342N Pseudoexam
1. Explain the meaning, or meanings, of the following:
(a) compatible,
(b) consistent,
(c) equilibrium,
(d) Euler method,
(e) explicit/implicit,
(f) Hopf bifurcation,
(g) incoming/outgoing solutions,
(h) linearisation,
(i) order,
(j) pitchfork bifurcation,
(k) resonance,
(l) saddle-node bifurcation,
(m) scattering matrix,
(n) spectral decomposition,
(o) stable/unstable/centre manifold,
(p) starting procedure
(q) stiff system
(r) transcritical bifurcation
(s) Wronsksian.
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2. (a) What are the solutions ϕ, θ, ψ− , ψ+ , χ− and χ+ to
y ′′(x) + λy(x) = q(x)y(x)?
For which λ, x are they defined?
(b) How are ψ+ and χ+ related?
(c) How is the scattering matrix defined? What properties does it
have, and why?
(d) Describe, in terms of the relations among these solutions, the spectral decomposition.
Note: There’s no need to give the derivation, just the formula.
(e) In the case q = 0 find the scattering matrix and the spectral
decomposition.
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3. (a) For the system
u′ = u + u2
v ′ = u + u2
and the equilibrium (0, 0):
i. Find the eigenvalues and eigenvectors of the linearised system.
ii. Find the dimensions of the stable, unstable and centre manifolds.
iii. What is the lowest order resonance?
iv. Find a quadratic change of variable which eliminates the quadratic
part of the nonlinearity.
(b) For the equation
y ′ = y 3 − y 2z − yz 2 + z 3 − y 2 + z 2 + y − z − 1
with parameter z:
i. Where are the equilibria?
ii. At what values of the parameter does the system have a bifurcation?
iii. Draw a bifurcation diagram for the system.
iv. What sort of bifurcation, or bifurcations, occur?
v. Are these bifurcations stable under small perturbations?
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4. (a) What is numerical instability? Why is it bad?
(b) Consider the numerical solution method
yn − 6yn−1 + 7yn−2 − 2yn−3 + 2hF (xn , yn ) = 0.
i. Is it explicit or implicit?
ii. Is is stable or unstable?
iii. What is its order?
(c) Give an example of a stable solution method of order at least 4.
Note: You should explain where any coefficients in this method
come from.
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