(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)


(* For problem 3 *)

g[x_]:= Piecewise[{{x^2 - 3x + 2, 0 <= x <= 1}, {x - 1, 1 < x <= 2}}]





(* For problem 5 *)

T[x_]:= Piecewise[{{2 x, 0 <= x < 1/2}, {2 - 2 x, 1/2 <= x <= 1}}]

(* This defines the tent map T *)


Reduce[T[T[x]]==x && Not[T[x]==x]]

(* This will find the period-2 points of T *)


perpts4 := List[ToRules[Reduce[T[T[T[T[x]]]] == x && Not[T[T[x]] == x]]]]

(* This will produce a list of the period-4 points of T, arranged as a Boolean list of "or" statements.  Reduce takes in a logical statement and reduces it to a simpler one, which in this case will make a list of all the values of x satisfying T^4[x]==x but not T^2[x]==x *)
(* The function ToRules will convert this output into a sequence of replacement rules, which is the output format given by functions like Solve; the function List converts this sequence into a list of replacement rules *)


percycs4 := NestList[T, x, 3] /. perpts4
percycs4

(* This will generate all the 4-cycles for T, though each of them will appear several times *)


perpts5 := perpts5 = 
  List[ToRules[
    Reduce[T[T[T[T[T[x]]]]] == x && Not[T[T[x]] == x] && 
      Not[T[x] == x]]]]
percycs5 := percycs5 = NestList[T, x, 5 - 1] /. perpts5
minptslist5 := minptslist5 = 
  DeleteDuplicates[Table[Min[percycs5[[n]]], {n, 1, Length[percycs5]}]]
cyclelist5 := cyclelist5 = 
  Table[NestList[T, minptslist5[[n]] , 5 - 1], {n, 1, 
    Length[minptslist5]}]
cyclelist5

(* Here is the same code modified to compute the 5-cycles.  The perpts5 := perpts5 = ... construction tells Mathematica to remember the values it computes, to prevent it from recomputing the list repeatedly as it performs the computations. *)