(a) Consider the map $G_{1}(x)=f(x+a)$, defined on $0 \leqslant x<1$, where $f(x)=x[\bmod 1]$, $0 \leqslant f<1$, and the constant $a$ satisfies $0 \leqslant a<1$. Give, with reasons, the values of $a$ (if any) for which the map has (i) a fixed point, (ii) a cycle of least period $n$, (iii) an aperiodic orbit. Does the map exhibit sensitive dependence on initial conditions?

Show (graphically if you wish) that if the map has an $n$-cycle then it has an infinite number of such cycles. Is this still true if $G_{1}$ is replaced by $f(c x+a), 0<c<1 ?$

(b) Consider the map

$$G_{2}(x)=f\left(x+a+\frac{b}{2 \pi} \sin 2 \pi x\right)$$

where $f(x)$ and $a$ are defined as in Part (a), and $b>0$ is a parameter.

Find the regions of the $(a, b)$ plane for which the map has (i) no fixed points, (ii) exactly two fixed points.

Now consider the possible existence of a 2-cycle of the map $G_{2}$ when $b \ll 1$, and suppose the elements of the cycle are $X, Y$ with $X<\frac{1}{2}$. By expanding $X, Y, a$ in powers of $b$, so that $X=X_{0}+b X_{1}+b^{2} X_{2}+O\left(b^{3}\right)$, and similarly for $Y$ and $a$, show that

$$a=\frac{1}{2}+\frac{b^{2}}{8 \pi} \sin 4 \pi X_{0}+O\left(b^{3}\right)$$

Use this result to sketch the region of the $(a, b)$ plane in which 2-cycles exist. How many distinct cycles are there for each value of $a$ in this region?