Unformatted text preview: 5.1 Equlllbdtnn Point Analysis 3Zl (f) (3) Solutions tend away from the origin along the yaxis in both systems. In the nonlinear
system, solutions approach the origin along the curve y = x3 which is tangent to the x
axis. For the linearized system. solutions tend to the origin along the xaxis. Near the
origin, the phase portraits are almost the same. 6. Warning: 'lhe system stated in the exercise in the ﬁrst two printings has a typo. The coefﬁcient to
the xyterm for dy/dt should be 2, not  I. This answer assumes that that coefﬁcient is —2. (a) The Jacobian is
2 — 2x  y —x
—2y 3 — 2y — 2x ‘ Evaluating at (0. 0). we get the linearized system dx
3 — 2x
dy
2; — 3y.
Evaluating at (0. 3), we get
‘2 — 1.
d: —
dy
E;  "GX  3y o
and evaluating at (2, 0). we get
dx
_ = 2x _
dz 2y
‘2 _
dt  (b) At (0, 0), the eigenvalues are 2 and 3, so (0, 0) is a source. At (0, 3), eigenvalues are —l
and —3, so (0. 3) is a sink. At (2, 0). the eigenvalues are —2 and — l . so (2. 0) is a sink. ...
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 Spring '08
 JU

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