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Numerical Methods of Detecting Chaos
MSc Examination, Summer 1993
Answer at least THREE questions.
1. Let f : M → M be a discrete time dynamical system on a compact manifold M and ϕ : M
→ IR a measurement function. Define the Takens embedding Φ : M → IR d, and give an
informal statement of the Takens embedding theorem.
Let zn = f(z n-1 ) be an orbit of f and x n = ϕ(zn) be an observed time series. Show that
Takens theorem implies the generic existence of a function Fϕ : Mϕ → Mϕ, where Mϕ is a
subset of IR d, such that Fϕ(xn,xn+1, … ,xn+d-1) = (xn+1,xn+2, … , xn+d).
Now suppose that ψ : M → IR is another measurement function, let yn = ψ(zn) and Fψ :
Mψ → Mψ be the corresponding function such that Fψ(yn,yn+1, … ,y n+d1) = (yn+1,yn+2,
… ,yn+d). Show that Fϕ and Fψ are smoothly conjugate, that is that there exists a smooth
1-1 map with smooth inverse H : Mϕ → Mψ such that Fψ = H°Fϕ°H-1.
Let µϕ be an ergodic invariant measure of Fϕ on M ϕ . Show how to define a corresponding invariant measure µψ of Fψ on Mψ . Show that µψ and µ ϕ have the same
correlation (Grassberger-Procaccia) dimension dc, i.e. that the choice of observable does
not affect the computation of dc.
2. Consider the ordinary differential equation
dx
dt
=
x∈ IR m
F(x)
Let A t be the matrix which describes the evolution of small perturbations about some
given trajectory xt. Write down a differential equation satisfied by At.
What does the Oseledec ergodic theorem imply about the limiting behaviour of At, as t
→ ∞ ? Explain how this can be used to define the Liapunov exponents λ 1, … , λ m. Show
that det At ~ e(λ1 + … +λm)t as t → ∞.
Using the fact that for any matrix A
det(I + εA)
=
1 + εtrace A + O(ε2)
where I is the identity matrix, show that
d
det At
dt
=
(∇.F) (det At)
where ∇.F = div F = trace DF. Assuming that ∇.F is bounded, show that
emin (∇.F).t
≤
det At
≤
emax (∇.F).t
≤
λ1 + … + λm
≤
max (∇.F)
and hence that
min (∇.F)
Give a classification of attractors based on the signs of the Lyapunov exponents when
∇.F < 0 everywhere.
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3. Answer either i) or ii) below
i) Let X be a closed bounded subset of IR m. Define the fractal (or box-counting) dimension
df(X) of X.
Let ft : IR m → IR m be a flow on IR m, and suppose that X is invariant under ft. Describe
how a finite segment of a trajectory of ft can be used to estimate df(X). What is the main
problem encountered in carrying this out in practice?
Let Σ be a Poincaré section in the neighbourhood of X and g : Σ → Σ the first return map
of ft. What is the relationship between df(X) and df(X∩Σ)? Justify your answer.
ii) Let f : IR m → IR m be a discrete time dynamical system on IR m and suppose that f(x) =
(f1(x1),f2(x2)), where x = (x1,x2) with x1∈ IR m1 , x2∈ IR m2 , m 1 + m2 = m and f1 : IR m1 →
IR m1 , f 2 : IR m2 → IR m2 . Let Xi, i=1,2 be invariant sets for f i and X = X 1 × X 2 be the
corresponding invariant set for f. Determine the relationship between the fractal
dimensions of X1 and X2 and the fractal dimension of X. Justify your answer.
Consider the map f : IR 4 → IR 4 given by
f(x,y,z,w)
=
(2x, 2y, 1 z, 1 w)
3 4

(2x,
2y
−
1, 31 z, 41 w + 43 )


1
2 1
(2x − 1, 2y, 3 z + 3 , 4 w)
(2x − 1, 2y − 1, 1 z + 2 , 1 w + 3 )
4
3
3 4

x < 21 , y < 21
x < 21 , y ≥ 21
x ≥ 21 , y < 21
x ≥ 21 , y ≥ 21
Compute the fractal dimension of the attractor of f.
Give an example of a countable set whose fractal dimension is 1/2, and hence or
otherwise deduce the existence of of a countable set whose fractal dimension is 1.
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4
Answer either i) or ii) below
i) Suppose that f : IR m → IR m is a discrete time dynamical system on IR m. Let xi = f(xi-1)
be an orbit of f. Describe how to estimate the derivative Df x of f at a point x∈ IR m from
a sample {xi : i = 1, … , n} of this orbit.
Briefly explain how this can be used to compute the Lyapunov exponents of f from such
a sample (assuming no knowledge of f itself), highlighting the numerical difficulties
encountered and possible ways of overcoming them.
Recently someone computed the Lyapunov exponents of an orbit segment of length n =
1000 and obtained values of 5,-0.5,-1,-2,-3 for the five largest exponents. At the same
time they estimated the correlation (Grassberger-Procaccia) dimension to be
approximately 5.9. Are these results reasonable?
ii) Let {xn} be a time series for which there exists some integer d and a function G : IR d →
IR such that xn+2d = G(x n,xn+1, … ,xn+d-1). Describe how a radial basis function approximation to G can be obtained from a finite sample {xi : i = 1, … , n} of the time series.
Explain how this can be used to predict the future behaviour of this time series, being
careful to distinguish between short and long term prediction. Derive bounds for the
maximum length of time over which one can expect to make meaningful predictions.
Suppose that {un} is a time series such that un = xn + sn, where sn = 0 if n is even and sn
is small if n is odd. Assuming that G is known, briefly describe how to decompose {un}
into {xn} and {sn}. What if sn = 0 whenever 0 ≤ n mod 4d < 2d?