5.
(a) Consider the dynamical system
$\dot{x} = (x - c)\sin(\pi x)$, $x \in [-1, 1]$,
where $\dot{x}$ means $\frac{dx}{dt}$ and $c \in \mathbb{R}$ is a control parameter of the system.
(i) Determine the number and location of the equilibrium points of the above system for
all values of the control parameter $c$.
(ii) Using linear stability analysis, determine the stability of the equilibrium points you
have found in part (i).
(iii) State the location of any bifurcations of the system.
(iv) Sketch the bifurcation diagram of the system.
