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C3 Theory of Computational Dynamics
MSc Examination, May 2000
2 hours.
You are advised to attempt FOUR questions, though credit will given for all material handed in.
To pass, you will need to correctly answer the equivalent of about two questions, whilst for a
distinction you will need to answer the equivalent of four questions.
1.
Suppose xt∈Rn satisfies the differential equation
dxt
dt
=
xt∈Rn
F(xt)
Write out the 2nd order Taylor expansion of xt+h in terms of F and its derivatives. Use this to derive the Euler method for integrating the above equation and give an explicit error estimate (in
terms of F and its derivatives) for one time step of this method. Explain why the Euler method is
called a 1st order method.
Consider a general one step Runge-Kutta type scheme of the form
xt+h
=
xt + h[αF(xt+h) + βF (xt)]
where α and β are constants. Show that for this to be a 1st order method α and β must satisfy α +
β = 1. Find a value of α such that the scheme is in fact 2nd order.
What is the difficulty in using this scheme in practice if α ≠ 0? By replacing F(xt+h) by its Euler
method approximation, or otherwise, suggest a 2nd order method that overcomes this difficulty.
2.
Describe Hénon’s method for the numerical computation of a Poincaré map for a differential
equation
dxt
dt
=
xt∈Rn
F(xt)
You need not give details of any standard algorithms for the numerical solution of differential
equations.
Consider the Duffing oscillator
..
.
x + δx - βx + x3
=
0
x∈R
Suppose that you wish to compute the Poincaré map corresponding to the plane x = 1. Give an
explicit formula for the differential equation that needs to be integrated in the final step of Hénon’s approach. Suggest where this method might run into difficulties, and hence or otherwise,
determine where the plane x = 1 fails to be a valid Poincaré section.
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3.
Derive the Newton method for finding the root x* of a function f : R → R. Show that it converges
quadratically when f '(x*) ≠ 0.
Now consider the function f (x) = xr for some r ≥ 2. What is the rate of convergence to the root x*
= 0, and why?
Explain how the Newton method can be used to find a periodic orbits of a map and discuss the
problems this will have at a saddle-node bifurcation.
4.
Explain how an LR decomposition can be used to invert a matrix. Find an LR decomposition of
 a
b

 aκ bλ
 aκ 2 bλ 2

c 

cµ 
c µ 2 
and compute its determinant. Hence, or otherwise, determine for which choices of κ , λ and µ
the equation
 1

κ
κ 2

1
λ
λ
2
1   x
µ   y 
µ 2   z 
=
1
 
1
1
 
has a solution, and find this solution when it exists.
5.
Let A(z) be a n×n matrix depending on z∈Rn. Briefly explain why one might want to compute
the derivative of det A(z) with respect to z, and show that this is given by
∂
det A(z)
∂z
=

∂A(z )
trace  adj(A(z ))†

∂z 

where adj(A(z)) is the transpose of the matrix of cofactors of A(z), and † denotes transpose. What
are the practical difficulties associated with the numerical evaluation of this formula? Assuming
that you can find vectors b,c ∈Rn such that the matrix
A0(z)
=
 A( z ) b 
 †

0
 c
is invertible show how to define a function u(z) such that u(z ) = 0 implies det A(z ) = 0 (you need
not show the converse). Describe how to efficiently compute the derivative of u(z) with respect
to z.
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6.
Explain how you would find a parameter value a 0 where the logistic map
xi+1
=
2
1 - axi
xi,a∈R
undergoes a saddle-node bifurcation for a period q orbit. Explain why you need the partial derivatives
∂xq ∂ 2 xq
∂a , ∂x 02
and give an explicit formula that allows their computation. State what other derivatives you
need to calculate, and outline how this can be done (you need not give explicit formulae for
these).
7.
State the implicit function theorem. Suppose that fαβ : R → R is a two parameter family of functions that has a double root at x = α = β = 0 and at least one of the matrices
 ∂f
 ∂x
 2
∂ f
 2
 ∂x
∂f 
∂α 

∂2f 

∂x∂α 
or
 ∂f
 ∂x

∂2f
 2
 ∂x
∂f 
∂β 

∂2f 

∂x∂β 
is invertible. Deduce that there is a curve in parameter space through x = α = β = 0 where fαβ has
a double root. Sketch out the various steps required to follow such a curve numerically. Describe the relevance of this to computational dynamics.
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