Phase planes

This module deals with autonomous systems of two equations, that is, systems of equations of the form

$$\frac{dx}{dt} = f(x,y), \quad \frac{dy}{dt} = g(x,y) .$$

The module shows the phase portrait associated with such a system, and it computes and plots numerical solutions of the system in this phase plane. It can also plot $x$ and $y$ versus $t$, either individually or together.

After selecting Phase planes from the Iode main menu, the Phase planes window will open up. It shows a plot of the phase portrait for the system, with the system itself written across the top of the plot.

When you first enter the module, you will find Iode has already chosen a system and some option settings. Next we explain how to change these settings and plot solutions.

Controls

These are the almost same as in the Direction fields module. See Chapter 1. The pop-up menu that appears upon right-clicking in the graph has one additional entry, for continuing plots.

Equation menu

This menu is very similar to the Equation menu in the Direction fields module. See Chapter 1. Below are noted a few differences.

• Enter differential equation: prompts you to enter the right hand side functions $f(x,y)$ and $g(x,y)$ for the system, using valid Matlab syntax (see Appendix C). You do not input the letters f or g: you just input the expressions that you want on the right hand side of the system.
  Remark 2.1 (Note for advanced users)   Iode can also handle nonautonomous systems, in which the expressions for $f$ and $g$ depend also on $t$. Iode will correctly compute numerical solutions for such systems, but will not display the full phase portrait (which is 3-dimensional); the phase portrait displayed is the ``slice'' at $t=t_0$. Also, see next item.

• Change plot duration: the plot of a solution curve can be thought of as tracing the path of a moving particle from time $t=t_{min}$ to time $t=t_{max}$. The time $t_0$ for the initial condition should fall somewhere between $t_{min}$ and $t_{max}$.

  The default setting in Iode is to take $t_{min}=t_0=0$ and $t_{max}=2\pi$, meaning the particle flows forwards from the initial point, starting at time 0, for a duration of $2\pi$ time units.

  If you want to plot more of the solution curve, just increase $t_{max}$. If you want to plot backwards in time from the initial point, choose $t_0=t_{max}$. To plot both backwards and forwards from the initial point, choose a value of $t_0$ strictly between $t_{min}$ and $t_{max}$.

Options menu

This is identical to the Options menu in the Direction fields module (see Chapter 1), except for one new option.

• Show coordinate maps: causes a new window to open up, containing plots of $x$ versus $t$, $y$ versus $t$, and $x$ and $y$ versus $t$. If you want to look at the 3-dimensional plot of $x$ and $y$ versus $t$ from a different angle, you can rotate it by dragging the mouse across the 3D plot.

Finally, as opposed to the behavior in the Direction fields module, zooming out is not exponential in the number of clicks because the duration of solution plots is not controlled by the size of the display.