# DiffEq Phase Plane Support Notes

DiffEq PhasePlane is one of the dynamical systems exploration tools in MacMath 9.2. MacMath 9.2 was developed locally by Professor John Hubbard and Beverly West, and has been commerically sold by Springer Verlag. It specializes in exploring two dimensional autonomous systems of differential equations dx/dt=f(x,y), dy/dt=g(x,y).

Related programs include DiffEq and DiffEq, 3D Views which specialize respectively in the one and three dimensional cases.

The program Planar Systems in MacMath 9.6 is an incomplete update of DiffEq, Phase Plane, with many nice additional features but at times incomplete support for basics like printing.

### Basic Operation

of the program involves entering a system of equations and then clicking at various locations to see the trajectories going through those points. By default trajectories first continue forward until a window boundary is encountered. Then  a trajectory backwards (negative time) from the initial point is traced. Clicking the mouse while a trajectory segment is being traversed terminates that direction of traversal immediately.

DiffEq Phase Plane gives very rich graphical displays but has no support for providing large amounts of numerical data.

The default stepsize is too large, often producing very jagged pictures. So stepsize should generally be reduced from under the Change menu.

Axes and Tickmarks
Equations
are also changed from below the Change menu.
Numerical Integration
methods available under the Change menu are Euler, Heun (midpoint Euler) , and Runge-Kutta.
Rescaling
Slope Marks
are available from the Settings menu. The program Planar Systems is notable for improving the algorithm used in drawing these.
Step Sizes
by default are very large because the program was written many years ago. Choose something smaller from below the Change menu.
Special Functions Supported include
• Option-p for pi.
• abs(),  sqrt(),  sgn(),  ln(),  exp(),  atan(),  and floor().
• ^ is exponentiation.
Zooming
is accomplished by selecting Blow Up fromt the Task menu and then selecting a rectangle while holding down the mouse button.
Equilibrium Points
can be found by selecting Locate Singularity from the Tasks menu and then clicking to set a starting point for the search.

### Location

In the folder  MacMath 9.2 toplevel on all Macintoshes in the lab.

### Printing

Most MacMath programs have some difficulty printing in the lab. Screen Dumps should always be viewed as the backup technique when difficulties are encountered. Turning background printing off may also help this program.

### Documentation

A manual (yellow) for MacMath 9.2 is available in the lab bookcase.