ME580
Nonlinear Systems
Home work #3
Due: October 7, 2010
1.
Three different systems have an equation of motion of the form dx(t)/dt = a(t)x(t), where a(t) is of
period T > 0. Find the period propagator K, and determine the stability of the equilibrium point for all
real values of the constants b and c, when a(t) is given by:
i) a(t) = 0
(0 < t < T/2),
= b
(T/2 < t < T);
ii) a(t) = 3b+ct
(0 < t < T);
iii) a(t) = b tT/2
( 0 < t < T).
2.
For the following systems,
a.
3
30
x
x
x
,
b.
3
x
x
x
,
c.
2
x
x
x
,
i)
Find the equilibrium points;
ii)
Find the local phase portraits around the equilibrium points, and characterize the
nature of the equilibrium points in terms of a center or a saddle;
iii)
Find the potential functions for the systems in (a), (b), and (c), and use them to draw the
complete phase portraits; and
iv)
For the regions in the phase plane where periodic motions are possible for these
systems, use integration of the equations describing the level curves to find the
relation
between
the
period
of
oscillation
and
the
corresponding
amplitude
of
motion.
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 Fall '10
 NA
 Equilibrium point, Stability theory, equilibrium points, phase plane, local phase portraits, complete phase portraits

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