Casio fx-CG50 field manual for IB Math
Phase portrait for a coupled system
Use sequence mode to plot one component of a coupled system against the other.
Plot a phase portrait for the coupled system x′=y, y′=−x−0.4y, x(0)=1, y(0)=0 using step size h=0.2.
From the MAIN MENU open Recursion. Press (TYPE), then (an+1) so both sequences are defined recursively.
Let an store x and bn store y. Sequence references come from the (n.an···) menu: inserts an and inserts bn. Enter an + 0.2bn on the first line and bn + 0.2(-an - 0.4bn) on the second, pressing after each.
Press (SET). Use Start = 0 and End = 40 so the trajectory runs long enough to show its shape, and set the initial terms a0 = 1, b0 = 0.
EXIT back and press (TABLE) to generate the table. Before plotting, set a known window: (V-Window) → (INITIAL), then . PHASE draws into whatever window is current, so a leftover one from another graph can put the trajectory off screen. Now press (PHASE): the calculator plots bn against an as points, the first sequence across and the second up.
The trajectory spirals inward toward the origin.
Your turn
Work each one on your calculator, then check the answer.
- 1
Plot the phase portrait of x′=y, y′=−2x−0.5y starting from (1.5,0) with h=0.2. Describe the behaviour.
- 2
Plot the phase portrait of x′=y, y′=x starting from (0.1,0.15) with h=0.2. Describe the behaviour.
- 3
Plot the phase portrait of x′=y, y′=−x+0.2y starting from (0.5,0) with h=0.2. Describe the behaviour.