Phase Portrait

To create a good picture of the phase plane we need to select a few more initial conditions. This is a process of trial and error. In general you should select the equilibrium solutions (or a few if there are an infinite number of these solutions) and other initial conditions that show all possible qualitative behavior of the solutions.

> IC1 := [ x(0) = 0, y(0) = 0 ];

> IC2 := [ x(0) = 1, y(0) = 0 ];

> IC3 := [ x(0) =-1, y(0) = 0 ];

> IC4 := [ x(0) = 0, y(0) = 1 ];

> IC5 := [ x(0) = 0, y(0) =-1 ];

> IC6 := [ x(0) = 0, y(0) = 0.5 ];

> IC7 := [ x(0) = 0, y(0) =-0.5 ];

> IC8 := [ x(0) = 0, y(0) = 1.5 ];

> IC9 := [ x(0) = 0, y(0) =-1.5 ];

> IC10 := [ x(0) = 0.5, y(0) =-1 ];

> IC11 := [ x(0) = -0.5, y(0) =-1 ];

> IC12 := [ x(0) = 1.5, y(0) =-1 ];

> IC13 := [ x(0) = -1.5, y(0) =-1 ];

> IC := [ IC1, IC2, IC3, IC4, IC5, IC6, IC7, IC8, IC9, IC10, IC11, IC12, IC13 ]:

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To see the role of time in the solution, choose a variety of domains - some short (less than 1), others medium (between 1 and 10), and some large (larger than 10). What difference does this make in the phase plane?

> DOMAIN := t = 0 .. 0.5 ;

> DEplot( MODEL, VAR, DOMAIN, IC, RANGE,

> scene=[ x, y ], arrows=NONE, linecolor=BLUE,

> title = `Phase Portrait for Problem 32 (Section 2.2)` );

Do you see all thirteen curves in the phase plane?

Vary the domain of the phase portrait to get a feel for time in the phase portrait.

What can you say about the period of the different solutions?

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