Section 6.3: 15

Section 6.5: 1, 13

Section 6.6: 1

Section 6.8: 8

Exercise A

Show that the system

x_ = x y x(x2 + y2);

y_ = x + y y(x2 + y2)

is given in polar coordinates (x; y) = (r cos ; r sin ) by

r_ = r(1 r2);

_ = 1:

Draw the associated phase portrait.

1

Exercise B

For r < 1 draw the phase portrait of

r_ = r;

_ =

1

ln r

:

Calculate the associated vector eld f in cartesian coordinates. Note that (by denition)

f(0; 0) = (0; 0): Show that the phase portrait of the linearization at the origin is the phase

portrait of a stable star.

Remark. In this example f 2 C1(B1(0)) nC2(B1(0)); where B1(0) = f(x; y)jx2+y2 < 1g

is the ball of radius r = 1 around the origin. This example is due to Perron.

Exercise C

(a) Use index theory to show that the system

x_ = x(4 y x2);

y_ = y(x 1)

has no closed orbit. You may assume that a branch of the unstable manifold of (2; 0)

approaches (1; 3):

(b) Draw a phase portrait with all of the following properties:

(1) it diers from the phase portrait of (a) only for x > 0 and y > 0,

(2) it has a closed orbit,

(3) it has a stable spiral at (1; 3):

6.3.15 Consider the system r..= r(1.r2 ), …. =1.cos.. where r, .. represent polar

coordinates. Sketch the phase portrait and thereby show that the fixed point

r* .. 1, ..* .. 0 is attracting but not Liapunov stable.

6.5.1 Consider the system x….= x3 .x.

a) Find all the equilibrium points and classify them.

b) Find a conserved quantity.

c) Sketch the phase portrait.

6.5.13 (Nonlinear centers)

a) Show that the Duffing equation x….+x+..x3 = 0 has a nonlinear center at the

origin for all .. .. 0.

b) If .. .. 0, show that all trajectories near the origin are closed. What about

trajectories that are far from the origin?

Show that each of the following systems is reversible, and sketch the phase portrait.

6.6.1 x = y(1−x2 ), y =1−y2

6.8.8 A smooth vector field on the phase plane is known to have exactly three

closed orbits. Two of the cycles, say C1 and C2, lie inside the third cycle C3.

However, C1 does not lie inside C2, nor vice-versa.

a) Sketch the arrangement of the three cycles.

b) Show that there must be at least one fixed point in the region bounded by C1, C2,

C3.

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