The program is able to display the solution curves (or "orbits") for initial value ODE of first or second order in one, two, or three dimensions. There is an ODE submenu of the Category menu to choose the order and number of dependent variables.
When you have chosen a particular category of ODE and selected either a particular canned example, or set up your own using the User Defined... feature, choosing the item "ODE Settings..." in the Settings menu will allow you to set the various data the program needs to compute and display an orbit, namely:
(Normally, for two-dimensional systems, the orbit is drawn dotted, with a constant time interval between dots. This gives a valuable visual clue concerning the velocity at which the orbit is traced out, but if you wish to turn this feature off, just set Dot Spacing to zero.)
Choosing Create from the Main menu will then
display the solution of the currently selected differential equation.
The latter should have been selected before
setting the above data by either:
1) Choosing one of the pre-programmed ODE
from the Main menu, or
2) Choosing "User Defined..." and then, in the dialog box that comes
up entering formulas for Xdot, Ydot (and, in the 3-dimensions, Zdot),
(i.e., for dx/dt, dy/dt, dz/dt), as functions of x,y,z,t, and the
usual nine parameters, aa,..., ii.
(When you select a canned ODE from the Main menu, then as usual, the program will choose reasonable default values of the initial time, the time-span, the step-size, and initial values of x,y,z and immediately draw an orbit.)
In any of the ODE categories the main display shows the evolution of an orbit in the phase space. By default, (except for a first order equation in one variable) the program also shows projections of the orbit on the various coordinate axes (using different colors to distinguish the projections). This display occurs in a second pane of the graphics window that opens automatically below the main pane. This pane can be hidden by choosing "No" from the Project ODE Orbits submenu of the View menu. There is a rectangular button at the right edge of the screen where the two panes meet. If you press on this button, the button itself will disappear and be replaced by a horizontal line. Drag the horizontal line to where you would like the new boundary between panes and release the mouse. (At least twenty percent of the total screen height must be devoted to each pane.)
For a first order ODE category in one or two variables and a second order ODE in one variable the program by default displays the direction field of the vector field defining the ODE. The display of the direction field can be turned off with the Show Direction Fields? submenu. (For the case of a first order ODE in three variables, showing the direction field is also possible when the orbit is being displayed in stereo vision. For second order ODE in three variables, for the special case of a charged particle in a magnetic field, it is possible to display the magnetic field using the Show Direction Fields? menu. But note, it is the direction of the magnetic field, not the Lorentz force that is displayed.)
Note that, even though all the canned ODE are autonomous (i.e., the "right-hand sides" of the ODE are not functions of the time t) the Runge-Kutta integration is set up to handle non-autonomous ODE too, and you may use t in the formulas of the User Defined... dialog box. In this case, you should turn of the display of the Direction Field, which will be pretty meaningless. (If you do not, you will see the direction field at time zero.)
It is fairly easy to do a rough "phase space analysis" by keeping the other data fixed and varying the initial values. This is particularly easy in the 2-dimensional case. There, if you select "IC by mouse" from the main menu, you will be able to click the mouse at a point on the screen, and the program will figure out the x and y coordinates of the mouse location and use these as the initial conditions for an orbit that it will immediately draw. (To speed things up even more, you can type Option-Command-C instead of selecting "IC by mouse" from the menu).
There is an ODE control panel now opens by default just below the Graphics Window. This has buttons to do more easily things you could do with the menus (Create, Erase, Continue, double or half the scale, and bring up the dialog to set initial conditions, step-size, time-span, and dottedness. In addition there are buttons for single-stepping the ODE forward or backward, and there is a read-out of the current time, position, and velocity. This control panel can be closed with the close box and re-opened using the Show ODE Controls submenu of the View menu