Paper 3, Section II, D

Dynamical Systems | Part II, 2014

Let f:IIf: I \rightarrow I be a continuous one-dimensional map of an interval IRI \subset \mathbb{R}. Explain what is meant by saying that ff has a horseshoe.

A map gg on the interval [a,b][a, b] is a tent map if

(i) g(a)=ag(a)=a and g(b)=ag(b)=a;

(ii) for some cc with a<c<b,ga<c<b, g is linear and increasing on the interval [a,c][a, c], linear and decreasing on the interval [c,b][c, b], and continuous at cc.

Consider the tent map defined on the interval [0,1][0,1] by

f(x)={μx0x12μ(1x)12x1f(x)= \begin{cases}\mu x & 0 \leqslant x \leqslant \frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}

with 1<μ21<\mu \leqslant 2. Find the corresponding expressions for f2(x)=f(f(x))f^{2}(x)=f(f(x)).

Find the non-zero fixed point x0x_{0} and the points x1<12<x2x_{-1}<\frac{1}{2}<x_{-2} that satisfy

f2(x2)=f(x1)=x0=f(x0).f^{2}\left(x_{-2}\right)=f\left(x_{-1}\right)=x_{0}=f\left(x_{0}\right) .

Sketch graphs of ff and f2f^{2} showing the points corresponding to x2,x1x_{-2}, x_{-1} and x0x_{0}. Indicate the values of ff and f2f^{2} at their maxima and minima and also the gradients of each piece of their graphs.

Identify a subinterval of [0,1][0,1] on which f2f^{2} is a tent map. Hence demonstrate that f2f^{2} has a horseshoe if μ21/2\mu \geqslant 2^{1 / 2}.

Explain briefly why f4f^{4} has a horseshoe when μ21/4\mu \geqslant 2^{1 / 4}.

Why are there periodic points of ff arbitrarily close to x0x_{0} for μ21/2\mu \geqslant 2^{1 / 2}, but no such points for 21/4μ<21/22^{1 / 4} \leqslant \mu<2^{1 / 2} ? Explain carefully any results or terms that you use.

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