Be sure this exam has 9 pages including the cover
The University of British Columbia
MATH 345, Section 201, Instructor Tai-Peng Tsai
Final Exam – April 2012
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A non-programmable calculator and one page of notes may be used.
No other aids are permitted.
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and shall be liable to disciplinary action;
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shall not be received;
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problem
max
1.
10
2.
15
3.
15
4.
10
5.
15
6.
20
7.
15
total
100
score
April 2012
Math 345 Section 201 Final Exam
Page 2 of 9
(10 pt) 1. Consider the following ODE for x(t) in the phase line R,
ẋ = x +
rx
.
1 + x2
(1)
Sketch all the qualitatively different vector fields that occur as r is varied. Show that a
pitchfork bifurcation occurs at a critical value of r and classify the bifurcation as supercritical
or subcritical. Finally, sketch the bifurcation diagram of fixed point x∗ versus r, indicating
the stable and unstable branches.
Normal forms for one dimensional bifurcation:
1. Saddle-node:
ẋ = r + x2 ,
2. Transcritical:
ẋ = rx − x2
3. Pitchfork:
ẋ = rx − x3 ,
ẋ = r − x2
ẋ = rx + x3 ,
ẋ = rx + x3 − x5
April 2012
Math 345 Section 201 Final Exam
Page 3 of 9
2. Consider the following ODE for x(t) in the phase line R,
ẋ = h + rx − x2 .
(10 pt)
(5 pt)
(a) Plot the bifurcation diagram of fixed point x∗ versus r for h < 0, h = 0, and h > 0.
(b) Sketch the regions in the (r, h) plane that correspond to qualitatively different vector
fields, and identify the bifurcation that occur on the boundaries of those regions.
(2)
April 2012
Math 345 Section 201 Final Exam
Page 4 of 9
3. Consider the two-dimensional system for (x(t), y(t)) in the phase plane R2 given by
ẋ = y,
ẏ = x2 − 4.
(3)
It has two fixed points (2, 0) and (−2, 0).
(5 pt)
(a) Use linear stability analysis to classify each fixed point.
(5 pt)
(b) Find a conserved quantity and verify that it is indeed conserved.
(5 pt)
(c) Sketch the phase portrait of (3). Indicate the (global) stable manifold and the (global)
unstable manifold of any saddle point. Indicate the homoclinic orbit(s), if any exists.
April 2012
Math 345 Section 201 Final Exam
Page 5 of 9
4. Consider the perturbed system with a small parameter δ > 0
ẋ = y,
ẏ = x2 − 4 − δy.
(4)
(2 pt)
(a) Use linear stability analysis to classify each fixed point.
(4 pt)
(b) Find a Liapunov function V (x, y) and verify that V decreases along all trajectories of (4)
except fixed points.
(4 pt)
(c) Are there any closed orbit of (4)? Explain.
April 2012
Math 345 Section 201 Final Exam
Page 6 of 9
5. A Hopf bifurcation occurs at the origin when µ = 0 for the following system
ẋ = µx + y − x3 ,
ẏ = −x + µy + 2y 3 .
(5)
(5 pt)
(a) Decide the stability of the origin when µ varies.
(5 pt)
(b) For both µ < 0 and µ > 0, with |µ| 1, sketch the nullclines and the flow directions.
(5 pt)
(c) By looking at the local phase portrait, determine and explain whether the bifurcation is
supercritical or subcritical. You need not construct a trapping or repelling region.
April 2012
Math 345 Section 201 Final Exam
(20 pt) 6. Consider the quadratic map xn+1 = x2n + c, where c is a real parameter.
(a) Find and classify all the fixed points as a function of c.
(b) Find the values of c at which the fixed points bifurcate.
Page 7 of 9
April 2012
Math 345 Section 201 Final Exam
(c) For which values of c is there is a 2-cycle?
(d) For which values of c is there is a stable 2-cycle?
Page 8 of 9
April 2012
Math 345 Section 201 Final Exam
Page 9 of 9
7. Multiple choice: Circle all the correct answers.
(6 pt)
(a) If the linearization of a 2D nonlinear system at a fixed point is a stable star, the local
portrait of the nonlinear system can be
1.
2.
3.
4.
(3 pt)
a
a
a
a
node
spiral
saddle
center
(b) Any closed orbit is a limit cycle.
1. True
2. False
(3 pt)
(c) Chaos may occur in a 2D flow.
1. True
2. False
(3 pt)
(d) When chaos occurs in the Lorentz system, the trajectory for the same initial condition
changes every time we compute it.
1. True
2. False