Section 5.1: 10
Section 5.2: 4, 9, 10
Section 6.1: 2
Section 6.3: 6
Remark. In exercise 6.3.3 you may draw the nullclines to check that the origin is the
only xed point.
Section 6.4: 1
Exercise A
For the nonlinear ODEs in (a)-(c), please…
… show that the origin is the only xed point. What type of phase portrait does the
linearization predict near the xed point?
… use a computer program to draw the actual phase portrait. Does it look like the
prediction of the linear system?
(a) x_ = x2, y_ = y
(b) x_ = y, y_ = x2
(c) x_ = x2 + xy, y_ = 1
2y2 + xy
5.1.10 (Attracting and Liapunov stable) Here are the official definitions of the
various types of stability. Consider a fixed point x* of a system x f(x).
We say that x* is attracting if there is a 0 such that lim
t→∞
x(t) = x* whenever
||x(0) x*|| . In other words, any trajectory that starts within a distance of x*
is guaranteed to converge to x* eventually. As shown schematically in Figure 1,
trajectories that start nearby are allowed to stray from x* in the short run, but they
must approach x* in the long run.
In contrast, Liapunov stability requires that nearby trajectories remain close for
all time. We say that x* is Liapunov stable if for each 0, there is a 0 such that
||x( t ) x*|| whenever t 0 and ||x(0) x*|| . Thus, trajectories that start
within of x* remain within of x* for all positive time (Figure 1):
radius radius
Attracting Liapunov stable
radius
(0) (0)
x x
x x
* *
= δ = ε = δ
Figure 1
Finally, x* is asymptotically stable if it is both attracting and Liapunov stable.
For each of the following systems, decide whether the origin is attracting,
Liapunov stable, asymptotically stable, or none of the above.
a) x = y, y =−4x. b) x 2y, y x
c) x 0, y x d) x = 0, y =−y
e) x =−x, y =−5y f) x x, y y
Plot the phase portrait and classify the fixed point of the following linear systems.
If the eigenvectors are real, indicate them in your sketch.
5.2.3 x.. = y, y.. =.2x.3y 5.2.4 x.. =5x+10y, y.. =.x.y
5.2.5 x.. =3x.4y, y.. = x.y 5.2.6 x.. =.3x+2y, y.. = x.2y
5.2.7 x.. =5x+2y, y.. =.17x.5y 5.2.8 x.. =.3x+4y, y.. =.2x+3y
5.2.9 x.. = 4x.3y, y.. =8x.6y 5.2.10 x.. = y, y.. =.x.2y.
6.1 Phase Portraits
For each of the following systems, find the fixed points. Then sketch the nullclines,
the vector field, and a plausible phase portrait.
6.1.2 x.. = x.x3 , y.. =.y
6.3 Fixed Points and Linearization
For each of the following systems, find the fixed points, classify them, sketch the
neighboring trajectories, and try to fill in the rest of the phase portrait.
6.3.6 x = xy−1, y = x−y3
6.4 Rabbits versus Sheep
Consider the following “rabbits vs. sheep” problems, where x, y 0. Find the fixed
points, investigate their stability, draw the nullclines, and sketch plausible phase
portraits. Indicate the basins of attraction of any stable fixed points.
6.4.1 x = x(3−x−y), y = y(2−x−y)
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