DiffEq, 3D Views

John H. Hubbard and Beverly H. West
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John H. Hubbard: Cornell University, Department of Mathematics
Beverly H. West: Cornell University, Department of Mathematics

Chapter 10 in MacMath 9.0, 1992, pp 67-75 from Springer

Abstract: Abstract This program draws three-dimensional graphs and relevant planar graphs for either of the following: An autonomous system of differential equations of the form $$\frac{\text{dx}}{\text{dt}}=f\text{(x,y,z)}, \;\;\;\ \frac{\text{dy}}{\text{dt}}=g\text{(x,y,z)},\;\;\;\text{and}\;\;\;\frac{\text{dz}}{\text{dt}}=h\text{(x,y,z)}$$ In this case the trajectories are drawn in xyz-space and the planar views are xy, xz, yz. The program also locates and analyzes singularities in xyz-space. (Three more planar views: xt, yt, and zt are not visible on the screen, but you can ask for the printouts to show them.) A nonautonomous system of differential equations of the form $$\frac{\text{dx}}{\text{dt}}=f\text{(t,x,y)} \;\;\;\text{and}\;\;\; \frac{\text{dy}}{\text{dt}}=g\text{(t,x,y)}$$ Here the trajectories are drawn in txy-space and the planar views are xy, tx, ty. In this case the program does not locate and analyze singularities since it is setting z = t ( hence dz/dt = 1 and there can be no 3D singularities). However it allows you the choice of investigating Poincaré sections provided the nonautonomous 2D differential equations are periodic in t. This will be explained at the end of this section.

Keywords: Planar View; Nonautonomous System; Poincare Section; Small Plane; Information Window (search for similar items in EconPapers)
Date: 1992
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DOI: 10.1007/978-1-4684-0390-9_10

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