1.685J/2.0345/18.377J Nonlinear Dynamics and Waves
Spring 2007
Problem Set No.
1
Out:
Thursday, February 15, 2007
Due:
Thursday, March
1,
2007
in
class
Problem 1
A
man rows a boat across a river of width
a
occupying the region 0
5
x
5
a
in the x, y-
plane, always rowing towards a fixed point on one bank, say (0,O). He rows at constant
speed
u
relative to the water, and the river flows at constant speed
v.
Show that
where (x,
y) are the coordinates of the boat. Show that the phase trajectories are given
by
where
a
=
vlu.
Sketch the phase diagram for
a
<
1
and interpret it. What kind of point
is the origin? What happens to the boat if
a
>
I?
Problem
2
Consider the system
Discuss the stability of the equilibrium point at the origin (i) by linearized thcory and (ii)
for the full nonlinear system.
(Hint:
consider x4
+
2y2)
