Mathematics 118A, Winter 1998

Final Exam

1. Recall that the trace of a matrix A, denoted by Tr A, is the sum of the diagonal elements of A. Set

|||A||| = | Tr A|.

(a) Prove that |||A + B||| ≤ |||A||| + |||B|||.

(b) Is ||| · ||| a norm? Prove the statement or give a counter example. (2+3)

2. Recall that a Hamiltonian system in the plane is defined through its Hamiltonian function H = H(p, q). The equations of motion are

ṗ = −\frac{∂H}{∂q} ,  q̇ = \frac{∂H}{∂p} .

Show that the phase flow corresponding to this system is volume preserving. (3)

3. Consider the second order equation

ẍ + x − \frac{1}{2}x^{2} = 0.

(a) Find and analyze the equilibrium points by linearization.

(b) Find a first integral.

(c) Use the first integral as a Lyapunov function to show that the origin of the phase plane is a Lyapunov stable critical point.

(d) Is the origin asymptotically stable? Explain your reasoning.

(e) Sketch the phase portrait of the system.

Hint: Recall that orbits are level curves of the first integral.

(f) Are there periodic orbits? How many? (Justify your answer. The hint to part (e) may be useful here, too.)

(4+2+2+2+2+2)

4. Consider the system

ẋ = (x − y)\sqrt{x^{2} + y^{2}} − x
ẏ = (x + y)\sqrt{x^{2} + y^{2}} − y.

(a) Use Bendixon’s criterion to show that if a periodic orbit exists, part or all of it must lie outside a circle of radius \frac{2}{3} centered at the origin.

(b) Find the periodic orbit. (There is only one—why?)

(c) Is the periodic orbit stable? Why or why not?

(d) Linearize the system about the periodic solution.

(e) State at least one solution to the linearized equation. (Explain your answer!)

(f) Recall that the Floquet exponents λ_{1}, …, λ_{n} of the linear equation

ż = A(t) z, where A(t) is a T -periodic n × n matrix, satisfy

∑_{j=0}^{n} λ_{j} = \frac{1}{T} ∫_{0}^{T} Tr A(t) dt (mod \frac{2πi}{T}).

Use this result to give an alternative proof of your answer to part (c).

(g) Sketch the phase portrait. (2+2+2+2+2+2+2)

5. The cuckoo is a bird that doesn’t breed its own eggs, but places them into the nests of other birds. When a young Cuckoo grows, it kills all of its step-siblings. A very simple model of a cuckoo population x and one host species y (the robin, say) is the system

ẋ = xy − kx
ẏ = y − y^{2} − xy,

where the positive parameter k is related to the natural death rate of the cuckoo. (For simplicity, other biologically relevant parameters have been set to one.)

(a) Explain briefly why this system models the situation described above. What is the role of each of the terms?

(b) Determine the critical points of the system. (There are three.) Are they all relevant, i.e., do they correspond to nonnegative bird populations?

(c) For which values of k can the two species co-exist? Can the cuckoo or the robin ever become extinct? To answer this question, analyze the stability of the critical points and sketch the phase portrait for different values of k. Hints:

• One of the critical points is associated with the eigenvalues

λ = \frac{−k ± \sqrt{5k^{2} − 4k}}{2} .

• Look very carefully at your answer to part (b) again!

• You may not need a full linear analysis of all the critical points as some

information can be gathered directly from the equations. (3+3+8)