Math-135: Final exam
Policies: 3 hours, open books and notes, NO calculators.
4 problems + 1 bonus problem. Point values given in parentheses, 65
maximum.
Do not work on the bonus problem before you finished and checked the
4 problems!
1. (20 points) Consider the system
ẋ = y
ẏ = x + x2 .
(a) Find all the equilibrium points and classify them.
(b) Find a conserved quantity for this system.
(c) Using reversibility arguments, show that the system has a homoclinic orbit in the half-plane x ≤ 0.
(d) Find an equation for the homoclinic orbit.
(e) Sketch the phase portrait.
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2. (15 points) Consider the system
ẋ = −y − x(1 − r2 )(1 + µ − r2 )
ẏ = x − y(1 − r2 )(1 + µ − r2 ),
where r2 = x2 + y 2 and µ denotes the bifurcation parameter.
Perform a bifurcation analysis for this system, that is:
(a) Compute the value(s) of µ at which a bifurcation occurs.
(b) Determine and draw a bifurcation diagram.
(c) Discuss the type(s) of bifurcation(s).
Hint: write the system in polar coordinates.
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3. (10 points) Consider the following system
ẋ = −2y + yz − x3
ẏ = x − xz − y 3
ż = xy − z 3 .
(a) Show that the system is dissipative, i.e. volumes in phase space
contract under the flow.
(b) Construct a Liapunov function for this system and conclude that
the origin is globally attracting: for all initial conditions, the solution (x(t), y(t), z(t)) approaches (0, 0, 0) as t → ∞.
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4. (20 points) Consider the map
xn+1 = f (xn ),
where
f (x) = x2 + x − r.
As usual, r is a real parameter.
(a) Find the fixed points. For which values of r do they exist?
(b) Study the linear stability of the fixed points as a function of r.
(c) Find the 2-cycles of the map. For which values of r do they exist?
(d) Determine the stability of the 2-cycles as a function of r.
(e) Plot a partial bifurcation diagram based on the information obtained. Discuss the type of bifurcations that occur.
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5. (bonus) Consider the system
ẋ = −y + x(x2 + y 2 ) sin
ẏ = x + y(x2 + y 2 ) sin
1
!
p
x2 + y 2
!
1
p
,
x2 + y 2
for x2 + y 2 6= 0 and ẋ = ẏ = 0 at (0, 0).
(a) Find the stable and unstable limit cycles of this system.
(b) Is the origin a fixed point of any of the types discussed in class?
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