Phase Lines

Select an autonomous ODE using the popdown menu [v] at bottom center.

The main graphing window displays the direction field and any equilibrium solutions. Stable equilibria are indicated in green, unstable in red, and semistable in cyan. When the cursor is placed over the graphing window a yellow vector parallel with the vector field is displayed, and the values of \(t, y,\) and \(\dot{y}\) are read out. When the mousekey is depressed, the solution though that point is displayed. When the mousekey is released, the solution remains. Solution curves can be cleared by pressing the [Clear] key.

Selecting the "\(\dot{y}\) as a function of \(y\)" dropdown reveals a graph of \(\dot{y}\) as a function of \(y\) at lower left. The zeros of \(\dot{y}\) are marked by green, red, or cyan disks, depending on the character of the critical point.

Each of the menu choices includes a parameter a which can be set using a slider at lower right. A toggle clears or displays the slope field.

Selecting the Phase Line checkbox displays the phase line. On it are drawn critical points (in green, red, or cyan) and a few direction arrows. As solutions are drawn on the graphing window, the corresponding trajectory is traced out on the phase line. Clicking on the phase line at a value y causes the solution through \((0,y)\) to be drawn on the graphing window. Clicking the [Clear] key under the phase line clears the trajectories.

Selecting the Bifurcation Diagram checkbox reveals the bifurcation diagram, which displays the locus of critical points, colored green, red, or cyan. The phase line for the selected value of \(a\) is also displayed.