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C3 Theory of Computational Dynamics
MSc Examination, May 1994
2 hours.
Full marks may be obtained by correct answers to FOUR questions.
1.
Suppose xt∈Rn satisfies the differential equation
dx t
dt
=
F(xt)
Write out the 2nd order Taylor expansion of x(t+h). Use this to derive the Euler method for
integrating the equation and to estimate the error over one time step for the Runge-Kutta
scheme given by
xt+h
≈
xt +
1
2h
(F(x )+ F (x +hF(x )))
t
t
t
Compare these two methods with the exact solution for the one dimensional equation
dxt
dt
=
axt + b
where a,b∈R are two constants.
2.
A forced pendulum is allowed to repeatedly collide with a fixed wall at θ = a. If the collisions are
perfectly elastic the motion may be described by
θ˙˙ + k θ̇ + sin θ
θ̇
=
→
cosωt
− θ̇
for θ < a
at θ = a
Describe an algorithm to numerically compute trajectories of this system (hint: recall Hénon’s
method for computing a Poincaré map).Where would you expect problems with your method to
occur?
3.
i)
Answer either i) or ii)
Describe and justify the Newton method for finding roots of a function g : R n → R n . For the
case n = 1 show that generically the method converges quadratically. Explain how it can be
used to find periodic orbits of a map f : Rn → R n (make sure that you indicate how the various
quantities required to implement the Newton method are calculated in this case).
ii) Sketch how you would find a periodic orbit of the Lorenz equations
ẋ
ẏ
ż
=
=
σ(y - x)
rx - y - xz
=
xy - bz
You need not give any derivations, but should give sufficient detail to allow a competent programmer to implement the algorithm from your description.
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4.
i)
Answer either i) or ii)
Describe a method of computing the LR decomposition of a matrix and briefly sketch how this
can be used to invert the matrix. Suppose a matrix has the block structure
 A11

 0

 0

 0
A12
0
A22
A23
0
A33
0
0
0 

0 

A34 

A44 
where each Aij is a 100×100 matrix. Indicate how you would numerically invert such a matrix.
ii) Find an LR decomposition of the matrix
1

1

1

1
1
1
2
1
3
2
2
3
a

b

c

d
and compute its determinant. Determine for which choices of a, b, c, d and e the equation
1

1

1

1
1
1
2
1
3
2
2
3
a

b

c

d
x
 
y
 
z
 
w 
=
 1
 
 1
 
 1
 
e 
has a solution, and find this solution when it exists.
5.
Suppose that the n×n matrix A has a real eigenvalue of maximum modulus of multiplicity 2, i.e.
if the eigenvalues of A are λ1, …, λn, then λ1 = λ2 and |λi| < | λ1| for i = 3, …, n. How might one
compute λ1 numerically? You may assume that A is diagonalizable.
Now suppose that λ1 and λ2 are a complex conjugate pair, so that λ1 = re iθ and λ2 = re -iθ . What
will happen when you try to use your method? Briefly suggest how you might modify your
algorithm to compute λ1 and λ2 in this case.
6.
Let
dx t
dt
=
F(xt,µ)
xt∈Rn, µ∈R
be a one parameter familiy of differential equations. Explain how you would find a parameter
value µ0 where this family undergoes a saddle node of equilibrium points.
End of paper