Dawnbridge

TI-Nspire CX II field manual for IB Math

AI HL only · 42 of 42

42

Phase portrait for a coupled system

AI HL

Use sequence mode to plot one component of a coupled system against the other.

Goal

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.

1

On a Graphs page, press → 3: Graph Entry/Edit → 7: Sequence → 1: Sequence. Let u1 store ​x​ and u2 store ​y: enter u1(n)=u1(n-1)+0.2·u2(n-1) with initial term 1, then repeat for u2(n)=u2(n-1)+0.2(-u1(n-1)-0.4·u2(n-1)) with initial term 0.

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u1(n)=u1(n−1)+0.2⋅u2(n−1)Initial Terms:=11≤n≤99 nstep=1
u2(n)=u2(n−1)+0.2(−u1(n−1)−0.4⋅u2(n−1))Initial Terms:=01≤n≤99 nstep=1
2

Press → 4: Window/Zoom → 1: Window Settings and set both axes to run from ​−1.2​ to ​1.2, with both scale fields 0.2, moving between fields with .

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Window Settings
XMin:
-1.2
XMax:
1.2
XScale:
0.2
YMin:
-1.2
YMax:
1.2
YScale:
0.2
OKCancel
3

Add the phase plot: press → 3: Graph Entry/Edit → 7: Sequence → 2: Custom, and define the plot as (u1(n), u2(n)) so ​y​ is plotted against ​x.

Result

The trajectory spirals inward toward the origin.

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Your turn

Work each one on your calculator, then check the answer.

  1. 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. 2

    Plot the phase portrait of ​x′=y,  ​y′=x​ starting from ​(0.1,0.15)​ with ​h=0.2. Describe the behaviour.

  3. 3

    Plot the phase portrait of ​x′=y,  ​y′=−x+0.2y​ starting from ​(0.5,0)​ with ​h=0.2. Describe the behaviour.