Consider the map fc(x) = x + c sin(x).

lConsider the map lm) = :1: + csin(:c). (a) Find all of the fixed points and all of the bifucations of these xed points analytically and deter- mine if the bifurcations are sadc'lle-nmle, pitchlorh transcriticaL or period doubling l'Jifurcatirms. Determine where the fixed points are attractive and where they are repulsive. (b) Consider the bifurcation diagram shown in Figure I. stable ' ' ' ' unstable Figure 1: Bifurcation diagram for Ida} = :1: + csin(;.-:}. Using a combination of analysis and numerics, explain with justification the bifurcations shown. Hint: "Which ones are pitchlork bifurcations of the map, period doublings of the map: pitchork bifurcations of a higher iterate of the map or period doublings of a higher iterate of the map. In particular. include an orbit diagram starting from different initial conditions for c E [3.514] and .1: E [T, 7]. Freeze with different colours to try to include all the attractive invariant sets that are shown in the bifurcation diagram. Hints: :- You will need to start from dilferent seeds. I You need to choose a negative number from -1 to -10 when you freeze. I Does your orbit diagram support your explanations in the previous part. If not, gure out what ou did wrong and revise