-order...

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Unformatted text preview: Systems of ﬁrst-order differential equations sol1=NDSolve[eq1,var,trange] sol2=NDSolve[eq2,var,trange] graph1=ParametricPlot[Evaluate[{x[t],y[t]} /.sol1], {t,0,20},PlotPoints->500, DisplayFunction->Identity]; graph2=ParametricPlot[Evaluate[{x[t],y[t]} /.sol2], {t,0,20},PlotPoints->500, DisplayFunction->Identity]; Show[{graph1,graph2},AxesLabel->{` ’,` ’}, `x’ `y’ DisplayFunction->\$DisplayFunction]; The more trajectories that are required the more cumbersome these instructions become. It is then that available packages, such as the one provided by Schwalbe and Wagon (1996), become useful. For instance, ﬁgure 4.36(b) can be produced using the programme provided by Schwalbe and Wagon with the following set of instructions: PhasePlot[{x’[t]==1-y[t],y’[t]==x[t]^2+y[t],^2}, {x[t],y[t]},{t,0,3},{x,-2,2},{y,-1,3}, InitialValues->{{-2,-1},{-1.75,-1},{-1.5,-1}, {-1,0},{-1,-1},{-0.5,-1},{0,-1},{-1.25,0}, {0.5,-1},{1,-1}}, PlotPoints->500, ShowInitialValues->False, DirectionArrows->False, AspectRatio->1, AxesLabel->{x,y}] When considering just one trajectory in the phase plane, the simple instructions given above can sufﬁce. For instance, consider the Lorenz curve, given in equation (4.34), with parametric 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: eqs:={x’[t]==10(y[t]-x[t]) ,y’[t]==28x[t]-y[t]-x[t]z[t], z’[t]==x[t]y[t]-(8/3)z[t], x[0]==5,y[0]==0,z[0]==0} var:={x,y,z} lorenzsol=NDSolve[eqs,var,{t,0,30},MaxSteps->3000] lorenzgraph=ParametricPlot3D[ Evaluate[x[t],y[t],z[t]} /.lorenzsol], {t,0,30},PlotPoints->2000,PlotRange->All]; The resulting phase line is shown in ﬁgure 4.37. This goes beyond the possibilities of a spreadsheet, and ﬁgure 4.37 should be compared with the three twodimensional plots given in ﬁgure 4.35. 191 ...
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